Research ArticleA Fairness Relation Based on the Asymmetric Choquet Integraland Its Application in Network Resource Allocation Problems

Aoi Honda and Mario Koumlppen

Kyushu Institute of Technology 680-2 Kawazu Iizuka Fukuoka 820-8502 Japan

Correspondence should be addressed to Mario Koppen mkoeppenieeeorg

Received 6 March 2014 Revised 3 June 2014 Accepted 5 June 2014 Published 17 July 2014

Academic Editor Wlodzimierz Ogryczak

Copyright copy 2014 A Honda and M Koppen This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The recent problem of network resource allocation is studied where pairs of users could be in a favourable situation given that theallocation scheme is refined by some add-on technologyThe general question here is whether the additional effort can be effectivewith regard to the users experience of fairnessThe computational approach proposed in this paper to handle this question is basedon the framework of relational optimization For representing different weightings for different pairs of users the use of a fuzzymeasure appears to be reasonable The generalized Choquet integrals are discussed from the viewpoint of representing fairnessand it is concluded that the asymmetric Choquet integral is the most suitable approach A binary relation using the asymmetricChoquet integral is proposed In case of a supermodular fuzzy measure this is a transitive and cycle-free relation The price offairness with regard to a wireless channel allocation problem taking channel interference into account is experimentally studiedand it can be seen that the asymmetric on relation actually selects allocations that perform on average between maxmin fairnessand proportional fairness and being more close to maxmin fairness as long as channel interference is not high

1 Introduction

The rapid spread of wireless communication poses manychallenges to the underlying networking technology andinfrastructure Daily experience of using wireless accessteaches us the increased efficiency of provider solutions Atthis stage of development the major technical demand is theefficient reuse of existing resources or their usage expansionbased on cost-efficient technical add-onsThat brings that themain valuation criterion today is the total utility of networkinfrastructure employment with a lower focus on fairnessaspects of resource allocation and distribution For now thestandard user of a wireless infrastructure does not have theimmediate experience of fairness taking the access vector ofother users into accountmdashthe needed information is simplynot available for the user This might change in the futureand fairness might become the primary not the secondaryaspect in comparison to efficiency Recently the relationbetween fairness and efficiency (in the sense of the total sumof per-user valuations) has become a relevant research issue

Obviously a fair allocation is not an ldquooptimalrdquo allocation sincethe total valuation will be lower than the maximum possibleHowever bounds do exist and in basic settings it is alreadyknown that the ldquoprice of fairnessrdquo is actually not so high [1]

In the specification of fairness the so-called ldquostandardsof comparisonrdquo (SoC) give an important formal instrumentfor the specification of fair solutions It refers to a modalityof comparing two solutions where a solution 119860 passes aspecific test with regard to another solution 119861 A classicalexample is the Nash standard of comparison for bargainingnegotiations [2] a solution is considered better than anothersolution if the relative wins are not outperformed by relativelosses If a solution 119860 appears to be better than any othersolution that is the relative losses are always larger thanor equal to the relative wins the solution is seen as astable point in negotiations and there is no incentive todeviate from this solution Later on a few issues with theNash SoC led to the formulation of an alternative Kalai-Smorodinsky SoC [3] This approach promotes the userthat receives the least allocation (which is also the base

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 725974 12 pageshttpdxdoiorg1011552014725974

2 Journal of Applied Mathematics

for the economical and political ideas of Rawls [4]) Bothapproaches originally developed in economical science havebeen rediscovered in network telecommunication theory andare very popular approaches in current research the NashSoC is now better known as proportional fairness [5] andthe Kalai-Smorodinsky SoC as maxmin fairness [6] It comesout that even historically the cultural distinction betweenthese two fundamental kinds of fairness existed for long time[7] In between a mediating approach that links proportionalfairness and maxmin fairness the so-called 120572-fairness hasbeen formulated as well [8] if 120572 = 0 the relation refers toproportional fairness and (with some caution) if 120572 rarr infinthe relation approximates maxmin fairness

However the fairness models so far are based on a per-user valuation of ldquowinsrdquo and ldquolossesrdquo where only a singleuser can decide what is in her interest and what not Forexample in proportional fairness the SoC is based on thecomparison between two allocation vectors 119909 and 119910 of thesame dimension

sum

119894

119910119894minus 119909

119894

119909119894

le 0 (1)

One can see that each user formally contributes a singleindex value to the comparison test However as mentionedabove for the case of wireless communication increaseddemand of reuse of existing resources by new technicalmeans and add-ons puts also groups of users into differentsituations that nevertheless need to be compared There aremany refined technologies explored these days that exhibitthis property One example is channel interference in awireless infrastructure a base station BS has to allocatetransmission channels (ie ranges of frequency bandwidth)to its subscriber stations (SS) The safe way is to allocate atmost one SS to a channel However if SSs are well separatedthat is the channel interference of two SSs using the samefrequency band is low the same channel can be assignedto more than one SS The question is how to know aboutthis opportunity by proper measurement and about the bestway to employ this Several studies have been devoted to thistechnical challenge but what they all have in common is theestablishment of a pairwise relation between users usuallyalso weighted as for example in [9ndash14]

Besides channel interference there are many other newernetworking technologies where this aspect of pairwise userweight appears in cognitive radio unused channel capacitiesof primary users can be on occasion allocated to secondaryusers The pairwise relation here is represented by a conflictmatrix that contains lower values where the chance that twousers (one from the group of primary users one from thegroup of secondary users) transmitting data in the sametime intervall is low The encounter probability also playsan important role in peer-to-peer networking opportunisticnetworking and vehicular networks

Now if considering above formalization of proportionalfairness how do we take this aspect into account that is thefact that a loss or win (relative or absolute) matters differentlyfor different pairs of users In other words if for exampleuser 1 and user 2 both experience losses when comparing twosolutions could be taken differently from a situation where

user 1 experiences a win and user 2 a loss So far no ldquopairwisestandard of comparisonrdquo is known and it does not seem tobe reasonable to define a different SoC for each differentsituation due to combinatorial explosion The related prob-lematic is about the price of fairness here if a provider willconsider a fair solution instead of the most efficient one andtaking the collision matrix information of a specific situationbetween users into account is it worthy of the effort toimplement the resource-reusing solution at all (ie to employlow channel interference or to set up a schedule for cognitiveradio secondary user channel assignment) if at the end noother allocation will be provided than without the technicaladd-on

The primary goal of this paper is to provide a for-mal fairness approach that can distinguish groups of usersand weight them differently where needed For this it isneeded to understand the SoC in terms of (set-theoretic)relations following [15] Then a specific relation will beintroduced based on fuzzy measure theory and relatedfuzzy integrals It happens that this relation comes outto be a transitive relation (allowing for ranking of solu-tions as well as fast search algorithms) and can be usedto valuate the efficiency losses related to making fairallocations

Fuzzy measures [16 17] which are called nonadditivemeasures [18] or capacity [19] are monotone and usuallynonnegative set function Since they can be used to expressinteractions between items that are not expressible by additivemeasures many studies have been done on their applicationin fields such as subjective evaluation problems decision-making support and pattern recognition The generaliza-tion with respect to fuzzy measures a generalization ofthe Lebesgue integral is proposed by several authors TheChoquet integral defined by Choquet [19] takes its placeas one of them and is most widely used The originalChoquet integral is only for nonnegative functions so thatseveral generalizations for general functions which are notnecessarily nonnegative have been given There are mainlytwo types of generalization one of which is called thesymmetric Choquet integral or the Sipos integral [20]and another one which is called the asymmetric Choquetintegral is given in [18] Concerning the details of theseintegrals the first attempt was made by Denneberg [18]This has been followed by Grabisch et al They studyproperties and application to multicriteria decision makingand so on for example in [21 22] Recent generalizationsof the symmetric Choquet integral employing averagesand balanced weights over permutations are the fusionChoquet integral [23] and the balancing Choquet integral[24]

On the other hand preference modelling and the relatedpreference prediction have become research fields of increas-ing importance The use of the Choquet integral as a basefor a preference relation has been the topic of [25] where itis used to solve combinatorial optimization problems Theapplication of a preference relation model based on Choquetintegral in multiobjective dynamic programming is the topicof [26] In these works the optimality is prespecified andgiven by independent means Here we shall focus on the

Journal of Applied Mathematics 3

formal representations of fairness in resource distributionas optimality issue by itself Our proposal is to evaluatedistributions from the aspect of fairness

In this paper we compare the generalized Choquet inte-gral the symmetric Choquet integral and the asymmetricChoquet integral from the viewpoint of representing fairnesswith the result that the asymmetric Choquet integral is moresuitable for describing fairness We also propose a fairnessrelation using the asymmetric Choquet integral

The paper is organized as follows Section 2 introducesnotations and necessary materials about fuzzy measure andthe Choquet integral and recalls the relational approach tofairnessThen Section 3 presents discussion about two gener-alized Choquet integrals The fairness relation is proposed inSection 4 and we give a sufficient condition that this relationis consistent and transitive Section 5 gives a demonstrationof how our fairness relation is used for decision makingSection 6 compares the solutions obtained by this fairnessrelation with the proportional and maxmin fair solutions byway of experiments

2 Preliminaries

21 Relational Optimization Speaking about optimality isspeaking about comparing things The basic mathematicalinstrument for the representation of optimality is the conceptof a binary relation Here we want to recall some basic termsand definitions formore details see for example [15] A (set-theoretic) binary relation119877 over a domain119860 is a subset of119860times119860 that is a set of ordered pairs (119909 119910) with 119909 119910 isin 119860 It is saidthat119909 is in relation119877 to119910 (sometimes alsowritten as119909119877119910 or ifthe focus is on comparison 119909ge

119877119910)There are alternative ways

to represent relations like a set comprehension a mapping orfunction from 119860 into the powerset of 119860 an incidence matrixor a directed graph Also relations can have properties Twoproperties are of particular interest in the following a relationis symmetric if from (119909 119910) isin 119877 it follows that also (119910 119909) isin 119877and it is asymmetric if from (119909 119910) isin 119877 it always follows that(119910 119909) does not belong to119877 Each relation can be decomposedinto a symmetric and an asymmetric part written as 119877 =

119868(119877)cup119875(119877) where 119868(119877) is a symmetric relation and119875(119877) is anasymmetric relation It is easy to decide to which part a pair(119909 119910) isin 119877 belongs in this decomposition if (119910 119909) is also in 119877it belongs to the symmetric part 119868(119877) if not it belongs to theasymmetric part 119875(119877) It means that each relation no matterwhat its domain is made of shows mixed aspects of equalityor similarity and a means of ldquobetternessrdquo or preferenceFor example the real-valued ldquogerdquo relation is decomposedinto an equivalence relation ldquo=rdquo as symmetric part and astrictly larger relation ldquogtrdquo as asymmetric part (which is wellsymbolized by ge) If we write 119909ge

119877119910 for a relation then 119909gt

119877119910

will denote the asymmetric part reflecting the ldquobetternessrdquoaspect of a relation 119877 over domain 119860

Any relation 119877 can expose specific elements of its domain119860 which are commonly called greatest maximal minimaland least elements A greatest element 119909lowast of the domain119860 with regard to 119877 has the property that for any 119909 isin 119860

(including 119909lowast) it holds that 119909lowastge119877119909 All greatest elements

comprise the best set Amaximal element 119909lowast has the propertythat there is no 119909 isin 119875(119877) such that 119909gt

119877119909lowast and all such 119909lowast

comprise themaximum set (of119860with respect to 119877) Minimaland least elements are defined correspondingly Note thatthey do not need to exist in general

One can easily see that a standard of comparison as itwas mentioned in the introduction section is the same asthe concept of a greatest element However often it is thecase that best sets are empty Therefore it is convenient torefer to the maximum set as optimization goal for severalreasons (1) In case of a finite domain the maximum set isnonempty if the asymmetric part of the relation is cycle-free(ie the corresponding directed graph does not contain anycycles) (2) If there is a greatest element it will also belongto the maximum set (3) The maximum set can be seen as aldquofrontierrdquo they are not greatest elements but at least no betterone is known

In [15] the special case of fairness relations has beenintensively discussed It is comparable and easy to representproportional fairness maxmin fairness leximin fairness 120572-fairness and so forth by this relational framework Then(Pareto-) efficient solutions can be found by finding themaximum sets for these relations But the advantage is alsothat the transition of a relational concept to another domainbecomes possible (by applying the same formal definition) orthat the relation can be specified in such away that completelynew aspects are taken into account (eg multiresource usagecollaboration between agents or vector-valued evaluations)This is because the concept of maximality applies to anyrelation no matter what its domain is Therefore if we wantto handle the above mentioned problem of pairwise weightswe just have to specify a corresponding relation (that ispreferably cycle-free and Pareto-efficient)

The last point that should be mentioned here is that thefinding of maximum sets can be a challenging task even ifthe complexity 119874(1198992) is not so high Thus exhaustive searchby a complete pairwise comparison can rapidly becomeintractable (eg a maximum set over a domain with onemillion elements would need 1012 pairwise comparisons)One possibility is to use metaheuristic search algorithms toapproximate maximum sets other alternatives are currentlyunder investigations In case the relation is transitive thesearch for maximum sets can be performed in ldquobatchesrdquo thatis decompose the domain into nonoverlapping subsets andfind themaximum set for each subset and then themaximumset of the union of the found maximum sets

22 Computational Fairness We have now used the termldquofairnessrdquo several times without providing a general defini-tion For example considering a given relation 119877 when is itsafe to say that this is a fairness relation Actually there is nocommon understanding of fairness to which all researchersand practitioners would fully agree It would go beyond thescope of this paper to consider the numerous considerationsthat have been done in order to provide a computationalmodel of fairness but at least we can summarize the basicideas The additional restriction here is that the focus is onthe modality of a distribution thus excluding other societal

4 Journal of Applied Mathematics

important concepts like procedural fairness or interactionfairness

The common theme of all approaches to (distributional)fairness is to divide a set of resources among a numberof agents As already mentioned in the introduction withfar longest cultural history we can find ideas about pro-portionality in the assignment (proportional according todemand ability suitability availability most available goodetc) or the resolution of conflicts about contended goodsin following an equity principle Besides a relation point ofview in order to compare different possible assignmentsthe aspect can be also seen in aggregating the distributionalaspects of all agents and thus keeping the spread of allocationdifferences small This idea becomes strongly related to thetheory of majorization [27] where numerical comparisonbetween different ways of distributing the same total isstrongly related to the Schur-convexity of the used compari-son measureThe application of this approach and its variousgeneralizations have been intensively studied for examplein [28]

Any continuation of a formal approach to fairness (inthe sense of a ldquodefinition of fairnessrdquo) would need thespecification of axioms that represent criteria derived fromspecific ways of how to balancemdashin a just waymdashthe conflictsbetween agents that are in a better and agents that are in aworse situationmdashkeeping inmind that fairness does generallynot stand for the simple requirement of complete even allo-cations Such axioms can refer to impartiality of distributionconsistency between solutions for different domains mathe-matical assumptions about the domain limiting behavioursstrategy-proofness of related procedures or efficiency of thedistribution

Saying this the considered aspect of fairness here is thebalance between users who would gain more and users whowould gain less in an alternative allocation We will useaggregation operators for representing the joint state of allwinning and all loosing agents thus basically following theldquoreciperdquo of proportional fairness But as a new aspect we alsowant to take a different weighting for groups of agents intoaccount Fuzzy integrals are a well-known formal approachfor such a representation

23 Fuzzy Measures and Integrals We now want to recallbasic issues of fuzzy measures and integrals in more detailMost of the material can be gathered together from variouspublications and textbooks but a comprehensive descriptionas it will be given now might be of advantage for theunderstanding of the main proposalmdasha fairness relationbased on the asymmetric Choquet integral for supermodularfuzzy measures

Throughout this paper we consider a finite universal set119873 = 1 2 119899 and 2119873 denotes the power set of119873

Definition 1 (Fuzzy measure [16 17]) A set function 120583

2119873rarr [0infin) is a fuzzy measure if it satisfies the following

conditions(i) 120583(0) = 0 and 120583(119873) lt +infin(ii) for any 119878 119879 isin 2119873 120583(119878) le 120583(119879) whenever 119878 sub 119879

Properties (i) and (ii) express boundedness and mono-tonicity respectively

Definition 2 (superadditivity subadditivity) A fuzzy mea-sure 120583 is superadditive if 120583 satisfies 120583(119878 cup 119879) ge 120583(119878) + 120583(119879)and subadditive if 120583 satisfies 120583(119878 cup 119879) le 120583(119878) + 120583(119879) for any119878 119879 isin 2

119873 satisfying 119878 cap 119879 = 0

Definition 3 (submodularity supermodularity) A fuzzymeasure 120583 is supermodular if 120583 satisfies 120583(119878 cup119879)+120583(119878 cap119879) ge120583(119878)+120583(119879) and submodular if 120583 satisfies 120583(119878cup119879)+120583(119878cap119879) le120583(119878) + 120583(119879) for any 119878 119879 isin 2119873

Definition 4 (dual measure) Let 120583 be a fuzzy measure Thedual measure of 120583 is defined by

119898119889(119878) = 120583 (119873) minus 120583 (119878

119888) (2)

The dual measure of a fuzzy measure is also a fuzzymeasure

Proposition 5 If 120583 is supermodular then 120583 is superadditiveIf 120583 is submodular then 120583 is subadditive

Proof When 120583 is supermodular 120583(119878 cup119879) + 120583(119878 cap119879) ge 120583(119878) +120583(119879) is satisfied for any 119878 119879 isin 2119873 and 119878 cap 119879 = 0 It implies120583(119878 cup 119879) ge 120583(119878) + 120583(119879) for any 119878 119879 isin 2119873 satisfying 119878 cap 119879 = 0The second assertion is obtained in a similar manner

Proposition 6 120583 is supermodular if and only if 120583119889 is sub-modular Similarly 120583 is submodular if and only if 120583119889 issupermodular

Proof Assume that 120583 is supermodularThen we have for any119878 119879 isin 2

119873

120583119889(119878 cup 119879) + 120583

119889(119878 cap 119879)

= 120583 (119873) minus 120583 ((119878 cup 119879)119888) + 120583 (119873) minus 120583 ((119878 cap 119879)

119888)

= 2120583 (119873) minus 120583 (119878119888cap 119879

119888) minus 120583 (119878

119888cup 119879

119888)

le 2120583 (119873) minus 120583 (119878119888) minus 120583 (119879

119888)

= 120583119889(119878) + 120583

119889(119879)

(3)

Following the above in reverse we obtain the converseReplacing le with ge we obtain the second assertion

Proposition 7 If 120583 is a superadditive fuzzy measure then120583119889(119878) ge 120583(119878) for any 119878 isin 2119873

Proof We have 120583(119878) + 120583(119878119888) le 120583(119873) by superadditivity of 120583Hence 120583(119878) le 120583(119873) minus 120583(119878119888) = 120583119889(119878)

Definition 8 (120582-fuzzy measure) A fuzzy measure 120583 is a 120582-fuzzy measure if there exists 120582 gt minus1 such that 120583(119878 cup 119879) =120583(119878)+120583(119879)+120582120583(119878)120583(119879) for any 119878 119879 isin 2119873 satisfying 119878cap119879 = 0and 120583(119873) = 1

Journal of Applied Mathematics 5

Definition 9 (120594-fuzzy measure) A fuzzy measure 120583 on finiteset119873 = 1 2 119899 is a 120594-fuzzy measure if there exists

120594 ge 1 minusmin

119894isin119873120583 (119894)

sum119894isin119873120583 (119894)

(4)

such that

120583 (119878) = 120594|119878|minus1

sum

119894isin119878

120583 (119894) (5)

for any 119878 isin 2119873 Here |119878| denotes the cardinal number of 119878

120582 ge 0 and 120594 ge 1 imply superadditivity and 120582 le 0 and120594 le 1 imply subadditivity The dual measure of a 120582-fuzzymeasure and of a 120594-fuzzy measure are not necessarily a 120582-fuzzy measure and a 120594-fuzzy measure

Proposition 10 Let 120583 be a 120582-fuzzy measure Then 120583 issuperadditive if and only if 120583 is supermodular Similarly 120583 issubadditive if and only if 120583 is submodular

Proof The sufficiencies hold by Proposition 5 We show thenecessities

Assume 120583 is a superadditive 120582-fuzzy measure that is 120582 ge0 Then we have for any 119878 119879 isin 2119873

120583 (119878 cup 119879) + 120583 (119878 cap 119879)

= 120583 (119878) + 120583 (119879 119878) + 120582120583 (119878) 120583 (119879 119878) + 120583 (119878 cap 119879)

ge 120583 (119878) + 120583 (119879 119878) + 120582120583 (119878 cap 119879) 120583 (119879 119878) + 120583 (119878 cap 119879)

= 120583 (119878) + 120583 (119879)

(6)

Replacing le with ge we obtain the second assertion

Definition 11 (Choquet integral [19]) Given a fuzzy measure120583 2

119873rarr [0 1] and a nonnegative function 119891 on 119873 the

Choquet integral of 119891 with respect to 120583 is defined by

(119862) int119891119889120583 = int

+infin

0

120583 (119909119891 (119909) gt 119903) 119889119903 (7)

In the case that119873 is a finite set (7) can be rewritten as

(119862) int119891119889120583 =

119899

sum

119894=1

(119886119894minus 119886

119894minus1) 120583 (119878

119894) (8)

where 119886119894= 119891(119894) 119886

0= 0 119886

1le 119886

2le sdot sdot sdot le 119886

119899 and

119878119894= 119894 119899 (cf Figure 1) Note that this computation will

include a permutation reordering of the 119886119894to ensure that they

are in nondecreasing orderThe Choquet integral can be generalized to general

functions which are not necessarily nonnegative in severalways Here we show the following two generalizations theasymmetric Choquet integral and the symmetric Choquetintegral

Definition 12 (asymmetric Choquet integral [18]) Given afuzzy measure 120583 2119873 rarr [0 1] and a function 119891 on 119873 the

an

anminus1

a3

a2

a1

a0

Sn

S2

S1

Figure 1 Choquet integral

asymmetric Choquet integral of119891with respect to 120583 is definedby

(ASC) int119891119889120583 = int

0

minusinfin

(120583 (119909 | 119891 (119909) gt 119903)

minus119898 (119873)) 119889119903

+int

+infin

0

120583 (119909 | 119891 (119909) gt 119903) 119889119903

(9)

Definition 13 (symmetric Choquet integral [20]) Given afuzzy measure 120583 2119873 rarr [0 1] and a function 119891 on 119873 thesymmetric Choquet integral of 119891 with respect to 120583 is definedby

(SC) int119891119889120583 = minusint+infin

0

120583 (119909 | 119891minus(119909) gt 119903) 119889119903

+ int

+infin

0

120583 (119909 | 119891+(119909) gt 119903) 119889119903

(10)

where 119891+= 119891 or 0 119891

minus= minus(119891 and 0)

Proposition 14 Consider

(ASC) int119891119889120583

= minusint

0

minusinfin

120583119889(119909 | 119891

minus(119909) gt 119903) 119889119903

+int

+infin

0

120583 (119909 | 119891+(119909) gt 119903) 119889119903

(11)

6 Journal of Applied Mathematics

Proof Because

int

0

minusinfin

(120583 (119909 | 119891 (119909) gt 119903) minus 120583 (119873)) 119889119903

= int (120583 (119909 | minus119891minus(119909) gt 119903) minus 120583 (119873)) 119889119903

= minusint (120583 (119909 | 119891minus(119909) le 119903) minus 120583 (119873)) 119889119903

= minusint120583119889(119909 | 119891

minus(119909) gt 119903) 119889119903

(12)

by this proposition we can say that the difference betweenthe asymmetricChoquet integral and the symmetricChoquetintegral is based on which measure are used for integratingthe negative part of 119891

3 Description of Fairness bythe Choquet Integral

In this section we discuss the generalized Choquet integralsin terms of describing fairness by the Choquet integral Let 120583be a fuzzy measure on 2119873 and 119891 = (119909

1 119909

2 119909

119899) a function

on 119873 For example 119873 corresponds to a set consisting of119899 players and 119891 corresponds to a distribution of resourceswhere 119891 can take negative values The more players aresatisfied the fairer the distribution is

Example 15 Let 119899 = 3 and there are 6 resources Then1198911= (2 2 2) is fairer than 119891

2= (4 2 0) and 119891

2is fairer

than 1198913= (6 0 0) To describe this we use the Choquet

integral with respect to a submodular function Suppose that120583(119878) takes the same values according to |119878| (note that in thiscase the Choquet integral becomes an OWA operator andproperties like submodularity depend on the choice of OWAweights) If 120583 is submodular then we have

(119862) int1198911119889120583 ge (119862)int1198912119889120583 ge (119862)int1198913119889120583 (13)

Next we consider the negative distribution

Example 16 Let 119899 = 3 and there are minus6 resources Then 1198911=

(minus2 minus2 minus2) is fairer than1198912= (minus4 minus2 0) and119891

2is fairer than

1198913= (minus6 0 0) Let 120583 be a submodular function If we use the

symmetric Choquet integral then we have

(SC) int1198911119889120583 le (SC) int1198912119889120583 le (SC) int1198913119889120583 (14)

where all integrals take negative values And if 1205831015840 would beanother but superadditive fuzzy measure then we obtain

(SC) int11989111198891205831015840ge (SC) int1198912119889120583

1015840ge (SC) int1198913119889120583

1015840 (15)

which is not what we wanted In other words we cannever describe both positive fairness and negative fairness

by one fuzzy measure using the symmetric Choquet inte-gral On the other hand using the asymmetric Choquetintegral with respect to 120583 where 120583119889 is superadditive weobtain

(ASC) int1198911119889120583 ge (ASC) int1198912119889120583 ge (ASC) int1198913119889120583 (16)

If 120583 is a supermodular fuzzy measure then 120583 is superadditiveand 120583119889 is subadditive Therefore using the asymmetric Cho-quet integral with respect to a supermodular fuzzy measurewe can describe both positive fairness and negative fairnessby one fuzzy measure

According to these examples the asymmetric Choquetintegral with respect to a superadditive fuzzy measure isbetter suited for describing fairness The Mobius transformenables us to construct fuzzy measure easily

Definition 17 (Mobius transform) The Mobius transform of120583 denoted by119898120583

2119873rarr [minus1 1] is defined by

119898120583(119878) = sum

119879sube119878

(minus1)|119878119879|

120583 (119879) (17)

for any 119878 isin 2119873 Conversely we obtain

120583 (119878) = sum

119879sube119878

119898120583(119879) (18)

by the inverse Mobius transform for any 119878 isin 2119873 and there isa one-to-one correspondence between 120583 and119898120583

Note that if the sum of all Mobius masses is 1 then thecorresponding fuzzy measure of the whole set is 1

Proposition 18 If 119898120583(119878) ge 0 for any 119878 isin 2

119873 then 120583 issupermodular

Proof Assume for any 119878 isin 2119873 119898120583(119878) ge 0 Then denoting

that 119903 = 119878 119879 119904 = 119878 cap 119879 119905 = 119879 119878 for 119878 119879 isin 2119873 we have

120583 (119878 cup 119879) + 120583 (119878 cap 119879)

= sum

119862sube119878cup119879

119898120583(119880) + sum

119880sube119878cap119879

119898120583(119880)

= (sum

119880sube119903

+ sum

119880sube119904

+ sum

119880sube119905

+ sum

119880sube119903cup119904119880cap119903 = 0

119880cap119904 = 0

Journal of Applied Mathematics 7

+ sum

119880sube119904cup119905119880cap119904 = 0

119880cap119905 = 0

+ sum

119880sube119903cup119905

119880cap119903 = 0119880cap119905 = 0

+ sum

119880sube119903cup119904cup119905119880cap119903 = 0

119880cap119904 = 0119880cap119905 = 0

)119898120583(119880) + sum

119880sube119904

119898120583(119880)

120583 (119878) + 120583 (119879) = sum

119880sube119878

119898120583(119880) + sum

119880sube119879

119898120583(119880)

= (sum

119880sube119903

+ sum

119880sube119904

+ sum

119880sube119903cup119904119880cap119903 = 0

119880cap119904 = 0

)119898120583(119880)

+ (sum

119880sube119904

+ sum

119880sube119905

+ sum

119880sube119904cup119905119880cap119904 = 0

119880cap119905 = 0

)119898120583(119880)

(19)

so that we obtain

(120583 (119878 cup 119879) + 120583 (119904)) minus (120583 (119878) + 120583 (119879))

= ( sum

119880sube119903cup119905

119880cap119903 = 0119880cap119905 = 0

+ sum

119880sube119903cup119904cup119905119880cap119903 = 0

119880cap119904 = 0119880cap119905 = 0

)119898120583(119880) ge 0

(20)

Remark 19 The converse of Proposition 18 is not necessarilytrue In fact define119898120583 on119873 = 1 2 3 by

119898120583(119878) =

0 119878 = 0

1 119878 = 0 1 2 3

minus1

2 119878 = 1 2 3

(21)

Then we obtain a supermodular fuzzy measure

120583 (0) = 0 120583 (1) = 120583 (2) = 120583 (3) = 1

120583 (1 2) = 120583 (1 3) = 120583 (2 3) = 3

120583 (1 2 3) =11

2

(22)

4 Fairness Relation Using the AsymmetricChoquet Integral

Definition 20 (Choquet Integral relation) Let 119909 = (1199091

1199092 119909

119899) let 119910 = (119910

1 119910

2 119910

119899) and let 120583 be a fuzzy

measure 119909 is said to CI-dominate 119910 denoted by 119909ge120583119910 if and

only if

(ASC) int (119909 minus 119910) 119889120583 ge 120579 (23)

Assume119873 is a set of all players 119909 and 119910 are distributionsof resources to all players Then 119909 minus 119910 means each degreeof satisfaction of the distribution 119909 compared with thedistribution 119910 Since the more players are satisfied the fairerthe distribution is 119909ge

120583119910 means 119909 is at least as fair as 119910 By

119909gt120583119910 we denote the corresponding strict relation

Proposition 21 Let120583 be a superadditive fuzzymeasure on 2119873Then it is an antisymmetric relation that is 119909ge

120583119910 and 119910ge

120583119909

implies 119909 = 119910 for any functions 119909 119910 on119873

Proposition 22 If 120583 is a supermodular measure on 2119873 thengt120583is a transitive relation that is if 119909ge

120583119910 and 119910ge

120583119911 then 119909ge

120583119911

for any functions 119909 119910 119911 on119873

By Propositions 21 and 22 using the asymmetric Choquetintegral with respect to supermodular functions we obtain afair ordering relation

The proofs for Propositions 21 and 22 will be given in theAppendix of this paper

5 Decision Making by CI-Fairness ANumerical Example

In this section we want to demonstrate how CI-fairnessrelation can be used for decisionmaking Consider a situationwhere resources have to be allocated to 3 agents 119860 119861 and 119862but sets of agents can utilize resources differently (eg givinga commodity to one agent how many items can she producewithin a time unit) Such resource utilization can be assessedby a fuzzy measure

subset 119878 120583 (119878)

0 0

119860 01

119861 03

119862 05

119860 119861 05

119860 119862 05

119861 119862 08

119860 119861 119862 1

(24)

Here for example 119862 appears to be the best in resourceutilization but 119860 is good in collaboration The measure issuperadditive We can also see that this is a supermodularmeasure There are 3 cases to check the other follow fromsubsethood or superadditivity

(1) 120583(119860 119861cup119860 119862)+120583(119860 119861cap119860 119862) = 120583(119860 119861 119862)+120583(119860) = 11 ge 120583(119860 119861) + 120583(119860 119862) = 10

(2) 120583(119860 119861cup 119861 119862)+120583(119860 119861cap 119861 119862) = 120583(119860 119861 119862)+120583(119861) = 13 ge 120583(119860 119861) + 120583(119861 119862) = 13

8 Journal of Applied Mathematics

(1 2 3)

(1 3 2)

(2 3 1) (3 1 2)(2 1 3)

(3 2 1)

Figure 2 CI-Fairness relation among six possible allocations Theconnections include transitivity

(3) 120583(119860 119862cup119861 119862)+120583(119860 119862cap119861 119862) = 120583(119860 119861 119862)+120583(119862) = 15 ge 120583(119860 119862) + 120583(119861 119862) = 13

(1 2 3) ge120583 (1 3 2)

(1 2 3) ge120583 (2 1 3)

(1 2 3) ge120583 (2 3 1)

(1 2 3) ge120583 (3 1 2)

(1 2 3) ge120583 (3 2 1)

(1 3 2) ge120583 (2 3 1)

(1 3 2) ge120583 (3 1 2)

(1 3 2) ge120583 (3 2 1)

(2 1 3) ge120583 (2 3 1)

(2 1 3) ge120583 (3 1 2)

(2 1 3) ge120583 (3 2 1)

(2 3 1) ge120583 (3 2 1)

(3 1 2) ge120583 (3 2 1)

(25)

We find the allocation (1 2 3) in relation to all other alloca-tions but no allocation is in relation to (1 2 3)This is naturalgiven the order of resource utilization weights for the threeagents By rank we can now also order the other 5 allocations

Now we want to allocate three shares of a resource of6 units (1 2 3) among 119860 119861 119862 and compare the 6 possiblepermutations of (1 2 3) by the Choquet integral relationwith120579 = 0 For example to see if allocation 119909 = (1 2 3) is inrelation to the allocation 119910 = (2 3 1) we have to test

(119862) int [119909 minus 119910]+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (26)

In this case [119909 minus 119910]+ = (0 0 2) and [119910 minus 119909]+ = (1 1 0) Sothe first integral becomes 2 times 120583(119862) = 10 and the secondintegral 1 times 120583119889(119860 119861) = (1 minus 120583(119862)) = 05 From 10 ge 05then 119909ge

120583119910 follows

Among all possible 30 pairs (119909 119910) with 119909 = 119910 we can find13 cases in total where the Choquet integral relation holds

Figure 2 shows the Hasse diagram of the relation for the 6possible allocations We can see that as an effect of the bettercollaboration weight of 119860 allocation (2 1 3) is promotedand appears on the same rank as the strong allocation(1 3 2) (strong with regard to single agent utilization ofa resource) and also of better rank than the allocation(2 3 1)mdashexpressing the preference for 119862 as partner of 119860when comparing possible wins and losses in the allocationFinally as expected there is no case to allocate 3 shares to 119860

6 Experiments

In this section we will show how the CI-fairness can beapplied in a practical situation We study wireless channelallocation in the network layer a base station BS that isconnected to the backbone can command over a number119898 of wireless transmission channels and allocates them to anumber 119899 of users (or subscriber stations mobile stationsrelay stations etcmdashdepending on the specific network archi-tecture) for uplink or downlink traffic In the most abstractnotation the BSmakes the allocation based on ameasured orestimated channel state information if it allocates a channel 119894to user 119895 there will be a channel coefficient 119862

119894119895from [0 1] that

represents (in a simplified form) to what degree the user canemploy that channel For example remote users are likely tohave smaller channel coefficients than users that are close tothe BS Here an allocation is seen as feasible if each channelis allocated to one user and to each user at least one channelis assigned

The sum of all channel coefficients of allocated channelsfor a specific user is seen as the performance of the allocationfor that user Then the optimality task is related to theperformance vector for all users This is a direct reference toa relation between performance vectors and various fairnessrelations can be easily and conveniently studied in thiscontext If the number of channels and users is not too largeexhaustive search for maximum sets is possible For examplefor 4 users and 6 channels there are 1560 feasible allocationswhich means about 24 Mio pairwise comparisons

In this model channel allocation ignores low channelinterferences between users and specific channels by therestriction that each channel is allocated to exactly one userHowever if implementing a wireless access where some userscan use the same channel for transmission it might be hardto decide the exact schedule of such multiple assignmentsor to decide if installing such a scheme at all Thereforewe follow a more simplified approach channel interferencewill be represented by a conflict matrix between userswhere low elements for pairs of users indicate the optionto use multichannel assignment Then we use the converseelements of the conflict matrix as masses in the Mobiustransform More specifically the Mobius masses for singleelement sets are set to equal values the masses for two-element are set to 1 minus the corresponding element of theconflictmatrix and all other are set to 0Then themassvectoris normalized to a total of all masses of 1 to ensure that

Journal of Applied Mathematics 9

Table 1 Comparison of total performances for capacity propor-tional and maxmin fairness

Fairness relation Average performanceConflict matrix range 0 to 1

CI 42882Proportional 454661maxmin 414688

Conflict matrix range 09 to 1CI 456051Proportional 451135maxmin 422561

Conflict matrix range 0 to 02CI 409365Proportional 441848maxmin 404879

for the fuzzy measure the measure for the whole set is 1The inverse Mobius transform of such a mass vector gives asupermodularmeasure that can be used to specify a CI-basedfairness relation according to Definition 20

Now this relation is applied to all feasible channelallocationsmdashit means we are not considering a specific use ofmultichannel assignment but we are looking for the influencethat such amultichannel allocation could have on the specificselection of maximal elements Then the focus is on thedisadvantage that some pairs of users can experience if theirfavoured situation (eg to be distant enough to use the samechannel) is not taken into account The similar approach canbe considered for cognitive radio opportunistic networkingP2P vehicular networks and so forth

We might especially look for the price of fairness [1]compared to a standard proportional fair or maxmin fairallocation

Some example results are shown in Table 1 There for thecase of 4 users and 6 cells the maximum sets over all feasibleallocations were computed for CI-fairness using a measure asdescribed above proportional fairness andmaxmin fairnessThe conflictmatrix elements for the CI-fairness were uniformrandomly set within specific ranges All results are averagedover 30 repetitions Note that average sizes of maximum setswere found to be 42 formaxmin fairness 61 for proportionalfairness and 419 for CI-fairness so the CI-fairness produceslarger maximum sets an issue that should be addressed infutureworksThe table now shows average performances overall maximal elements

We see with regard to efficiency the well-known fact thatproportional-fair allocations are in average more efficientthan maxmin fair allocations The CI-fairness appears toselect maximal relations with a total performance betweenproportional fairness and maxmin fairness In case of stronginterferences (where a multichannel assignment is not rea-sonable) it is very close to proportional fairness while in caseof low interference it appears to be more close to maxminfairness that is the fairness relation favouring least elementsThus such an empirical result can be understood in the sensethat the neglecting of the potential multichannel assignment

corresponds with the neglection of strong users This is alsoparticularly appearing for the general case where all conflictmatrix elements were randomly selected between 0 and 1 alsohere the average performance appearsmore close tomaxminfairness than proportional fairness

7 Conclusion

A fairness relation among vectors based on the asymmetricChoquet integral was studied It formally follows the Nashstandard of comparison where the relative losses and winsare replaced by absolute losses and wins but instead takingweights for groups of vector components into account Thusit can represent for example pairs of users that are ina favourable situation regarding resource allocation Theappealing points of this CI-relation are as follows (1) Incase a supermodular measure is used for the integrationthe relation will be transitive which suits faster search formaximal elements (2) it can be parametrized by givinga weight to each subset of vector components (while inpractice this might be relevant only for smaller subsets) Therepresentation of the fuzzy measure by its Mobius transformappears to be a convenient way to yield a supermodularfuzzy measure the masses just need to be all nonnegativeThus the relation can be conveniently applied in many prac-tical applications to specify optimality Slight disadvantagesare with the complexity of the involved calculations andsome empirical evidence for specifying larger maximum setsthan other fairness relations like proportional fairness andmaxmin fairness

In case of wireless channel allocation how CI-fairnesscan be used to help deciding whether implementations ofmore complex allocation schemes are indeed worth the effortwas suggested This method can be applied to many otherproblems of higher efficiency of network resource utilizationIn future work we will consider the use of advanced fuzzyintegrals like balancing and fusion Choquet integral andthe adjustment of related CI-fairness parameters to specificsituations and what can be concluded from such parametersWe will also more intensively study the performance issueswith regard to computational effort and search for maximalelements and study conditions where also greatest elementsexist

Appendix

Proofs of Theorems

In the following we provide the proofs for Propositions 21and 22 For formal convenience of the proofs the notationwill be a little bit changing therefore we repeat the basicdefinitions

We define the Choquet-integral-based (fairness) relationas follows be 119909 and 119910 two vectors from 119877

119899 By [119909]+ weindicate the positive support of a vector 119909 that is the 119894thcomponent of [119909]+ is 119909

119894if 119909

119894ge 0 or 0 if 119909

119894lt 0 The Choquet

integral is denoted by (119862) int119909 119889120583 There 120583 is a fuzzy measure

10 Journal of Applied Mathematics

Also 120583119889 denotes its dual measure (ie 120583119889(119860) = 120583119883minus 120583(119860

119862))

Then 119909 is said to CI-dominate 119910 if and only if

119909ge120583119910 larrrarr (119862)int [119909 minus 119910]

+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (A1)

We can alternatively rewrite this as the asymmetric Choquetintegral

119909ge120583119910 larrrarr (ASC) int (119909 minus 119910) 119889120583 ge 0 (A2)

since the right-hand side condition of (A1) exactly cor-responds with the definition of the asymmetric Choquetintegral

First we can show the following

Lemma 23 If 120583 is a superadditive fuzzy measure where foreach 119860 119861 = 0 and 119860 cap 119861 = 0 120583

119860+ 120583

119861lt 120583

119860cup119861 then for

119860 = 0 120583119889

119860gt 120583

119860

Proof Since 120583 is superadditive for119860 = 0 120583119860+120583

119860119862 lt 120583

119860cup119860119862 =

120583119883 Then 120583

119860lt 120583

119883minus 120583

119860119862 = 120583

119889

119860follows directly

Theorem 24 If 120583 is superadditive then from 119909ge120583119910 and

119910ge120583119909119909 = 119910 follows

Proof 119909ge120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889 If

119909 = 119910 we can use Lemma 23 to obtain

(119862) int [119909 minus 119910]+119889120583

119889gt (119862)int [119909 minus 119910]

+119889120583

ge (119862)int [119910 minus 119909]+119889120583

119889gt (119862)int [119910 minus 119909]

+119889120583

(A3)

that is (119862) int [119910 minus 119909]+119889120583 ge (119862) int [119909 minus 119910]+119889120583

119889 This wouldmean that 119910ge

120583119909 would not hold Therefore it must be 119909 =

119910

For a supermodular measure and nonnegative vectors 119909and 119910 we have the known property (119862) int 119909 119889120583+ (119862) int 119910 119889120583 le(119862) int(119909 + 119910)119889120583 Since the dual measure of a supermodularmeasure is submodular it also holds that (119862) int 119909 119889120583 +(119862) int 119910 119889120583 ge (119862) int(119909 + 119910)119889120583 Using this the following canbe shown

Theorem25 If120583 is supermodular then the relationgt120583is cycle-

free

Proof We will only show the case of nonexistence of a 3-cycle here the concept generalizes directly to any other caseAssume that 119909gt

120583119910 and 119910gt

120583119911 First we note that

[119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+

= [119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+

(A4)

and therefore

(119862) int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

= (119862)int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

(A5)

Using Lemma 23 we yield

(119862) int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

119889

gt (119862)int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

(A6)

and from supermodularity of 120583 and submodularity of 120583119889

(119862) int [119910 minus 119909]+119889120583

119889+ (119862)int [119911 minus 119910]

+119889120583

119889+ (119862)int [119909 minus 119911]

+119889120583

119889

gt (119862)int [119909 minus 119910]+119889120583 + (119862)int [119910 minus 119911]

+119889120583

+ (119862)int [119911 minus 119909]+119889120583

(A7)

However 119909gt120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889

and 119910gt120583119911 means (119862) int [119910 minus 119911]+119889120583 ge (119862) int [119911 minus 119910]

+119889120583

119889Therefore above inequality can only hold if

(119862) int [119911 minus 119909]+119889120583 lt (119862)int [119909 minus 119911]

+119889120583

119889 (A8)

which means 119911 gt120583119909

If the measure 120583 is supermodular also transitivity ofthe relation can be shown Before we can show this weneed to introduce a notation and a corresponding lemmaThe notation refers to computing the (asymmetric) Choquetintegral of 119909 by measure 120583 in a ldquodifferent orderrdquo If 119909

119909(119894)

indicates the 119894th largest element of 119909 then the Choquetintegral is calculated as

(119862) int 119909119889120583 = 119909119909(1)120583119909(1)

+

119899

sum

119894=2

119909119909(119894)[120583

119909(1)119909(2)sdotsdotsdot119909(119894)minus 120583

119909(1)sdotsdotsdot119909(119894minus1)]

(A9)

For any other119910 we can compute the same expression justfollowing the order of the elements of 119910 instead of 119909 Then119909119910(119894)

indicates the element of119909with the index of the 119894th largestelement of 119910 and we define (for convenience we still use thesymbol for the Choquet integral keeping in mind that it isnot a real fuzzy integral anymore)

(119862) int(119910)

119909119889120583 = 119909119910(1)120583119910(1)

+

119899

sum

119894=2

119909119910(119894)[120583

119910(1)119910(2)sdotsdotsdot119910(119894)minus 120583

119910(1)sdotsdotsdot119910(119894minus1)]

(A10)

Journal of Applied Mathematics 11

Consequently we could also write (119862) int(119909)119909119889120583 for the

ldquooriginalrdquo Choquet integral We can do the same for theasymmetric Choquet integral by applying the resorting to thenegative components of the vector 119909 as well and will use thenotation (ASC) int

(119910)119909 119889120583 here Now we have the following

Lemma 26 If 120583 is a supermodular measure then for any119910 ( ASC ) int

(119910)119909 119889120583 ge ( ASC ) int

(119909)119909119889120583

Proof A moment of reasoning gives that any sorting of theindizes can be achieved by suitable application of a sequenceof three kinds of ldquoneighbour swapsrdquo starting from the order119909(1)119909(2) sdot sdot sdot 119909(119899))

Swap 1 In the computation of the integral a larger orequal nonnegative element 119889

119894at index 119894 is swapped with its

immediate nonnegative neighbour 119889119894+1

to the right Thenwhat was computed as 119889

119894(120583

119886119894minus120583

119886) +119889

119894+1(120583

119886119894119894+1minus120583

119886119894) before

the swap changes to 119889119894+1(120583

119886119894+1minus120583

119886) + 119889

119894(120583

119886119894119894+1minus120583

119886119894+1) after

the swap all other parts of the resorted asymmetric Choquetintegral expression remain the same (by 119886 we indicate theindex order before index 119894) Thus the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886119894)

minus 119889119894+1(120583

119886119894119894+1minus 120583

119886119894minus 120583

119886119894+1+ 120583

119886119894)

(A11)

and from 119889119894ge 119889

119894+1and 120583

119886119894119894+1+ 120583

119886ge 120583

119886119894+1+ 120583

119886119894

(supermodularity of 120583) it directly follows that the computedvalue will not decrease by this swap

Swap 2 In the computation of the integral a larger orequal negative element 119889

119894+1at index 119894 + 1 is swapped with

its immediate negative left neighbour 119889119894 From a similar

evaluation as for Swap 1 it can be seen that also here the valueof the computed expression will not decrease after the swap

Swap 3 A nonnegative element at index 119894 is swapped withits immediate negative neighbor to the right with index(119894 + 1) Then before the swap we compute 119889

119894(120583

119886119894minus 120583

119886) minus

119889119894+1(120583

119886119894119894+1minus 120583

119886119894) (119889

119894+1is the absolute value of the negative-

valued neighbour of 119889119894) and after the swap this part of the

expression changes to minus119889119894+1(120583

119886119894+1minus 120583

119886) + 119889

119894(120583

119886119894119894+1minus 120583

119886119894+1)

Then the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886)

minus 119889119894+1(120583

119886119894+1minus 120583

119886minus 120583

119886119894119894+1+ 120583

119886119894)

(A12)

Supermodularity of 120583 gives 120583119886119894+ 120583

119886119894+1le 120583

119886+ 120583

119886119894119894+1and

therefore the factor of 119889119894in former expression is nonnegative

and the factor of 119889119894+1

(which is the negation of the factor of119889119894) is negative or 0Therefore the total change is nonnegative

as wellThus with each of these swap operations we will never

decrease the value of the expression and starting from theasymmetric Choquet integral for 119909 upon reaching the finalorder the final value will be larger or equal

An example might be helpful Assume that the order in 119909is (1 2 3 minus 4 minus 5) (a negative index should indicate that

the corresponding element of 119909 is negative) and in 119910 we havethe order (minus5 3 1 2 minus4) (so 119910

5is the largest component of

119910 and 1199104the smallest) Then the order of swaps is as follows

Swap 1 applied as (1 2 3larr997888rarr

minus4 minus5) gives (1 3 2 minus4 minus5)Swap 1 applied as (1 3

larr997888rarr2 minus4 minus5) gives (3 1 2 minus4 minus5)

Next Swap 2 applied as (3 1 2 minus4 minus 5larr997888997888997888997888rarr

) gives (3 1 2 minus

5 minus 4) Swap 3 applied as (3 1 2 minus 5larr997888997888997888rarr

minus 4) gives (3 1 minus

5 2 minus 4) Swap 3 applied as (3 1 minus 5larr997888997888997888rarr

2 minus 4) gives (3 minus

5 1 2 minus4) and finally Swap 3 applied as (3 minus 5larr997888997888997888rarr

1 2 minus4)

gives the order (minus5 3 1 2 minus 4) of 119910 Note also that in casethe elements of 119909 are not in that order we can always relabelthe indizes and measures correspondingly

This means if we compute (using 119909 =

(1199091 119909

2 119909

3 minus119909

4 minus119909

5))

(ASC) int(119910)

119909119889120583

= minus11990951205835+ 119909

3(120583

35minus 120583

5) + 119909

1(120583

135minus 120583

35)

+ 1199092(120583

1235minus 120583

135) minus 119909

4(120583

12345minus 120583

1235)

(A13)

instead of

(ASC) int 119909 119889120583

= 11990911205831+ 119909

2(120583

12minus 120583

1) + 119909

3(120583

123minus 120583

12)

minus 1199094(120583

1234minus 120583

123) minus 119909

5(120583

12345minus 120583

1234)

(A14)

then (ASC) int(119910)119909 119889120583 ge (ASC) int 119909 119889120583

Using this Lemma we can now easily show the following

Theorem 27 If 120583 is a supermodular measure then ge120583is a

transitive relation

Proof For transitivity we have to show that for any 119909 119910 and119911 such that 119909ge

120583119910 and 119910ge

120583119911 it follows that 119909ge

120583119911 Now 119909ge

120583119910

means (ASC) int(119909 minus 119910)119889120583 ge 0 and 119910ge120583119911 means (ASC) int(119910 minus

119911)119889120583 ge 0 From supermodularity of 120583 and Lemma 26 then italso follows that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 ge (ASC) int (119909 minus 119910) 119889120583 ge 0

(ASC) int(119909minus119911)

(119910 minus 119911) 119889120583 ge (ASC) int (119910 minus 119911) 119889120583 ge 0(A15)

However by using the fact that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 + (ASC) int(119909minus119911)

(119910 minus 119911) 119889120583

= (ASC) int (119909 minus 119911) 119889120583(A16)

we see that also

(ASC) int (119909 minus 119911) 119889120583 ge 0 (A17)

which means 119909ge120583119911

12 Journal of Applied Mathematics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Bertsimas V F Farias and N Trichakis ldquoThe price offairnessrdquo Operations Research vol 59 no 1 pp 17ndash31 2011

[2] J F Nash Jr ldquoThe bargaining problemrdquo Econometrica vol 18pp 155ndash162 1950

[3] E Kalai and M Smorodinsky ldquoOther solutions to Nashrsquosbargaining problemrdquo Econometrica vol 43 pp 513ndash518 1975

[4] J Rawls ATheory of Justice Harvard University Press 1999[5] F Kelly ldquoCharging and rate control for elastic trafficrdquo European

Transactions onTelecommunications vol 8 no 1 pp 33ndash37 1997[6] D P Bertsekas R G Gallager and P Humblet Data Networks

vol 2 Prentice-Hall International 1992[7] H P Young Equity In Theory and Practice Princeton Univer-

sity Press Princeton NJ USA 1995[8] J Mo and J Walrand ldquoFair end-to-end window-based conges-

tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[9] M Alicherry R Bhatia and L E Li ldquoJoint channel assignmentand routing for throughput optimization inmulti-radiowirelessmesh networksrdquo in Proceedings of the 11th Annual InternationalConference on Mobile Computing and Networking (MobiCom05) pp 58ndash72 ACM Cologne Germany August-September2005

[10] A Iyer C Rosenberg and A Karnik ldquoWhat is the rightmodel for wireless channel interferencerdquo IEEE Transactions onWireless Communications vol 8 no 5 pp 2662ndash2671 2009

[11] A Mishra V Shrivastava S Banerjee and W ArbaughldquoPartially overlapped channels not considered harmfulrdquo ACMSIGMETRICS Performance Evaluation Review vol 34 no 1 pp63ndash74 2006

[12] N Nie and C Comaniciu ldquoAdaptive channel allocation spec-trum etiquette for cognitive radio networksrdquo Mobile Networksand Applications vol 11 no 6 pp 779ndash797 2006

[13] C Peng H Zheng and B Y Zhao ldquoUtilization and fairnessin spectrum assignment for opportunistic spectrum accessrdquoMobile Networks and Applications vol 11 no 4 pp 555ndash5762006

[14] X Yang andA P Petropulu ldquoCo-channel interferencemodelingand analysis in a Poisson field of interferers in wireless commu-nicationsrdquo IEEE Transactions on Signal Processing vol 51 no 1pp 64ndash76 2003

[15] M Koppen ldquoRelational optimization and its application frombottleneck flow control to wireless channel allocationrdquo Infor-matica vol 24 no 3 pp 413ndash433 2013

[16] M Sugeno Theory of fuzzy integrals and its applications [PhDthesis] 1974

[17] S Michio ldquoFuzzymeasures and fuzzy integrals a surveyrdquo FuzzyAutomata and Decision Processes vol 78 no 33 pp 89ndash1021977

[18] D Denneberg Non-Additive Measure and Integral vol 27Springer 1994

[19] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[20] J Sipos ldquoIntegral with respect to a pre-measurerdquo MathematicaSlovaca vol 29 no 2 pp 141ndash155 1979

[21] M Grabisch and C Labreuche ldquoThe symmetric and asymmet-ric Choquet integrals on finite spaces for decision makingrdquoStatistical Papers vol 43 no 1 pp 37ndash52 2002

[22] M Grabisch C Labreuche and J-C Vansnick ldquoOn theextension of pseudo-Boolean functions for the aggregation ofinteracting criteriardquo European Journal of Operational Researchvol 148 no 1 pp 28ndash47 2003

[23] R Mesiar A Mesiarova-Zemankova and K Ahmad ldquoDiscreteChoquet integral and some of its symmetric extensionsrdquo FuzzySets and Systems vol 184 pp 148ndash155 2011

[24] A Mesiarova-Zemankova R Mesiar and K Ahmad ldquoThebalancing Choquet integralrdquo Fuzzy Sets and Systems vol 161no 17 pp 2243ndash2255 2010

[25] H Fouchal X Gandibleux and F Lehuede ldquoA lower bound ofthe Choquet integral integrated within Martinsalgorithmrdquo inNewState ofMCDM in the 21st Century vol 648 ofLectureNotesin Economics and Mathematical Systems pp 79ndash89 SpringerHeidelberg Germany 2011

[26] L Galand L Julien and P Patrice ldquoDominance rules forthe choquet integral in multiobjective dynamic programmingrdquoin Proceedings of the 23rd International Joint Conference onArtificial Intelligence pp 538ndash544 AAAI Press 2013

[27] A W Marshall I Olkin and B C Arnold Inequalities TheoryofMajorization and Its Applications Springer 2nd edition 2010

[28] W Ogryczak AWierzbicki andMMilewski ldquoAmulti-criteriaapproach to fair and efficient bandwidth allocationrdquoOmega vol36 no 3 pp 451ndash463 2008

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

formal representations of fairness in resource distributionas optimality issue by itself Our proposal is to evaluatedistributions from the aspect of fairness

In this paper we compare the generalized Choquet inte-gral the symmetric Choquet integral and the asymmetricChoquet integral from the viewpoint of representing fairnesswith the result that the asymmetric Choquet integral is moresuitable for describing fairness We also propose a fairnessrelation using the asymmetric Choquet integral

The paper is organized as follows Section 2 introducesnotations and necessary materials about fuzzy measure andthe Choquet integral and recalls the relational approach tofairnessThen Section 3 presents discussion about two gener-alized Choquet integrals The fairness relation is proposed inSection 4 and we give a sufficient condition that this relationis consistent and transitive Section 5 gives a demonstrationof how our fairness relation is used for decision makingSection 6 compares the solutions obtained by this fairnessrelation with the proportional and maxmin fair solutions byway of experiments

2 Preliminaries

21 Relational Optimization Speaking about optimality isspeaking about comparing things The basic mathematicalinstrument for the representation of optimality is the conceptof a binary relation Here we want to recall some basic termsand definitions formore details see for example [15] A (set-theoretic) binary relation119877 over a domain119860 is a subset of119860times119860 that is a set of ordered pairs (119909 119910) with 119909 119910 isin 119860 It is saidthat119909 is in relation119877 to119910 (sometimes alsowritten as119909119877119910 or ifthe focus is on comparison 119909ge

119877119910)There are alternative ways

to represent relations like a set comprehension a mapping orfunction from 119860 into the powerset of 119860 an incidence matrixor a directed graph Also relations can have properties Twoproperties are of particular interest in the following a relationis symmetric if from (119909 119910) isin 119877 it follows that also (119910 119909) isin 119877and it is asymmetric if from (119909 119910) isin 119877 it always follows that(119910 119909) does not belong to119877 Each relation can be decomposedinto a symmetric and an asymmetric part written as 119877 =

119868(119877)cup119875(119877) where 119868(119877) is a symmetric relation and119875(119877) is anasymmetric relation It is easy to decide to which part a pair(119909 119910) isin 119877 belongs in this decomposition if (119910 119909) is also in 119877it belongs to the symmetric part 119868(119877) if not it belongs to theasymmetric part 119875(119877) It means that each relation no matterwhat its domain is made of shows mixed aspects of equalityor similarity and a means of ldquobetternessrdquo or preferenceFor example the real-valued ldquogerdquo relation is decomposedinto an equivalence relation ldquo=rdquo as symmetric part and astrictly larger relation ldquogtrdquo as asymmetric part (which is wellsymbolized by ge) If we write 119909ge

119877119910 for a relation then 119909gt

119877119910

will denote the asymmetric part reflecting the ldquobetternessrdquoaspect of a relation 119877 over domain 119860

Any relation 119877 can expose specific elements of its domain119860 which are commonly called greatest maximal minimaland least elements A greatest element 119909lowast of the domain119860 with regard to 119877 has the property that for any 119909 isin 119860

(including 119909lowast) it holds that 119909lowastge119877119909 All greatest elements

comprise the best set Amaximal element 119909lowast has the propertythat there is no 119909 isin 119875(119877) such that 119909gt

119877119909lowast and all such 119909lowast

comprise themaximum set (of119860with respect to 119877) Minimaland least elements are defined correspondingly Note thatthey do not need to exist in general

One can easily see that a standard of comparison as itwas mentioned in the introduction section is the same asthe concept of a greatest element However often it is thecase that best sets are empty Therefore it is convenient torefer to the maximum set as optimization goal for severalreasons (1) In case of a finite domain the maximum set isnonempty if the asymmetric part of the relation is cycle-free(ie the corresponding directed graph does not contain anycycles) (2) If there is a greatest element it will also belongto the maximum set (3) The maximum set can be seen as aldquofrontierrdquo they are not greatest elements but at least no betterone is known

In [15] the special case of fairness relations has beenintensively discussed It is comparable and easy to representproportional fairness maxmin fairness leximin fairness 120572-fairness and so forth by this relational framework Then(Pareto-) efficient solutions can be found by finding themaximum sets for these relations But the advantage is alsothat the transition of a relational concept to another domainbecomes possible (by applying the same formal definition) orthat the relation can be specified in such away that completelynew aspects are taken into account (eg multiresource usagecollaboration between agents or vector-valued evaluations)This is because the concept of maximality applies to anyrelation no matter what its domain is Therefore if we wantto handle the above mentioned problem of pairwise weightswe just have to specify a corresponding relation (that ispreferably cycle-free and Pareto-efficient)

The last point that should be mentioned here is that thefinding of maximum sets can be a challenging task even ifthe complexity 119874(1198992) is not so high Thus exhaustive searchby a complete pairwise comparison can rapidly becomeintractable (eg a maximum set over a domain with onemillion elements would need 1012 pairwise comparisons)One possibility is to use metaheuristic search algorithms toapproximate maximum sets other alternatives are currentlyunder investigations In case the relation is transitive thesearch for maximum sets can be performed in ldquobatchesrdquo thatis decompose the domain into nonoverlapping subsets andfind themaximum set for each subset and then themaximumset of the union of the found maximum sets

22 Computational Fairness We have now used the termldquofairnessrdquo several times without providing a general defini-tion For example considering a given relation 119877 when is itsafe to say that this is a fairness relation Actually there is nocommon understanding of fairness to which all researchersand practitioners would fully agree It would go beyond thescope of this paper to consider the numerous considerationsthat have been done in order to provide a computationalmodel of fairness but at least we can summarize the basicideas The additional restriction here is that the focus is onthe modality of a distribution thus excluding other societal

4 Journal of Applied Mathematics

important concepts like procedural fairness or interactionfairness

The common theme of all approaches to (distributional)fairness is to divide a set of resources among a numberof agents As already mentioned in the introduction withfar longest cultural history we can find ideas about pro-portionality in the assignment (proportional according todemand ability suitability availability most available goodetc) or the resolution of conflicts about contended goodsin following an equity principle Besides a relation point ofview in order to compare different possible assignmentsthe aspect can be also seen in aggregating the distributionalaspects of all agents and thus keeping the spread of allocationdifferences small This idea becomes strongly related to thetheory of majorization [27] where numerical comparisonbetween different ways of distributing the same total isstrongly related to the Schur-convexity of the used compari-son measureThe application of this approach and its variousgeneralizations have been intensively studied for examplein [28]

Any continuation of a formal approach to fairness (inthe sense of a ldquodefinition of fairnessrdquo) would need thespecification of axioms that represent criteria derived fromspecific ways of how to balancemdashin a just waymdashthe conflictsbetween agents that are in a better and agents that are in aworse situationmdashkeeping inmind that fairness does generallynot stand for the simple requirement of complete even allo-cations Such axioms can refer to impartiality of distributionconsistency between solutions for different domains mathe-matical assumptions about the domain limiting behavioursstrategy-proofness of related procedures or efficiency of thedistribution

Saying this the considered aspect of fairness here is thebalance between users who would gain more and users whowould gain less in an alternative allocation We will useaggregation operators for representing the joint state of allwinning and all loosing agents thus basically following theldquoreciperdquo of proportional fairness But as a new aspect we alsowant to take a different weighting for groups of agents intoaccount Fuzzy integrals are a well-known formal approachfor such a representation

23 Fuzzy Measures and Integrals We now want to recallbasic issues of fuzzy measures and integrals in more detailMost of the material can be gathered together from variouspublications and textbooks but a comprehensive descriptionas it will be given now might be of advantage for theunderstanding of the main proposalmdasha fairness relationbased on the asymmetric Choquet integral for supermodularfuzzy measures

Throughout this paper we consider a finite universal set119873 = 1 2 119899 and 2119873 denotes the power set of119873

Definition 1 (Fuzzy measure [16 17]) A set function 120583

2119873rarr [0infin) is a fuzzy measure if it satisfies the following

conditions(i) 120583(0) = 0 and 120583(119873) lt +infin(ii) for any 119878 119879 isin 2119873 120583(119878) le 120583(119879) whenever 119878 sub 119879

Properties (i) and (ii) express boundedness and mono-tonicity respectively

Definition 2 (superadditivity subadditivity) A fuzzy mea-sure 120583 is superadditive if 120583 satisfies 120583(119878 cup 119879) ge 120583(119878) + 120583(119879)and subadditive if 120583 satisfies 120583(119878 cup 119879) le 120583(119878) + 120583(119879) for any119878 119879 isin 2

119873 satisfying 119878 cap 119879 = 0

Definition 3 (submodularity supermodularity) A fuzzymeasure 120583 is supermodular if 120583 satisfies 120583(119878 cup119879)+120583(119878 cap119879) ge120583(119878)+120583(119879) and submodular if 120583 satisfies 120583(119878cup119879)+120583(119878cap119879) le120583(119878) + 120583(119879) for any 119878 119879 isin 2119873

Definition 4 (dual measure) Let 120583 be a fuzzy measure Thedual measure of 120583 is defined by

119898119889(119878) = 120583 (119873) minus 120583 (119878

119888) (2)

The dual measure of a fuzzy measure is also a fuzzymeasure

Proposition 5 If 120583 is supermodular then 120583 is superadditiveIf 120583 is submodular then 120583 is subadditive

Proof When 120583 is supermodular 120583(119878 cup119879) + 120583(119878 cap119879) ge 120583(119878) +120583(119879) is satisfied for any 119878 119879 isin 2119873 and 119878 cap 119879 = 0 It implies120583(119878 cup 119879) ge 120583(119878) + 120583(119879) for any 119878 119879 isin 2119873 satisfying 119878 cap 119879 = 0The second assertion is obtained in a similar manner

Proposition 6 120583 is supermodular if and only if 120583119889 is sub-modular Similarly 120583 is submodular if and only if 120583119889 issupermodular

Proof Assume that 120583 is supermodularThen we have for any119878 119879 isin 2

119873

120583119889(119878 cup 119879) + 120583

119889(119878 cap 119879)

= 120583 (119873) minus 120583 ((119878 cup 119879)119888) + 120583 (119873) minus 120583 ((119878 cap 119879)

119888)

= 2120583 (119873) minus 120583 (119878119888cap 119879

119888) minus 120583 (119878

119888cup 119879

119888)

le 2120583 (119873) minus 120583 (119878119888) minus 120583 (119879

119888)

= 120583119889(119878) + 120583

119889(119879)

(3)

Following the above in reverse we obtain the converseReplacing le with ge we obtain the second assertion

Proposition 7 If 120583 is a superadditive fuzzy measure then120583119889(119878) ge 120583(119878) for any 119878 isin 2119873

Proof We have 120583(119878) + 120583(119878119888) le 120583(119873) by superadditivity of 120583Hence 120583(119878) le 120583(119873) minus 120583(119878119888) = 120583119889(119878)

Definition 8 (120582-fuzzy measure) A fuzzy measure 120583 is a 120582-fuzzy measure if there exists 120582 gt minus1 such that 120583(119878 cup 119879) =120583(119878)+120583(119879)+120582120583(119878)120583(119879) for any 119878 119879 isin 2119873 satisfying 119878cap119879 = 0and 120583(119873) = 1

Journal of Applied Mathematics 5

Definition 9 (120594-fuzzy measure) A fuzzy measure 120583 on finiteset119873 = 1 2 119899 is a 120594-fuzzy measure if there exists

120594 ge 1 minusmin

119894isin119873120583 (119894)

sum119894isin119873120583 (119894)

(4)

such that

120583 (119878) = 120594|119878|minus1

sum

119894isin119878

120583 (119894) (5)

for any 119878 isin 2119873 Here |119878| denotes the cardinal number of 119878

120582 ge 0 and 120594 ge 1 imply superadditivity and 120582 le 0 and120594 le 1 imply subadditivity The dual measure of a 120582-fuzzymeasure and of a 120594-fuzzy measure are not necessarily a 120582-fuzzy measure and a 120594-fuzzy measure

Proposition 10 Let 120583 be a 120582-fuzzy measure Then 120583 issuperadditive if and only if 120583 is supermodular Similarly 120583 issubadditive if and only if 120583 is submodular

Proof The sufficiencies hold by Proposition 5 We show thenecessities

Assume 120583 is a superadditive 120582-fuzzy measure that is 120582 ge0 Then we have for any 119878 119879 isin 2119873

120583 (119878 cup 119879) + 120583 (119878 cap 119879)

= 120583 (119878) + 120583 (119879 119878) + 120582120583 (119878) 120583 (119879 119878) + 120583 (119878 cap 119879)

ge 120583 (119878) + 120583 (119879 119878) + 120582120583 (119878 cap 119879) 120583 (119879 119878) + 120583 (119878 cap 119879)

= 120583 (119878) + 120583 (119879)

(6)

Replacing le with ge we obtain the second assertion

Definition 11 (Choquet integral [19]) Given a fuzzy measure120583 2

119873rarr [0 1] and a nonnegative function 119891 on 119873 the

Choquet integral of 119891 with respect to 120583 is defined by

(119862) int119891119889120583 = int

+infin

0

120583 (119909119891 (119909) gt 119903) 119889119903 (7)

In the case that119873 is a finite set (7) can be rewritten as

(119862) int119891119889120583 =

119899

sum

119894=1

(119886119894minus 119886

119894minus1) 120583 (119878

119894) (8)

where 119886119894= 119891(119894) 119886

0= 0 119886

1le 119886

2le sdot sdot sdot le 119886

119899 and

119878119894= 119894 119899 (cf Figure 1) Note that this computation will

include a permutation reordering of the 119886119894to ensure that they

are in nondecreasing orderThe Choquet integral can be generalized to general

functions which are not necessarily nonnegative in severalways Here we show the following two generalizations theasymmetric Choquet integral and the symmetric Choquetintegral

Definition 12 (asymmetric Choquet integral [18]) Given afuzzy measure 120583 2119873 rarr [0 1] and a function 119891 on 119873 the

an

anminus1

a3

a2

a1

a0

Sn

S2

S1

Figure 1 Choquet integral

asymmetric Choquet integral of119891with respect to 120583 is definedby

(ASC) int119891119889120583 = int

0

minusinfin

(120583 (119909 | 119891 (119909) gt 119903)

minus119898 (119873)) 119889119903

+int

+infin

0

120583 (119909 | 119891 (119909) gt 119903) 119889119903

(9)

Definition 13 (symmetric Choquet integral [20]) Given afuzzy measure 120583 2119873 rarr [0 1] and a function 119891 on 119873 thesymmetric Choquet integral of 119891 with respect to 120583 is definedby

(SC) int119891119889120583 = minusint+infin

0

120583 (119909 | 119891minus(119909) gt 119903) 119889119903

+ int

+infin

0

120583 (119909 | 119891+(119909) gt 119903) 119889119903

(10)

where 119891+= 119891 or 0 119891

minus= minus(119891 and 0)

Proposition 14 Consider

(ASC) int119891119889120583

= minusint

0

minusinfin

120583119889(119909 | 119891

minus(119909) gt 119903) 119889119903

+int

+infin

0

120583 (119909 | 119891+(119909) gt 119903) 119889119903

(11)

6 Journal of Applied Mathematics

Proof Because

int

0

minusinfin

(120583 (119909 | 119891 (119909) gt 119903) minus 120583 (119873)) 119889119903

= int (120583 (119909 | minus119891minus(119909) gt 119903) minus 120583 (119873)) 119889119903

= minusint (120583 (119909 | 119891minus(119909) le 119903) minus 120583 (119873)) 119889119903

= minusint120583119889(119909 | 119891

minus(119909) gt 119903) 119889119903

(12)

by this proposition we can say that the difference betweenthe asymmetricChoquet integral and the symmetricChoquetintegral is based on which measure are used for integratingthe negative part of 119891

3 Description of Fairness bythe Choquet Integral

In this section we discuss the generalized Choquet integralsin terms of describing fairness by the Choquet integral Let 120583be a fuzzy measure on 2119873 and 119891 = (119909

1 119909

2 119909

119899) a function

on 119873 For example 119873 corresponds to a set consisting of119899 players and 119891 corresponds to a distribution of resourceswhere 119891 can take negative values The more players aresatisfied the fairer the distribution is

Example 15 Let 119899 = 3 and there are 6 resources Then1198911= (2 2 2) is fairer than 119891

2= (4 2 0) and 119891

2is fairer

than 1198913= (6 0 0) To describe this we use the Choquet

integral with respect to a submodular function Suppose that120583(119878) takes the same values according to |119878| (note that in thiscase the Choquet integral becomes an OWA operator andproperties like submodularity depend on the choice of OWAweights) If 120583 is submodular then we have

(119862) int1198911119889120583 ge (119862)int1198912119889120583 ge (119862)int1198913119889120583 (13)

Next we consider the negative distribution

Example 16 Let 119899 = 3 and there are minus6 resources Then 1198911=

(minus2 minus2 minus2) is fairer than1198912= (minus4 minus2 0) and119891

2is fairer than

1198913= (minus6 0 0) Let 120583 be a submodular function If we use the

symmetric Choquet integral then we have

(SC) int1198911119889120583 le (SC) int1198912119889120583 le (SC) int1198913119889120583 (14)

where all integrals take negative values And if 1205831015840 would beanother but superadditive fuzzy measure then we obtain

(SC) int11989111198891205831015840ge (SC) int1198912119889120583

1015840ge (SC) int1198913119889120583

1015840 (15)

which is not what we wanted In other words we cannever describe both positive fairness and negative fairness

by one fuzzy measure using the symmetric Choquet inte-gral On the other hand using the asymmetric Choquetintegral with respect to 120583 where 120583119889 is superadditive weobtain

(ASC) int1198911119889120583 ge (ASC) int1198912119889120583 ge (ASC) int1198913119889120583 (16)

If 120583 is a supermodular fuzzy measure then 120583 is superadditiveand 120583119889 is subadditive Therefore using the asymmetric Cho-quet integral with respect to a supermodular fuzzy measurewe can describe both positive fairness and negative fairnessby one fuzzy measure

According to these examples the asymmetric Choquetintegral with respect to a superadditive fuzzy measure isbetter suited for describing fairness The Mobius transformenables us to construct fuzzy measure easily

Definition 17 (Mobius transform) The Mobius transform of120583 denoted by119898120583

2119873rarr [minus1 1] is defined by

119898120583(119878) = sum

119879sube119878

(minus1)|119878119879|

120583 (119879) (17)

for any 119878 isin 2119873 Conversely we obtain

120583 (119878) = sum

119879sube119878

119898120583(119879) (18)

by the inverse Mobius transform for any 119878 isin 2119873 and there isa one-to-one correspondence between 120583 and119898120583

Note that if the sum of all Mobius masses is 1 then thecorresponding fuzzy measure of the whole set is 1

Proposition 18 If 119898120583(119878) ge 0 for any 119878 isin 2

119873 then 120583 issupermodular

Proof Assume for any 119878 isin 2119873 119898120583(119878) ge 0 Then denoting

that 119903 = 119878 119879 119904 = 119878 cap 119879 119905 = 119879 119878 for 119878 119879 isin 2119873 we have

120583 (119878 cup 119879) + 120583 (119878 cap 119879)

= sum

119862sube119878cup119879

119898120583(119880) + sum

119880sube119878cap119879

119898120583(119880)

= (sum

119880sube119903

+ sum

119880sube119904

+ sum

119880sube119905

+ sum

119880sube119903cup119904119880cap119903 = 0

119880cap119904 = 0

Journal of Applied Mathematics 7

+ sum

119880sube119904cup119905119880cap119904 = 0

119880cap119905 = 0

+ sum

119880sube119903cup119905

119880cap119903 = 0119880cap119905 = 0

+ sum

119880sube119903cup119904cup119905119880cap119903 = 0

119880cap119904 = 0119880cap119905 = 0

)119898120583(119880) + sum

119880sube119904

119898120583(119880)

120583 (119878) + 120583 (119879) = sum

119880sube119878

119898120583(119880) + sum

119880sube119879

119898120583(119880)

= (sum

119880sube119903

+ sum

119880sube119904

+ sum

119880sube119903cup119904119880cap119903 = 0

119880cap119904 = 0

)119898120583(119880)

+ (sum

119880sube119904

+ sum

119880sube119905

+ sum

119880sube119904cup119905119880cap119904 = 0

119880cap119905 = 0

)119898120583(119880)

(19)

so that we obtain

(120583 (119878 cup 119879) + 120583 (119904)) minus (120583 (119878) + 120583 (119879))

= ( sum

119880sube119903cup119905

119880cap119903 = 0119880cap119905 = 0

+ sum

119880sube119903cup119904cup119905119880cap119903 = 0

119880cap119904 = 0119880cap119905 = 0

)119898120583(119880) ge 0

(20)

Remark 19 The converse of Proposition 18 is not necessarilytrue In fact define119898120583 on119873 = 1 2 3 by

119898120583(119878) =

0 119878 = 0

1 119878 = 0 1 2 3

minus1

2 119878 = 1 2 3

(21)

Then we obtain a supermodular fuzzy measure

120583 (0) = 0 120583 (1) = 120583 (2) = 120583 (3) = 1

120583 (1 2) = 120583 (1 3) = 120583 (2 3) = 3

120583 (1 2 3) =11

2

(22)

4 Fairness Relation Using the AsymmetricChoquet Integral

Definition 20 (Choquet Integral relation) Let 119909 = (1199091

1199092 119909

119899) let 119910 = (119910

1 119910

2 119910

119899) and let 120583 be a fuzzy

measure 119909 is said to CI-dominate 119910 denoted by 119909ge120583119910 if and

only if

(ASC) int (119909 minus 119910) 119889120583 ge 120579 (23)

Assume119873 is a set of all players 119909 and 119910 are distributionsof resources to all players Then 119909 minus 119910 means each degreeof satisfaction of the distribution 119909 compared with thedistribution 119910 Since the more players are satisfied the fairerthe distribution is 119909ge

120583119910 means 119909 is at least as fair as 119910 By

119909gt120583119910 we denote the corresponding strict relation

Proposition 21 Let120583 be a superadditive fuzzymeasure on 2119873Then it is an antisymmetric relation that is 119909ge

120583119910 and 119910ge

120583119909

implies 119909 = 119910 for any functions 119909 119910 on119873

Proposition 22 If 120583 is a supermodular measure on 2119873 thengt120583is a transitive relation that is if 119909ge

120583119910 and 119910ge

120583119911 then 119909ge

120583119911

for any functions 119909 119910 119911 on119873

By Propositions 21 and 22 using the asymmetric Choquetintegral with respect to supermodular functions we obtain afair ordering relation

The proofs for Propositions 21 and 22 will be given in theAppendix of this paper

5 Decision Making by CI-Fairness ANumerical Example

In this section we want to demonstrate how CI-fairnessrelation can be used for decisionmaking Consider a situationwhere resources have to be allocated to 3 agents 119860 119861 and 119862but sets of agents can utilize resources differently (eg givinga commodity to one agent how many items can she producewithin a time unit) Such resource utilization can be assessedby a fuzzy measure

subset 119878 120583 (119878)

0 0

119860 01

119861 03

119862 05

119860 119861 05

119860 119862 05

119861 119862 08

119860 119861 119862 1

(24)

Here for example 119862 appears to be the best in resourceutilization but 119860 is good in collaboration The measure issuperadditive We can also see that this is a supermodularmeasure There are 3 cases to check the other follow fromsubsethood or superadditivity

(1) 120583(119860 119861cup119860 119862)+120583(119860 119861cap119860 119862) = 120583(119860 119861 119862)+120583(119860) = 11 ge 120583(119860 119861) + 120583(119860 119862) = 10

(2) 120583(119860 119861cup 119861 119862)+120583(119860 119861cap 119861 119862) = 120583(119860 119861 119862)+120583(119861) = 13 ge 120583(119860 119861) + 120583(119861 119862) = 13

8 Journal of Applied Mathematics

(1 2 3)

(1 3 2)

(2 3 1) (3 1 2)(2 1 3)

(3 2 1)

Figure 2 CI-Fairness relation among six possible allocations Theconnections include transitivity

(3) 120583(119860 119862cup119861 119862)+120583(119860 119862cap119861 119862) = 120583(119860 119861 119862)+120583(119862) = 15 ge 120583(119860 119862) + 120583(119861 119862) = 13

(1 2 3) ge120583 (1 3 2)

(1 2 3) ge120583 (2 1 3)

(1 2 3) ge120583 (2 3 1)

(1 2 3) ge120583 (3 1 2)

(1 2 3) ge120583 (3 2 1)

(1 3 2) ge120583 (2 3 1)

(1 3 2) ge120583 (3 1 2)

(1 3 2) ge120583 (3 2 1)

(2 1 3) ge120583 (2 3 1)

(2 1 3) ge120583 (3 1 2)

(2 1 3) ge120583 (3 2 1)

(2 3 1) ge120583 (3 2 1)

(3 1 2) ge120583 (3 2 1)

(25)

We find the allocation (1 2 3) in relation to all other alloca-tions but no allocation is in relation to (1 2 3)This is naturalgiven the order of resource utilization weights for the threeagents By rank we can now also order the other 5 allocations

Now we want to allocate three shares of a resource of6 units (1 2 3) among 119860 119861 119862 and compare the 6 possiblepermutations of (1 2 3) by the Choquet integral relationwith120579 = 0 For example to see if allocation 119909 = (1 2 3) is inrelation to the allocation 119910 = (2 3 1) we have to test

(119862) int [119909 minus 119910]+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (26)

In this case [119909 minus 119910]+ = (0 0 2) and [119910 minus 119909]+ = (1 1 0) Sothe first integral becomes 2 times 120583(119862) = 10 and the secondintegral 1 times 120583119889(119860 119861) = (1 minus 120583(119862)) = 05 From 10 ge 05then 119909ge

120583119910 follows

Among all possible 30 pairs (119909 119910) with 119909 = 119910 we can find13 cases in total where the Choquet integral relation holds

Figure 2 shows the Hasse diagram of the relation for the 6possible allocations We can see that as an effect of the bettercollaboration weight of 119860 allocation (2 1 3) is promotedand appears on the same rank as the strong allocation(1 3 2) (strong with regard to single agent utilization ofa resource) and also of better rank than the allocation(2 3 1)mdashexpressing the preference for 119862 as partner of 119860when comparing possible wins and losses in the allocationFinally as expected there is no case to allocate 3 shares to 119860

6 Experiments

In this section we will show how the CI-fairness can beapplied in a practical situation We study wireless channelallocation in the network layer a base station BS that isconnected to the backbone can command over a number119898 of wireless transmission channels and allocates them to anumber 119899 of users (or subscriber stations mobile stationsrelay stations etcmdashdepending on the specific network archi-tecture) for uplink or downlink traffic In the most abstractnotation the BSmakes the allocation based on ameasured orestimated channel state information if it allocates a channel 119894to user 119895 there will be a channel coefficient 119862

119894119895from [0 1] that

represents (in a simplified form) to what degree the user canemploy that channel For example remote users are likely tohave smaller channel coefficients than users that are close tothe BS Here an allocation is seen as feasible if each channelis allocated to one user and to each user at least one channelis assigned

The sum of all channel coefficients of allocated channelsfor a specific user is seen as the performance of the allocationfor that user Then the optimality task is related to theperformance vector for all users This is a direct reference toa relation between performance vectors and various fairnessrelations can be easily and conveniently studied in thiscontext If the number of channels and users is not too largeexhaustive search for maximum sets is possible For examplefor 4 users and 6 channels there are 1560 feasible allocationswhich means about 24 Mio pairwise comparisons

In this model channel allocation ignores low channelinterferences between users and specific channels by therestriction that each channel is allocated to exactly one userHowever if implementing a wireless access where some userscan use the same channel for transmission it might be hardto decide the exact schedule of such multiple assignmentsor to decide if installing such a scheme at all Thereforewe follow a more simplified approach channel interferencewill be represented by a conflict matrix between userswhere low elements for pairs of users indicate the optionto use multichannel assignment Then we use the converseelements of the conflict matrix as masses in the Mobiustransform More specifically the Mobius masses for singleelement sets are set to equal values the masses for two-element are set to 1 minus the corresponding element of theconflictmatrix and all other are set to 0Then themassvectoris normalized to a total of all masses of 1 to ensure that

Journal of Applied Mathematics 9

Table 1 Comparison of total performances for capacity propor-tional and maxmin fairness

Fairness relation Average performanceConflict matrix range 0 to 1

CI 42882Proportional 454661maxmin 414688

Conflict matrix range 09 to 1CI 456051Proportional 451135maxmin 422561

Conflict matrix range 0 to 02CI 409365Proportional 441848maxmin 404879

for the fuzzy measure the measure for the whole set is 1The inverse Mobius transform of such a mass vector gives asupermodularmeasure that can be used to specify a CI-basedfairness relation according to Definition 20

Now this relation is applied to all feasible channelallocationsmdashit means we are not considering a specific use ofmultichannel assignment but we are looking for the influencethat such amultichannel allocation could have on the specificselection of maximal elements Then the focus is on thedisadvantage that some pairs of users can experience if theirfavoured situation (eg to be distant enough to use the samechannel) is not taken into account The similar approach canbe considered for cognitive radio opportunistic networkingP2P vehicular networks and so forth

We might especially look for the price of fairness [1]compared to a standard proportional fair or maxmin fairallocation

Some example results are shown in Table 1 There for thecase of 4 users and 6 cells the maximum sets over all feasibleallocations were computed for CI-fairness using a measure asdescribed above proportional fairness andmaxmin fairnessThe conflictmatrix elements for the CI-fairness were uniformrandomly set within specific ranges All results are averagedover 30 repetitions Note that average sizes of maximum setswere found to be 42 formaxmin fairness 61 for proportionalfairness and 419 for CI-fairness so the CI-fairness produceslarger maximum sets an issue that should be addressed infutureworksThe table now shows average performances overall maximal elements

We see with regard to efficiency the well-known fact thatproportional-fair allocations are in average more efficientthan maxmin fair allocations The CI-fairness appears toselect maximal relations with a total performance betweenproportional fairness and maxmin fairness In case of stronginterferences (where a multichannel assignment is not rea-sonable) it is very close to proportional fairness while in caseof low interference it appears to be more close to maxminfairness that is the fairness relation favouring least elementsThus such an empirical result can be understood in the sensethat the neglecting of the potential multichannel assignment

corresponds with the neglection of strong users This is alsoparticularly appearing for the general case where all conflictmatrix elements were randomly selected between 0 and 1 alsohere the average performance appearsmore close tomaxminfairness than proportional fairness

7 Conclusion

A fairness relation among vectors based on the asymmetricChoquet integral was studied It formally follows the Nashstandard of comparison where the relative losses and winsare replaced by absolute losses and wins but instead takingweights for groups of vector components into account Thusit can represent for example pairs of users that are ina favourable situation regarding resource allocation Theappealing points of this CI-relation are as follows (1) Incase a supermodular measure is used for the integrationthe relation will be transitive which suits faster search formaximal elements (2) it can be parametrized by givinga weight to each subset of vector components (while inpractice this might be relevant only for smaller subsets) Therepresentation of the fuzzy measure by its Mobius transformappears to be a convenient way to yield a supermodularfuzzy measure the masses just need to be all nonnegativeThus the relation can be conveniently applied in many prac-tical applications to specify optimality Slight disadvantagesare with the complexity of the involved calculations andsome empirical evidence for specifying larger maximum setsthan other fairness relations like proportional fairness andmaxmin fairness

In case of wireless channel allocation how CI-fairnesscan be used to help deciding whether implementations ofmore complex allocation schemes are indeed worth the effortwas suggested This method can be applied to many otherproblems of higher efficiency of network resource utilizationIn future work we will consider the use of advanced fuzzyintegrals like balancing and fusion Choquet integral andthe adjustment of related CI-fairness parameters to specificsituations and what can be concluded from such parametersWe will also more intensively study the performance issueswith regard to computational effort and search for maximalelements and study conditions where also greatest elementsexist

Appendix

Proofs of Theorems

In the following we provide the proofs for Propositions 21and 22 For formal convenience of the proofs the notationwill be a little bit changing therefore we repeat the basicdefinitions

We define the Choquet-integral-based (fairness) relationas follows be 119909 and 119910 two vectors from 119877

119899 By [119909]+ weindicate the positive support of a vector 119909 that is the 119894thcomponent of [119909]+ is 119909

119894if 119909

119894ge 0 or 0 if 119909

119894lt 0 The Choquet

integral is denoted by (119862) int119909 119889120583 There 120583 is a fuzzy measure

10 Journal of Applied Mathematics

Also 120583119889 denotes its dual measure (ie 120583119889(119860) = 120583119883minus 120583(119860

119862))

Then 119909 is said to CI-dominate 119910 if and only if

119909ge120583119910 larrrarr (119862)int [119909 minus 119910]

+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (A1)

We can alternatively rewrite this as the asymmetric Choquetintegral

119909ge120583119910 larrrarr (ASC) int (119909 minus 119910) 119889120583 ge 0 (A2)

since the right-hand side condition of (A1) exactly cor-responds with the definition of the asymmetric Choquetintegral

First we can show the following

Lemma 23 If 120583 is a superadditive fuzzy measure where foreach 119860 119861 = 0 and 119860 cap 119861 = 0 120583

119860+ 120583

119861lt 120583

119860cup119861 then for

119860 = 0 120583119889

119860gt 120583

119860

Proof Since 120583 is superadditive for119860 = 0 120583119860+120583

119860119862 lt 120583

119860cup119860119862 =

120583119883 Then 120583

119860lt 120583

119883minus 120583

119860119862 = 120583

119889

119860follows directly

Theorem 24 If 120583 is superadditive then from 119909ge120583119910 and

119910ge120583119909119909 = 119910 follows

Proof 119909ge120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889 If

119909 = 119910 we can use Lemma 23 to obtain

(119862) int [119909 minus 119910]+119889120583

119889gt (119862)int [119909 minus 119910]

+119889120583

ge (119862)int [119910 minus 119909]+119889120583

119889gt (119862)int [119910 minus 119909]

+119889120583

(A3)

that is (119862) int [119910 minus 119909]+119889120583 ge (119862) int [119909 minus 119910]+119889120583

119889 This wouldmean that 119910ge

120583119909 would not hold Therefore it must be 119909 =

119910

For a supermodular measure and nonnegative vectors 119909and 119910 we have the known property (119862) int 119909 119889120583+ (119862) int 119910 119889120583 le(119862) int(119909 + 119910)119889120583 Since the dual measure of a supermodularmeasure is submodular it also holds that (119862) int 119909 119889120583 +(119862) int 119910 119889120583 ge (119862) int(119909 + 119910)119889120583 Using this the following canbe shown

Theorem25 If120583 is supermodular then the relationgt120583is cycle-

free

Proof We will only show the case of nonexistence of a 3-cycle here the concept generalizes directly to any other caseAssume that 119909gt

120583119910 and 119910gt

120583119911 First we note that

[119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+

= [119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+

(A4)

and therefore

(119862) int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

= (119862)int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

(A5)

Using Lemma 23 we yield

(119862) int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

119889

gt (119862)int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

(A6)

and from supermodularity of 120583 and submodularity of 120583119889

(119862) int [119910 minus 119909]+119889120583

119889+ (119862)int [119911 minus 119910]

+119889120583

119889+ (119862)int [119909 minus 119911]

+119889120583

119889

gt (119862)int [119909 minus 119910]+119889120583 + (119862)int [119910 minus 119911]

+119889120583

+ (119862)int [119911 minus 119909]+119889120583

(A7)

However 119909gt120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889

and 119910gt120583119911 means (119862) int [119910 minus 119911]+119889120583 ge (119862) int [119911 minus 119910]

+119889120583

119889Therefore above inequality can only hold if

(119862) int [119911 minus 119909]+119889120583 lt (119862)int [119909 minus 119911]

+119889120583

119889 (A8)

which means 119911 gt120583119909

If the measure 120583 is supermodular also transitivity ofthe relation can be shown Before we can show this weneed to introduce a notation and a corresponding lemmaThe notation refers to computing the (asymmetric) Choquetintegral of 119909 by measure 120583 in a ldquodifferent orderrdquo If 119909

119909(119894)

indicates the 119894th largest element of 119909 then the Choquetintegral is calculated as

(119862) int 119909119889120583 = 119909119909(1)120583119909(1)

+

119899

sum

119894=2

119909119909(119894)[120583

119909(1)119909(2)sdotsdotsdot119909(119894)minus 120583

119909(1)sdotsdotsdot119909(119894minus1)]

(A9)

For any other119910 we can compute the same expression justfollowing the order of the elements of 119910 instead of 119909 Then119909119910(119894)

indicates the element of119909with the index of the 119894th largestelement of 119910 and we define (for convenience we still use thesymbol for the Choquet integral keeping in mind that it isnot a real fuzzy integral anymore)

(119862) int(119910)

119909119889120583 = 119909119910(1)120583119910(1)

+

119899

sum

119894=2

119909119910(119894)[120583

119910(1)119910(2)sdotsdotsdot119910(119894)minus 120583

119910(1)sdotsdotsdot119910(119894minus1)]

(A10)

Journal of Applied Mathematics 11

Consequently we could also write (119862) int(119909)119909119889120583 for the

ldquooriginalrdquo Choquet integral We can do the same for theasymmetric Choquet integral by applying the resorting to thenegative components of the vector 119909 as well and will use thenotation (ASC) int

(119910)119909 119889120583 here Now we have the following

Lemma 26 If 120583 is a supermodular measure then for any119910 ( ASC ) int

(119910)119909 119889120583 ge ( ASC ) int

(119909)119909119889120583

Proof A moment of reasoning gives that any sorting of theindizes can be achieved by suitable application of a sequenceof three kinds of ldquoneighbour swapsrdquo starting from the order119909(1)119909(2) sdot sdot sdot 119909(119899))

Swap 1 In the computation of the integral a larger orequal nonnegative element 119889

119894at index 119894 is swapped with its

immediate nonnegative neighbour 119889119894+1

to the right Thenwhat was computed as 119889

119894(120583

119886119894minus120583

119886) +119889

119894+1(120583

119886119894119894+1minus120583

119886119894) before

the swap changes to 119889119894+1(120583

119886119894+1minus120583

119886) + 119889

119894(120583

119886119894119894+1minus120583

119886119894+1) after

the swap all other parts of the resorted asymmetric Choquetintegral expression remain the same (by 119886 we indicate theindex order before index 119894) Thus the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886119894)

minus 119889119894+1(120583

119886119894119894+1minus 120583

119886119894minus 120583

119886119894+1+ 120583

119886119894)

(A11)

and from 119889119894ge 119889

119894+1and 120583

119886119894119894+1+ 120583

119886ge 120583

119886119894+1+ 120583

119886119894

(supermodularity of 120583) it directly follows that the computedvalue will not decrease by this swap

Swap 2 In the computation of the integral a larger orequal negative element 119889

119894+1at index 119894 + 1 is swapped with

its immediate negative left neighbour 119889119894 From a similar

evaluation as for Swap 1 it can be seen that also here the valueof the computed expression will not decrease after the swap

Swap 3 A nonnegative element at index 119894 is swapped withits immediate negative neighbor to the right with index(119894 + 1) Then before the swap we compute 119889

119894(120583

119886119894minus 120583

119886) minus

119889119894+1(120583

119886119894119894+1minus 120583

119886119894) (119889

119894+1is the absolute value of the negative-

valued neighbour of 119889119894) and after the swap this part of the

expression changes to minus119889119894+1(120583

119886119894+1minus 120583

119886) + 119889

119894(120583

119886119894119894+1minus 120583

119886119894+1)

Then the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886)

minus 119889119894+1(120583

119886119894+1minus 120583

119886minus 120583

119886119894119894+1+ 120583

119886119894)

(A12)

Supermodularity of 120583 gives 120583119886119894+ 120583

119886119894+1le 120583

119886+ 120583

119886119894119894+1and

therefore the factor of 119889119894in former expression is nonnegative

and the factor of 119889119894+1

(which is the negation of the factor of119889119894) is negative or 0Therefore the total change is nonnegative

as wellThus with each of these swap operations we will never

decrease the value of the expression and starting from theasymmetric Choquet integral for 119909 upon reaching the finalorder the final value will be larger or equal

An example might be helpful Assume that the order in 119909is (1 2 3 minus 4 minus 5) (a negative index should indicate that

the corresponding element of 119909 is negative) and in 119910 we havethe order (minus5 3 1 2 minus4) (so 119910

5is the largest component of

119910 and 1199104the smallest) Then the order of swaps is as follows

Swap 1 applied as (1 2 3larr997888rarr

minus4 minus5) gives (1 3 2 minus4 minus5)Swap 1 applied as (1 3

larr997888rarr2 minus4 minus5) gives (3 1 2 minus4 minus5)

Next Swap 2 applied as (3 1 2 minus4 minus 5larr997888997888997888997888rarr

) gives (3 1 2 minus

5 minus 4) Swap 3 applied as (3 1 2 minus 5larr997888997888997888rarr

minus 4) gives (3 1 minus

5 2 minus 4) Swap 3 applied as (3 1 minus 5larr997888997888997888rarr

2 minus 4) gives (3 minus

5 1 2 minus4) and finally Swap 3 applied as (3 minus 5larr997888997888997888rarr

1 2 minus4)

gives the order (minus5 3 1 2 minus 4) of 119910 Note also that in casethe elements of 119909 are not in that order we can always relabelthe indizes and measures correspondingly

This means if we compute (using 119909 =

(1199091 119909

2 119909

3 minus119909

4 minus119909

5))

(ASC) int(119910)

119909119889120583

= minus11990951205835+ 119909

3(120583

35minus 120583

5) + 119909

1(120583

135minus 120583

35)

+ 1199092(120583

1235minus 120583

135) minus 119909

4(120583

12345minus 120583

1235)

(A13)

instead of

(ASC) int 119909 119889120583

= 11990911205831+ 119909

2(120583

12minus 120583

1) + 119909

3(120583

123minus 120583

12)

minus 1199094(120583

1234minus 120583

123) minus 119909

5(120583

12345minus 120583

1234)

(A14)

then (ASC) int(119910)119909 119889120583 ge (ASC) int 119909 119889120583

Using this Lemma we can now easily show the following

Theorem 27 If 120583 is a supermodular measure then ge120583is a

transitive relation

Proof For transitivity we have to show that for any 119909 119910 and119911 such that 119909ge

120583119910 and 119910ge

120583119911 it follows that 119909ge

120583119911 Now 119909ge

120583119910

means (ASC) int(119909 minus 119910)119889120583 ge 0 and 119910ge120583119911 means (ASC) int(119910 minus

119911)119889120583 ge 0 From supermodularity of 120583 and Lemma 26 then italso follows that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 ge (ASC) int (119909 minus 119910) 119889120583 ge 0

(ASC) int(119909minus119911)

(119910 minus 119911) 119889120583 ge (ASC) int (119910 minus 119911) 119889120583 ge 0(A15)

However by using the fact that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 + (ASC) int(119909minus119911)

(119910 minus 119911) 119889120583

= (ASC) int (119909 minus 119911) 119889120583(A16)

we see that also

(ASC) int (119909 minus 119911) 119889120583 ge 0 (A17)

which means 119909ge120583119911

12 Journal of Applied Mathematics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Bertsimas V F Farias and N Trichakis ldquoThe price offairnessrdquo Operations Research vol 59 no 1 pp 17ndash31 2011

[2] J F Nash Jr ldquoThe bargaining problemrdquo Econometrica vol 18pp 155ndash162 1950

[3] E Kalai and M Smorodinsky ldquoOther solutions to Nashrsquosbargaining problemrdquo Econometrica vol 43 pp 513ndash518 1975

[4] J Rawls ATheory of Justice Harvard University Press 1999[5] F Kelly ldquoCharging and rate control for elastic trafficrdquo European

Transactions onTelecommunications vol 8 no 1 pp 33ndash37 1997[6] D P Bertsekas R G Gallager and P Humblet Data Networks

vol 2 Prentice-Hall International 1992[7] H P Young Equity In Theory and Practice Princeton Univer-

sity Press Princeton NJ USA 1995[8] J Mo and J Walrand ldquoFair end-to-end window-based conges-

tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[9] M Alicherry R Bhatia and L E Li ldquoJoint channel assignmentand routing for throughput optimization inmulti-radiowirelessmesh networksrdquo in Proceedings of the 11th Annual InternationalConference on Mobile Computing and Networking (MobiCom05) pp 58ndash72 ACM Cologne Germany August-September2005

[10] A Iyer C Rosenberg and A Karnik ldquoWhat is the rightmodel for wireless channel interferencerdquo IEEE Transactions onWireless Communications vol 8 no 5 pp 2662ndash2671 2009

[11] A Mishra V Shrivastava S Banerjee and W ArbaughldquoPartially overlapped channels not considered harmfulrdquo ACMSIGMETRICS Performance Evaluation Review vol 34 no 1 pp63ndash74 2006

[12] N Nie and C Comaniciu ldquoAdaptive channel allocation spec-trum etiquette for cognitive radio networksrdquo Mobile Networksand Applications vol 11 no 6 pp 779ndash797 2006

[13] C Peng H Zheng and B Y Zhao ldquoUtilization and fairnessin spectrum assignment for opportunistic spectrum accessrdquoMobile Networks and Applications vol 11 no 4 pp 555ndash5762006

[14] X Yang andA P Petropulu ldquoCo-channel interferencemodelingand analysis in a Poisson field of interferers in wireless commu-nicationsrdquo IEEE Transactions on Signal Processing vol 51 no 1pp 64ndash76 2003

[15] M Koppen ldquoRelational optimization and its application frombottleneck flow control to wireless channel allocationrdquo Infor-matica vol 24 no 3 pp 413ndash433 2013

[16] M Sugeno Theory of fuzzy integrals and its applications [PhDthesis] 1974

[17] S Michio ldquoFuzzymeasures and fuzzy integrals a surveyrdquo FuzzyAutomata and Decision Processes vol 78 no 33 pp 89ndash1021977

[18] D Denneberg Non-Additive Measure and Integral vol 27Springer 1994

[19] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[20] J Sipos ldquoIntegral with respect to a pre-measurerdquo MathematicaSlovaca vol 29 no 2 pp 141ndash155 1979

[21] M Grabisch and C Labreuche ldquoThe symmetric and asymmet-ric Choquet integrals on finite spaces for decision makingrdquoStatistical Papers vol 43 no 1 pp 37ndash52 2002

[22] M Grabisch C Labreuche and J-C Vansnick ldquoOn theextension of pseudo-Boolean functions for the aggregation ofinteracting criteriardquo European Journal of Operational Researchvol 148 no 1 pp 28ndash47 2003

[23] R Mesiar A Mesiarova-Zemankova and K Ahmad ldquoDiscreteChoquet integral and some of its symmetric extensionsrdquo FuzzySets and Systems vol 184 pp 148ndash155 2011

[24] A Mesiarova-Zemankova R Mesiar and K Ahmad ldquoThebalancing Choquet integralrdquo Fuzzy Sets and Systems vol 161no 17 pp 2243ndash2255 2010

[25] H Fouchal X Gandibleux and F Lehuede ldquoA lower bound ofthe Choquet integral integrated within Martinsalgorithmrdquo inNewState ofMCDM in the 21st Century vol 648 ofLectureNotesin Economics and Mathematical Systems pp 79ndash89 SpringerHeidelberg Germany 2011

[26] L Galand L Julien and P Patrice ldquoDominance rules forthe choquet integral in multiobjective dynamic programmingrdquoin Proceedings of the 23rd International Joint Conference onArtificial Intelligence pp 538ndash544 AAAI Press 2013

[27] A W Marshall I Olkin and B C Arnold Inequalities TheoryofMajorization and Its Applications Springer 2nd edition 2010

[28] W Ogryczak AWierzbicki andMMilewski ldquoAmulti-criteriaapproach to fair and efficient bandwidth allocationrdquoOmega vol36 no 3 pp 451ndash463 2008

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

important concepts like procedural fairness or interactionfairness

The common theme of all approaches to (distributional)fairness is to divide a set of resources among a numberof agents As already mentioned in the introduction withfar longest cultural history we can find ideas about pro-portionality in the assignment (proportional according todemand ability suitability availability most available goodetc) or the resolution of conflicts about contended goodsin following an equity principle Besides a relation point ofview in order to compare different possible assignmentsthe aspect can be also seen in aggregating the distributionalaspects of all agents and thus keeping the spread of allocationdifferences small This idea becomes strongly related to thetheory of majorization [27] where numerical comparisonbetween different ways of distributing the same total isstrongly related to the Schur-convexity of the used compari-son measureThe application of this approach and its variousgeneralizations have been intensively studied for examplein [28]

Any continuation of a formal approach to fairness (inthe sense of a ldquodefinition of fairnessrdquo) would need thespecification of axioms that represent criteria derived fromspecific ways of how to balancemdashin a just waymdashthe conflictsbetween agents that are in a better and agents that are in aworse situationmdashkeeping inmind that fairness does generallynot stand for the simple requirement of complete even allo-cations Such axioms can refer to impartiality of distributionconsistency between solutions for different domains mathe-matical assumptions about the domain limiting behavioursstrategy-proofness of related procedures or efficiency of thedistribution

Saying this the considered aspect of fairness here is thebalance between users who would gain more and users whowould gain less in an alternative allocation We will useaggregation operators for representing the joint state of allwinning and all loosing agents thus basically following theldquoreciperdquo of proportional fairness But as a new aspect we alsowant to take a different weighting for groups of agents intoaccount Fuzzy integrals are a well-known formal approachfor such a representation

23 Fuzzy Measures and Integrals We now want to recallbasic issues of fuzzy measures and integrals in more detailMost of the material can be gathered together from variouspublications and textbooks but a comprehensive descriptionas it will be given now might be of advantage for theunderstanding of the main proposalmdasha fairness relationbased on the asymmetric Choquet integral for supermodularfuzzy measures

Throughout this paper we consider a finite universal set119873 = 1 2 119899 and 2119873 denotes the power set of119873

Definition 1 (Fuzzy measure [16 17]) A set function 120583

2119873rarr [0infin) is a fuzzy measure if it satisfies the following

conditions(i) 120583(0) = 0 and 120583(119873) lt +infin(ii) for any 119878 119879 isin 2119873 120583(119878) le 120583(119879) whenever 119878 sub 119879

Properties (i) and (ii) express boundedness and mono-tonicity respectively

Definition 2 (superadditivity subadditivity) A fuzzy mea-sure 120583 is superadditive if 120583 satisfies 120583(119878 cup 119879) ge 120583(119878) + 120583(119879)and subadditive if 120583 satisfies 120583(119878 cup 119879) le 120583(119878) + 120583(119879) for any119878 119879 isin 2

119873 satisfying 119878 cap 119879 = 0

Definition 3 (submodularity supermodularity) A fuzzymeasure 120583 is supermodular if 120583 satisfies 120583(119878 cup119879)+120583(119878 cap119879) ge120583(119878)+120583(119879) and submodular if 120583 satisfies 120583(119878cup119879)+120583(119878cap119879) le120583(119878) + 120583(119879) for any 119878 119879 isin 2119873

Definition 4 (dual measure) Let 120583 be a fuzzy measure Thedual measure of 120583 is defined by

119898119889(119878) = 120583 (119873) minus 120583 (119878

119888) (2)

The dual measure of a fuzzy measure is also a fuzzymeasure

Proposition 5 If 120583 is supermodular then 120583 is superadditiveIf 120583 is submodular then 120583 is subadditive

Proof When 120583 is supermodular 120583(119878 cup119879) + 120583(119878 cap119879) ge 120583(119878) +120583(119879) is satisfied for any 119878 119879 isin 2119873 and 119878 cap 119879 = 0 It implies120583(119878 cup 119879) ge 120583(119878) + 120583(119879) for any 119878 119879 isin 2119873 satisfying 119878 cap 119879 = 0The second assertion is obtained in a similar manner

Proposition 6 120583 is supermodular if and only if 120583119889 is sub-modular Similarly 120583 is submodular if and only if 120583119889 issupermodular

Proof Assume that 120583 is supermodularThen we have for any119878 119879 isin 2

119873

120583119889(119878 cup 119879) + 120583

119889(119878 cap 119879)

= 120583 (119873) minus 120583 ((119878 cup 119879)119888) + 120583 (119873) minus 120583 ((119878 cap 119879)

119888)

= 2120583 (119873) minus 120583 (119878119888cap 119879

119888) minus 120583 (119878

119888cup 119879

119888)

le 2120583 (119873) minus 120583 (119878119888) minus 120583 (119879

119888)

= 120583119889(119878) + 120583

119889(119879)

(3)

Following the above in reverse we obtain the converseReplacing le with ge we obtain the second assertion

Proposition 7 If 120583 is a superadditive fuzzy measure then120583119889(119878) ge 120583(119878) for any 119878 isin 2119873

Proof We have 120583(119878) + 120583(119878119888) le 120583(119873) by superadditivity of 120583Hence 120583(119878) le 120583(119873) minus 120583(119878119888) = 120583119889(119878)

Definition 8 (120582-fuzzy measure) A fuzzy measure 120583 is a 120582-fuzzy measure if there exists 120582 gt minus1 such that 120583(119878 cup 119879) =120583(119878)+120583(119879)+120582120583(119878)120583(119879) for any 119878 119879 isin 2119873 satisfying 119878cap119879 = 0and 120583(119873) = 1

Journal of Applied Mathematics 5

Definition 9 (120594-fuzzy measure) A fuzzy measure 120583 on finiteset119873 = 1 2 119899 is a 120594-fuzzy measure if there exists

120594 ge 1 minusmin

119894isin119873120583 (119894)

sum119894isin119873120583 (119894)

(4)

such that

120583 (119878) = 120594|119878|minus1

sum

119894isin119878

120583 (119894) (5)

for any 119878 isin 2119873 Here |119878| denotes the cardinal number of 119878

120582 ge 0 and 120594 ge 1 imply superadditivity and 120582 le 0 and120594 le 1 imply subadditivity The dual measure of a 120582-fuzzymeasure and of a 120594-fuzzy measure are not necessarily a 120582-fuzzy measure and a 120594-fuzzy measure

Proposition 10 Let 120583 be a 120582-fuzzy measure Then 120583 issuperadditive if and only if 120583 is supermodular Similarly 120583 issubadditive if and only if 120583 is submodular

Proof The sufficiencies hold by Proposition 5 We show thenecessities

Assume 120583 is a superadditive 120582-fuzzy measure that is 120582 ge0 Then we have for any 119878 119879 isin 2119873

120583 (119878 cup 119879) + 120583 (119878 cap 119879)

= 120583 (119878) + 120583 (119879 119878) + 120582120583 (119878) 120583 (119879 119878) + 120583 (119878 cap 119879)

ge 120583 (119878) + 120583 (119879 119878) + 120582120583 (119878 cap 119879) 120583 (119879 119878) + 120583 (119878 cap 119879)

= 120583 (119878) + 120583 (119879)

(6)

Replacing le with ge we obtain the second assertion

Definition 11 (Choquet integral [19]) Given a fuzzy measure120583 2

119873rarr [0 1] and a nonnegative function 119891 on 119873 the

Choquet integral of 119891 with respect to 120583 is defined by

(119862) int119891119889120583 = int

+infin

0

120583 (119909119891 (119909) gt 119903) 119889119903 (7)

In the case that119873 is a finite set (7) can be rewritten as

(119862) int119891119889120583 =

119899

sum

119894=1

(119886119894minus 119886

119894minus1) 120583 (119878

119894) (8)

where 119886119894= 119891(119894) 119886

0= 0 119886

1le 119886

2le sdot sdot sdot le 119886

119899 and

119878119894= 119894 119899 (cf Figure 1) Note that this computation will

include a permutation reordering of the 119886119894to ensure that they

are in nondecreasing orderThe Choquet integral can be generalized to general

functions which are not necessarily nonnegative in severalways Here we show the following two generalizations theasymmetric Choquet integral and the symmetric Choquetintegral

Definition 12 (asymmetric Choquet integral [18]) Given afuzzy measure 120583 2119873 rarr [0 1] and a function 119891 on 119873 the

an

anminus1

a3

a2

a1

a0

Sn

S2

S1

Figure 1 Choquet integral

asymmetric Choquet integral of119891with respect to 120583 is definedby

(ASC) int119891119889120583 = int

0

minusinfin

(120583 (119909 | 119891 (119909) gt 119903)

minus119898 (119873)) 119889119903

+int

+infin

0

120583 (119909 | 119891 (119909) gt 119903) 119889119903

(9)

Definition 13 (symmetric Choquet integral [20]) Given afuzzy measure 120583 2119873 rarr [0 1] and a function 119891 on 119873 thesymmetric Choquet integral of 119891 with respect to 120583 is definedby

(SC) int119891119889120583 = minusint+infin

0

120583 (119909 | 119891minus(119909) gt 119903) 119889119903

+ int

+infin

0

120583 (119909 | 119891+(119909) gt 119903) 119889119903

(10)

where 119891+= 119891 or 0 119891

minus= minus(119891 and 0)

Proposition 14 Consider

(ASC) int119891119889120583

= minusint

0

minusinfin

120583119889(119909 | 119891

minus(119909) gt 119903) 119889119903

+int

+infin

0

120583 (119909 | 119891+(119909) gt 119903) 119889119903

(11)

6 Journal of Applied Mathematics

Proof Because

int

0

minusinfin

(120583 (119909 | 119891 (119909) gt 119903) minus 120583 (119873)) 119889119903

= int (120583 (119909 | minus119891minus(119909) gt 119903) minus 120583 (119873)) 119889119903

= minusint (120583 (119909 | 119891minus(119909) le 119903) minus 120583 (119873)) 119889119903

= minusint120583119889(119909 | 119891

minus(119909) gt 119903) 119889119903

(12)

by this proposition we can say that the difference betweenthe asymmetricChoquet integral and the symmetricChoquetintegral is based on which measure are used for integratingthe negative part of 119891

3 Description of Fairness bythe Choquet Integral

In this section we discuss the generalized Choquet integralsin terms of describing fairness by the Choquet integral Let 120583be a fuzzy measure on 2119873 and 119891 = (119909

1 119909

2 119909

119899) a function

on 119873 For example 119873 corresponds to a set consisting of119899 players and 119891 corresponds to a distribution of resourceswhere 119891 can take negative values The more players aresatisfied the fairer the distribution is

Example 15 Let 119899 = 3 and there are 6 resources Then1198911= (2 2 2) is fairer than 119891

2= (4 2 0) and 119891

2is fairer

than 1198913= (6 0 0) To describe this we use the Choquet

integral with respect to a submodular function Suppose that120583(119878) takes the same values according to |119878| (note that in thiscase the Choquet integral becomes an OWA operator andproperties like submodularity depend on the choice of OWAweights) If 120583 is submodular then we have

(119862) int1198911119889120583 ge (119862)int1198912119889120583 ge (119862)int1198913119889120583 (13)

Next we consider the negative distribution

Example 16 Let 119899 = 3 and there are minus6 resources Then 1198911=

(minus2 minus2 minus2) is fairer than1198912= (minus4 minus2 0) and119891

2is fairer than

1198913= (minus6 0 0) Let 120583 be a submodular function If we use the

symmetric Choquet integral then we have

(SC) int1198911119889120583 le (SC) int1198912119889120583 le (SC) int1198913119889120583 (14)

where all integrals take negative values And if 1205831015840 would beanother but superadditive fuzzy measure then we obtain

(SC) int11989111198891205831015840ge (SC) int1198912119889120583

1015840ge (SC) int1198913119889120583

1015840 (15)

which is not what we wanted In other words we cannever describe both positive fairness and negative fairness

by one fuzzy measure using the symmetric Choquet inte-gral On the other hand using the asymmetric Choquetintegral with respect to 120583 where 120583119889 is superadditive weobtain

(ASC) int1198911119889120583 ge (ASC) int1198912119889120583 ge (ASC) int1198913119889120583 (16)

If 120583 is a supermodular fuzzy measure then 120583 is superadditiveand 120583119889 is subadditive Therefore using the asymmetric Cho-quet integral with respect to a supermodular fuzzy measurewe can describe both positive fairness and negative fairnessby one fuzzy measure

According to these examples the asymmetric Choquetintegral with respect to a superadditive fuzzy measure isbetter suited for describing fairness The Mobius transformenables us to construct fuzzy measure easily

Definition 17 (Mobius transform) The Mobius transform of120583 denoted by119898120583

2119873rarr [minus1 1] is defined by

119898120583(119878) = sum

119879sube119878

(minus1)|119878119879|

120583 (119879) (17)

for any 119878 isin 2119873 Conversely we obtain

120583 (119878) = sum

119879sube119878

119898120583(119879) (18)

by the inverse Mobius transform for any 119878 isin 2119873 and there isa one-to-one correspondence between 120583 and119898120583

Note that if the sum of all Mobius masses is 1 then thecorresponding fuzzy measure of the whole set is 1

Proposition 18 If 119898120583(119878) ge 0 for any 119878 isin 2

119873 then 120583 issupermodular

Proof Assume for any 119878 isin 2119873 119898120583(119878) ge 0 Then denoting

that 119903 = 119878 119879 119904 = 119878 cap 119879 119905 = 119879 119878 for 119878 119879 isin 2119873 we have

120583 (119878 cup 119879) + 120583 (119878 cap 119879)

= sum

119862sube119878cup119879

119898120583(119880) + sum

119880sube119878cap119879

119898120583(119880)

= (sum

119880sube119903

+ sum

119880sube119904

+ sum

119880sube119905

+ sum

119880sube119903cup119904119880cap119903 = 0

119880cap119904 = 0

Journal of Applied Mathematics 7

+ sum

119880sube119904cup119905119880cap119904 = 0

119880cap119905 = 0

+ sum

119880sube119903cup119905

119880cap119903 = 0119880cap119905 = 0

+ sum

119880sube119903cup119904cup119905119880cap119903 = 0

119880cap119904 = 0119880cap119905 = 0

)119898120583(119880) + sum

119880sube119904

119898120583(119880)

120583 (119878) + 120583 (119879) = sum

119880sube119878

119898120583(119880) + sum

119880sube119879

119898120583(119880)

= (sum

119880sube119903

+ sum

119880sube119904

+ sum

119880sube119903cup119904119880cap119903 = 0

119880cap119904 = 0

)119898120583(119880)

+ (sum

119880sube119904

+ sum

119880sube119905

+ sum

119880sube119904cup119905119880cap119904 = 0

119880cap119905 = 0

)119898120583(119880)

(19)

so that we obtain

(120583 (119878 cup 119879) + 120583 (119904)) minus (120583 (119878) + 120583 (119879))

= ( sum

119880sube119903cup119905

119880cap119903 = 0119880cap119905 = 0

+ sum

119880sube119903cup119904cup119905119880cap119903 = 0

119880cap119904 = 0119880cap119905 = 0

)119898120583(119880) ge 0

(20)

Remark 19 The converse of Proposition 18 is not necessarilytrue In fact define119898120583 on119873 = 1 2 3 by

119898120583(119878) =

0 119878 = 0

1 119878 = 0 1 2 3

minus1

2 119878 = 1 2 3

(21)

Then we obtain a supermodular fuzzy measure

120583 (0) = 0 120583 (1) = 120583 (2) = 120583 (3) = 1

120583 (1 2) = 120583 (1 3) = 120583 (2 3) = 3

120583 (1 2 3) =11

2

(22)

4 Fairness Relation Using the AsymmetricChoquet Integral

Definition 20 (Choquet Integral relation) Let 119909 = (1199091

1199092 119909

119899) let 119910 = (119910

1 119910

2 119910

119899) and let 120583 be a fuzzy

measure 119909 is said to CI-dominate 119910 denoted by 119909ge120583119910 if and

only if

(ASC) int (119909 minus 119910) 119889120583 ge 120579 (23)

Assume119873 is a set of all players 119909 and 119910 are distributionsof resources to all players Then 119909 minus 119910 means each degreeof satisfaction of the distribution 119909 compared with thedistribution 119910 Since the more players are satisfied the fairerthe distribution is 119909ge

120583119910 means 119909 is at least as fair as 119910 By

119909gt120583119910 we denote the corresponding strict relation

Proposition 21 Let120583 be a superadditive fuzzymeasure on 2119873Then it is an antisymmetric relation that is 119909ge

120583119910 and 119910ge

120583119909

implies 119909 = 119910 for any functions 119909 119910 on119873

Proposition 22 If 120583 is a supermodular measure on 2119873 thengt120583is a transitive relation that is if 119909ge

120583119910 and 119910ge

120583119911 then 119909ge

120583119911

for any functions 119909 119910 119911 on119873

By Propositions 21 and 22 using the asymmetric Choquetintegral with respect to supermodular functions we obtain afair ordering relation

The proofs for Propositions 21 and 22 will be given in theAppendix of this paper

5 Decision Making by CI-Fairness ANumerical Example

In this section we want to demonstrate how CI-fairnessrelation can be used for decisionmaking Consider a situationwhere resources have to be allocated to 3 agents 119860 119861 and 119862but sets of agents can utilize resources differently (eg givinga commodity to one agent how many items can she producewithin a time unit) Such resource utilization can be assessedby a fuzzy measure

subset 119878 120583 (119878)

0 0

119860 01

119861 03

119862 05

119860 119861 05

119860 119862 05

119861 119862 08

119860 119861 119862 1

(24)

Here for example 119862 appears to be the best in resourceutilization but 119860 is good in collaboration The measure issuperadditive We can also see that this is a supermodularmeasure There are 3 cases to check the other follow fromsubsethood or superadditivity

(1) 120583(119860 119861cup119860 119862)+120583(119860 119861cap119860 119862) = 120583(119860 119861 119862)+120583(119860) = 11 ge 120583(119860 119861) + 120583(119860 119862) = 10

(2) 120583(119860 119861cup 119861 119862)+120583(119860 119861cap 119861 119862) = 120583(119860 119861 119862)+120583(119861) = 13 ge 120583(119860 119861) + 120583(119861 119862) = 13

8 Journal of Applied Mathematics

(1 2 3)

(1 3 2)

(2 3 1) (3 1 2)(2 1 3)

(3 2 1)

Figure 2 CI-Fairness relation among six possible allocations Theconnections include transitivity

(3) 120583(119860 119862cup119861 119862)+120583(119860 119862cap119861 119862) = 120583(119860 119861 119862)+120583(119862) = 15 ge 120583(119860 119862) + 120583(119861 119862) = 13

(1 2 3) ge120583 (1 3 2)

(1 2 3) ge120583 (2 1 3)

(1 2 3) ge120583 (2 3 1)

(1 2 3) ge120583 (3 1 2)

(1 2 3) ge120583 (3 2 1)

(1 3 2) ge120583 (2 3 1)

(1 3 2) ge120583 (3 1 2)

(1 3 2) ge120583 (3 2 1)

(2 1 3) ge120583 (2 3 1)

(2 1 3) ge120583 (3 1 2)

(2 1 3) ge120583 (3 2 1)

(2 3 1) ge120583 (3 2 1)

(3 1 2) ge120583 (3 2 1)

(25)

We find the allocation (1 2 3) in relation to all other alloca-tions but no allocation is in relation to (1 2 3)This is naturalgiven the order of resource utilization weights for the threeagents By rank we can now also order the other 5 allocations

Now we want to allocate three shares of a resource of6 units (1 2 3) among 119860 119861 119862 and compare the 6 possiblepermutations of (1 2 3) by the Choquet integral relationwith120579 = 0 For example to see if allocation 119909 = (1 2 3) is inrelation to the allocation 119910 = (2 3 1) we have to test

(119862) int [119909 minus 119910]+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (26)

In this case [119909 minus 119910]+ = (0 0 2) and [119910 minus 119909]+ = (1 1 0) Sothe first integral becomes 2 times 120583(119862) = 10 and the secondintegral 1 times 120583119889(119860 119861) = (1 minus 120583(119862)) = 05 From 10 ge 05then 119909ge

120583119910 follows

Among all possible 30 pairs (119909 119910) with 119909 = 119910 we can find13 cases in total where the Choquet integral relation holds

Figure 2 shows the Hasse diagram of the relation for the 6possible allocations We can see that as an effect of the bettercollaboration weight of 119860 allocation (2 1 3) is promotedand appears on the same rank as the strong allocation(1 3 2) (strong with regard to single agent utilization ofa resource) and also of better rank than the allocation(2 3 1)mdashexpressing the preference for 119862 as partner of 119860when comparing possible wins and losses in the allocationFinally as expected there is no case to allocate 3 shares to 119860

6 Experiments

In this section we will show how the CI-fairness can beapplied in a practical situation We study wireless channelallocation in the network layer a base station BS that isconnected to the backbone can command over a number119898 of wireless transmission channels and allocates them to anumber 119899 of users (or subscriber stations mobile stationsrelay stations etcmdashdepending on the specific network archi-tecture) for uplink or downlink traffic In the most abstractnotation the BSmakes the allocation based on ameasured orestimated channel state information if it allocates a channel 119894to user 119895 there will be a channel coefficient 119862

119894119895from [0 1] that

represents (in a simplified form) to what degree the user canemploy that channel For example remote users are likely tohave smaller channel coefficients than users that are close tothe BS Here an allocation is seen as feasible if each channelis allocated to one user and to each user at least one channelis assigned

The sum of all channel coefficients of allocated channelsfor a specific user is seen as the performance of the allocationfor that user Then the optimality task is related to theperformance vector for all users This is a direct reference toa relation between performance vectors and various fairnessrelations can be easily and conveniently studied in thiscontext If the number of channels and users is not too largeexhaustive search for maximum sets is possible For examplefor 4 users and 6 channels there are 1560 feasible allocationswhich means about 24 Mio pairwise comparisons

In this model channel allocation ignores low channelinterferences between users and specific channels by therestriction that each channel is allocated to exactly one userHowever if implementing a wireless access where some userscan use the same channel for transmission it might be hardto decide the exact schedule of such multiple assignmentsor to decide if installing such a scheme at all Thereforewe follow a more simplified approach channel interferencewill be represented by a conflict matrix between userswhere low elements for pairs of users indicate the optionto use multichannel assignment Then we use the converseelements of the conflict matrix as masses in the Mobiustransform More specifically the Mobius masses for singleelement sets are set to equal values the masses for two-element are set to 1 minus the corresponding element of theconflictmatrix and all other are set to 0Then themassvectoris normalized to a total of all masses of 1 to ensure that

Journal of Applied Mathematics 9

Table 1 Comparison of total performances for capacity propor-tional and maxmin fairness

Fairness relation Average performanceConflict matrix range 0 to 1

CI 42882Proportional 454661maxmin 414688

Conflict matrix range 09 to 1CI 456051Proportional 451135maxmin 422561

Conflict matrix range 0 to 02CI 409365Proportional 441848maxmin 404879

for the fuzzy measure the measure for the whole set is 1The inverse Mobius transform of such a mass vector gives asupermodularmeasure that can be used to specify a CI-basedfairness relation according to Definition 20

Now this relation is applied to all feasible channelallocationsmdashit means we are not considering a specific use ofmultichannel assignment but we are looking for the influencethat such amultichannel allocation could have on the specificselection of maximal elements Then the focus is on thedisadvantage that some pairs of users can experience if theirfavoured situation (eg to be distant enough to use the samechannel) is not taken into account The similar approach canbe considered for cognitive radio opportunistic networkingP2P vehicular networks and so forth

We might especially look for the price of fairness [1]compared to a standard proportional fair or maxmin fairallocation

Some example results are shown in Table 1 There for thecase of 4 users and 6 cells the maximum sets over all feasibleallocations were computed for CI-fairness using a measure asdescribed above proportional fairness andmaxmin fairnessThe conflictmatrix elements for the CI-fairness were uniformrandomly set within specific ranges All results are averagedover 30 repetitions Note that average sizes of maximum setswere found to be 42 formaxmin fairness 61 for proportionalfairness and 419 for CI-fairness so the CI-fairness produceslarger maximum sets an issue that should be addressed infutureworksThe table now shows average performances overall maximal elements

We see with regard to efficiency the well-known fact thatproportional-fair allocations are in average more efficientthan maxmin fair allocations The CI-fairness appears toselect maximal relations with a total performance betweenproportional fairness and maxmin fairness In case of stronginterferences (where a multichannel assignment is not rea-sonable) it is very close to proportional fairness while in caseof low interference it appears to be more close to maxminfairness that is the fairness relation favouring least elementsThus such an empirical result can be understood in the sensethat the neglecting of the potential multichannel assignment

corresponds with the neglection of strong users This is alsoparticularly appearing for the general case where all conflictmatrix elements were randomly selected between 0 and 1 alsohere the average performance appearsmore close tomaxminfairness than proportional fairness

7 Conclusion

A fairness relation among vectors based on the asymmetricChoquet integral was studied It formally follows the Nashstandard of comparison where the relative losses and winsare replaced by absolute losses and wins but instead takingweights for groups of vector components into account Thusit can represent for example pairs of users that are ina favourable situation regarding resource allocation Theappealing points of this CI-relation are as follows (1) Incase a supermodular measure is used for the integrationthe relation will be transitive which suits faster search formaximal elements (2) it can be parametrized by givinga weight to each subset of vector components (while inpractice this might be relevant only for smaller subsets) Therepresentation of the fuzzy measure by its Mobius transformappears to be a convenient way to yield a supermodularfuzzy measure the masses just need to be all nonnegativeThus the relation can be conveniently applied in many prac-tical applications to specify optimality Slight disadvantagesare with the complexity of the involved calculations andsome empirical evidence for specifying larger maximum setsthan other fairness relations like proportional fairness andmaxmin fairness

In case of wireless channel allocation how CI-fairnesscan be used to help deciding whether implementations ofmore complex allocation schemes are indeed worth the effortwas suggested This method can be applied to many otherproblems of higher efficiency of network resource utilizationIn future work we will consider the use of advanced fuzzyintegrals like balancing and fusion Choquet integral andthe adjustment of related CI-fairness parameters to specificsituations and what can be concluded from such parametersWe will also more intensively study the performance issueswith regard to computational effort and search for maximalelements and study conditions where also greatest elementsexist

Appendix

Proofs of Theorems

In the following we provide the proofs for Propositions 21and 22 For formal convenience of the proofs the notationwill be a little bit changing therefore we repeat the basicdefinitions

We define the Choquet-integral-based (fairness) relationas follows be 119909 and 119910 two vectors from 119877

119899 By [119909]+ weindicate the positive support of a vector 119909 that is the 119894thcomponent of [119909]+ is 119909

119894if 119909

119894ge 0 or 0 if 119909

119894lt 0 The Choquet

integral is denoted by (119862) int119909 119889120583 There 120583 is a fuzzy measure

10 Journal of Applied Mathematics

Also 120583119889 denotes its dual measure (ie 120583119889(119860) = 120583119883minus 120583(119860

119862))

Then 119909 is said to CI-dominate 119910 if and only if

119909ge120583119910 larrrarr (119862)int [119909 minus 119910]

+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (A1)

We can alternatively rewrite this as the asymmetric Choquetintegral

119909ge120583119910 larrrarr (ASC) int (119909 minus 119910) 119889120583 ge 0 (A2)

since the right-hand side condition of (A1) exactly cor-responds with the definition of the asymmetric Choquetintegral

First we can show the following

Lemma 23 If 120583 is a superadditive fuzzy measure where foreach 119860 119861 = 0 and 119860 cap 119861 = 0 120583

119860+ 120583

119861lt 120583

119860cup119861 then for

119860 = 0 120583119889

119860gt 120583

119860

Proof Since 120583 is superadditive for119860 = 0 120583119860+120583

119860119862 lt 120583

119860cup119860119862 =

120583119883 Then 120583

119860lt 120583

119883minus 120583

119860119862 = 120583

119889

119860follows directly

Theorem 24 If 120583 is superadditive then from 119909ge120583119910 and

119910ge120583119909119909 = 119910 follows

Proof 119909ge120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889 If

119909 = 119910 we can use Lemma 23 to obtain

(119862) int [119909 minus 119910]+119889120583

119889gt (119862)int [119909 minus 119910]

+119889120583

ge (119862)int [119910 minus 119909]+119889120583

119889gt (119862)int [119910 minus 119909]

+119889120583

(A3)

that is (119862) int [119910 minus 119909]+119889120583 ge (119862) int [119909 minus 119910]+119889120583

119889 This wouldmean that 119910ge

120583119909 would not hold Therefore it must be 119909 =

119910

For a supermodular measure and nonnegative vectors 119909and 119910 we have the known property (119862) int 119909 119889120583+ (119862) int 119910 119889120583 le(119862) int(119909 + 119910)119889120583 Since the dual measure of a supermodularmeasure is submodular it also holds that (119862) int 119909 119889120583 +(119862) int 119910 119889120583 ge (119862) int(119909 + 119910)119889120583 Using this the following canbe shown

Theorem25 If120583 is supermodular then the relationgt120583is cycle-

free

Proof We will only show the case of nonexistence of a 3-cycle here the concept generalizes directly to any other caseAssume that 119909gt

120583119910 and 119910gt

120583119911 First we note that

[119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+

= [119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+

(A4)

and therefore

(119862) int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

= (119862)int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

(A5)

Using Lemma 23 we yield

(119862) int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

119889

gt (119862)int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

(A6)

and from supermodularity of 120583 and submodularity of 120583119889

(119862) int [119910 minus 119909]+119889120583

119889+ (119862)int [119911 minus 119910]

+119889120583

119889+ (119862)int [119909 minus 119911]

+119889120583

119889

gt (119862)int [119909 minus 119910]+119889120583 + (119862)int [119910 minus 119911]

+119889120583

+ (119862)int [119911 minus 119909]+119889120583

(A7)

However 119909gt120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889

and 119910gt120583119911 means (119862) int [119910 minus 119911]+119889120583 ge (119862) int [119911 minus 119910]

+119889120583

119889Therefore above inequality can only hold if

(119862) int [119911 minus 119909]+119889120583 lt (119862)int [119909 minus 119911]

+119889120583

119889 (A8)

which means 119911 gt120583119909

If the measure 120583 is supermodular also transitivity ofthe relation can be shown Before we can show this weneed to introduce a notation and a corresponding lemmaThe notation refers to computing the (asymmetric) Choquetintegral of 119909 by measure 120583 in a ldquodifferent orderrdquo If 119909

119909(119894)

indicates the 119894th largest element of 119909 then the Choquetintegral is calculated as

(119862) int 119909119889120583 = 119909119909(1)120583119909(1)

+

119899

sum

119894=2

119909119909(119894)[120583

119909(1)119909(2)sdotsdotsdot119909(119894)minus 120583

119909(1)sdotsdotsdot119909(119894minus1)]

(A9)

For any other119910 we can compute the same expression justfollowing the order of the elements of 119910 instead of 119909 Then119909119910(119894)

indicates the element of119909with the index of the 119894th largestelement of 119910 and we define (for convenience we still use thesymbol for the Choquet integral keeping in mind that it isnot a real fuzzy integral anymore)

(119862) int(119910)

119909119889120583 = 119909119910(1)120583119910(1)

+

119899

sum

119894=2

119909119910(119894)[120583

119910(1)119910(2)sdotsdotsdot119910(119894)minus 120583

119910(1)sdotsdotsdot119910(119894minus1)]

(A10)

Journal of Applied Mathematics 11

Consequently we could also write (119862) int(119909)119909119889120583 for the

ldquooriginalrdquo Choquet integral We can do the same for theasymmetric Choquet integral by applying the resorting to thenegative components of the vector 119909 as well and will use thenotation (ASC) int

(119910)119909 119889120583 here Now we have the following

Lemma 26 If 120583 is a supermodular measure then for any119910 ( ASC ) int

(119910)119909 119889120583 ge ( ASC ) int

(119909)119909119889120583

Proof A moment of reasoning gives that any sorting of theindizes can be achieved by suitable application of a sequenceof three kinds of ldquoneighbour swapsrdquo starting from the order119909(1)119909(2) sdot sdot sdot 119909(119899))

Swap 1 In the computation of the integral a larger orequal nonnegative element 119889

119894at index 119894 is swapped with its

immediate nonnegative neighbour 119889119894+1

to the right Thenwhat was computed as 119889

119894(120583

119886119894minus120583

119886) +119889

119894+1(120583

119886119894119894+1minus120583

119886119894) before

the swap changes to 119889119894+1(120583

119886119894+1minus120583

119886) + 119889

119894(120583

119886119894119894+1minus120583

119886119894+1) after

the swap all other parts of the resorted asymmetric Choquetintegral expression remain the same (by 119886 we indicate theindex order before index 119894) Thus the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886119894)

minus 119889119894+1(120583

119886119894119894+1minus 120583

119886119894minus 120583

119886119894+1+ 120583

119886119894)

(A11)

and from 119889119894ge 119889

119894+1and 120583

119886119894119894+1+ 120583

119886ge 120583

119886119894+1+ 120583

119886119894

(supermodularity of 120583) it directly follows that the computedvalue will not decrease by this swap

Swap 2 In the computation of the integral a larger orequal negative element 119889

119894+1at index 119894 + 1 is swapped with

its immediate negative left neighbour 119889119894 From a similar

evaluation as for Swap 1 it can be seen that also here the valueof the computed expression will not decrease after the swap

Swap 3 A nonnegative element at index 119894 is swapped withits immediate negative neighbor to the right with index(119894 + 1) Then before the swap we compute 119889

119894(120583

119886119894minus 120583

119886) minus

119889119894+1(120583

119886119894119894+1minus 120583

119886119894) (119889

119894+1is the absolute value of the negative-

valued neighbour of 119889119894) and after the swap this part of the

expression changes to minus119889119894+1(120583

119886119894+1minus 120583

119886) + 119889

119894(120583

119886119894119894+1minus 120583

119886119894+1)

Then the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886)

minus 119889119894+1(120583

119886119894+1minus 120583

119886minus 120583

119886119894119894+1+ 120583

119886119894)

(A12)

Supermodularity of 120583 gives 120583119886119894+ 120583

119886119894+1le 120583

119886+ 120583

119886119894119894+1and

therefore the factor of 119889119894in former expression is nonnegative

and the factor of 119889119894+1

(which is the negation of the factor of119889119894) is negative or 0Therefore the total change is nonnegative

as wellThus with each of these swap operations we will never

decrease the value of the expression and starting from theasymmetric Choquet integral for 119909 upon reaching the finalorder the final value will be larger or equal

An example might be helpful Assume that the order in 119909is (1 2 3 minus 4 minus 5) (a negative index should indicate that

the corresponding element of 119909 is negative) and in 119910 we havethe order (minus5 3 1 2 minus4) (so 119910

5is the largest component of

119910 and 1199104the smallest) Then the order of swaps is as follows

Swap 1 applied as (1 2 3larr997888rarr

minus4 minus5) gives (1 3 2 minus4 minus5)Swap 1 applied as (1 3

larr997888rarr2 minus4 minus5) gives (3 1 2 minus4 minus5)

Next Swap 2 applied as (3 1 2 minus4 minus 5larr997888997888997888997888rarr

) gives (3 1 2 minus

5 minus 4) Swap 3 applied as (3 1 2 minus 5larr997888997888997888rarr

minus 4) gives (3 1 minus

5 2 minus 4) Swap 3 applied as (3 1 minus 5larr997888997888997888rarr

2 minus 4) gives (3 minus

5 1 2 minus4) and finally Swap 3 applied as (3 minus 5larr997888997888997888rarr

1 2 minus4)

gives the order (minus5 3 1 2 minus 4) of 119910 Note also that in casethe elements of 119909 are not in that order we can always relabelthe indizes and measures correspondingly

This means if we compute (using 119909 =

(1199091 119909

2 119909

3 minus119909

4 minus119909

5))

(ASC) int(119910)

119909119889120583

= minus11990951205835+ 119909

3(120583

35minus 120583

5) + 119909

1(120583

135minus 120583

35)

+ 1199092(120583

1235minus 120583

135) minus 119909

4(120583

12345minus 120583

1235)

(A13)

instead of

(ASC) int 119909 119889120583

= 11990911205831+ 119909

2(120583

12minus 120583

1) + 119909

3(120583

123minus 120583

12)

minus 1199094(120583

1234minus 120583

123) minus 119909

5(120583

12345minus 120583

1234)

(A14)

then (ASC) int(119910)119909 119889120583 ge (ASC) int 119909 119889120583

Using this Lemma we can now easily show the following

Theorem 27 If 120583 is a supermodular measure then ge120583is a

transitive relation

Proof For transitivity we have to show that for any 119909 119910 and119911 such that 119909ge

120583119910 and 119910ge

120583119911 it follows that 119909ge

120583119911 Now 119909ge

120583119910

means (ASC) int(119909 minus 119910)119889120583 ge 0 and 119910ge120583119911 means (ASC) int(119910 minus

119911)119889120583 ge 0 From supermodularity of 120583 and Lemma 26 then italso follows that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 ge (ASC) int (119909 minus 119910) 119889120583 ge 0

(ASC) int(119909minus119911)

(119910 minus 119911) 119889120583 ge (ASC) int (119910 minus 119911) 119889120583 ge 0(A15)

However by using the fact that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 + (ASC) int(119909minus119911)

(119910 minus 119911) 119889120583

= (ASC) int (119909 minus 119911) 119889120583(A16)

we see that also

(ASC) int (119909 minus 119911) 119889120583 ge 0 (A17)

which means 119909ge120583119911

12 Journal of Applied Mathematics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Bertsimas V F Farias and N Trichakis ldquoThe price offairnessrdquo Operations Research vol 59 no 1 pp 17ndash31 2011

[2] J F Nash Jr ldquoThe bargaining problemrdquo Econometrica vol 18pp 155ndash162 1950

[3] E Kalai and M Smorodinsky ldquoOther solutions to Nashrsquosbargaining problemrdquo Econometrica vol 43 pp 513ndash518 1975

[4] J Rawls ATheory of Justice Harvard University Press 1999[5] F Kelly ldquoCharging and rate control for elastic trafficrdquo European

Transactions onTelecommunications vol 8 no 1 pp 33ndash37 1997[6] D P Bertsekas R G Gallager and P Humblet Data Networks

vol 2 Prentice-Hall International 1992[7] H P Young Equity In Theory and Practice Princeton Univer-

sity Press Princeton NJ USA 1995[8] J Mo and J Walrand ldquoFair end-to-end window-based conges-

tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[9] M Alicherry R Bhatia and L E Li ldquoJoint channel assignmentand routing for throughput optimization inmulti-radiowirelessmesh networksrdquo in Proceedings of the 11th Annual InternationalConference on Mobile Computing and Networking (MobiCom05) pp 58ndash72 ACM Cologne Germany August-September2005

[10] A Iyer C Rosenberg and A Karnik ldquoWhat is the rightmodel for wireless channel interferencerdquo IEEE Transactions onWireless Communications vol 8 no 5 pp 2662ndash2671 2009

[11] A Mishra V Shrivastava S Banerjee and W ArbaughldquoPartially overlapped channels not considered harmfulrdquo ACMSIGMETRICS Performance Evaluation Review vol 34 no 1 pp63ndash74 2006

[12] N Nie and C Comaniciu ldquoAdaptive channel allocation spec-trum etiquette for cognitive radio networksrdquo Mobile Networksand Applications vol 11 no 6 pp 779ndash797 2006

[13] C Peng H Zheng and B Y Zhao ldquoUtilization and fairnessin spectrum assignment for opportunistic spectrum accessrdquoMobile Networks and Applications vol 11 no 4 pp 555ndash5762006

[14] X Yang andA P Petropulu ldquoCo-channel interferencemodelingand analysis in a Poisson field of interferers in wireless commu-nicationsrdquo IEEE Transactions on Signal Processing vol 51 no 1pp 64ndash76 2003

[15] M Koppen ldquoRelational optimization and its application frombottleneck flow control to wireless channel allocationrdquo Infor-matica vol 24 no 3 pp 413ndash433 2013

[16] M Sugeno Theory of fuzzy integrals and its applications [PhDthesis] 1974

[17] S Michio ldquoFuzzymeasures and fuzzy integrals a surveyrdquo FuzzyAutomata and Decision Processes vol 78 no 33 pp 89ndash1021977

[18] D Denneberg Non-Additive Measure and Integral vol 27Springer 1994

[19] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[20] J Sipos ldquoIntegral with respect to a pre-measurerdquo MathematicaSlovaca vol 29 no 2 pp 141ndash155 1979

[21] M Grabisch and C Labreuche ldquoThe symmetric and asymmet-ric Choquet integrals on finite spaces for decision makingrdquoStatistical Papers vol 43 no 1 pp 37ndash52 2002

[22] M Grabisch C Labreuche and J-C Vansnick ldquoOn theextension of pseudo-Boolean functions for the aggregation ofinteracting criteriardquo European Journal of Operational Researchvol 148 no 1 pp 28ndash47 2003

[23] R Mesiar A Mesiarova-Zemankova and K Ahmad ldquoDiscreteChoquet integral and some of its symmetric extensionsrdquo FuzzySets and Systems vol 184 pp 148ndash155 2011

[24] A Mesiarova-Zemankova R Mesiar and K Ahmad ldquoThebalancing Choquet integralrdquo Fuzzy Sets and Systems vol 161no 17 pp 2243ndash2255 2010

[25] H Fouchal X Gandibleux and F Lehuede ldquoA lower bound ofthe Choquet integral integrated within Martinsalgorithmrdquo inNewState ofMCDM in the 21st Century vol 648 ofLectureNotesin Economics and Mathematical Systems pp 79ndash89 SpringerHeidelberg Germany 2011

[26] L Galand L Julien and P Patrice ldquoDominance rules forthe choquet integral in multiobjective dynamic programmingrdquoin Proceedings of the 23rd International Joint Conference onArtificial Intelligence pp 538ndash544 AAAI Press 2013

[27] A W Marshall I Olkin and B C Arnold Inequalities TheoryofMajorization and Its Applications Springer 2nd edition 2010

[28] W Ogryczak AWierzbicki andMMilewski ldquoAmulti-criteriaapproach to fair and efficient bandwidth allocationrdquoOmega vol36 no 3 pp 451ndash463 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

Definition 9 (120594-fuzzy measure) A fuzzy measure 120583 on finiteset119873 = 1 2 119899 is a 120594-fuzzy measure if there exists

120594 ge 1 minusmin

119894isin119873120583 (119894)

sum119894isin119873120583 (119894)

(4)

such that

120583 (119878) = 120594|119878|minus1

sum

119894isin119878

120583 (119894) (5)

for any 119878 isin 2119873 Here |119878| denotes the cardinal number of 119878

120582 ge 0 and 120594 ge 1 imply superadditivity and 120582 le 0 and120594 le 1 imply subadditivity The dual measure of a 120582-fuzzymeasure and of a 120594-fuzzy measure are not necessarily a 120582-fuzzy measure and a 120594-fuzzy measure

Proposition 10 Let 120583 be a 120582-fuzzy measure Then 120583 issuperadditive if and only if 120583 is supermodular Similarly 120583 issubadditive if and only if 120583 is submodular

Proof The sufficiencies hold by Proposition 5 We show thenecessities

Assume 120583 is a superadditive 120582-fuzzy measure that is 120582 ge0 Then we have for any 119878 119879 isin 2119873

120583 (119878 cup 119879) + 120583 (119878 cap 119879)

= 120583 (119878) + 120583 (119879 119878) + 120582120583 (119878) 120583 (119879 119878) + 120583 (119878 cap 119879)

ge 120583 (119878) + 120583 (119879 119878) + 120582120583 (119878 cap 119879) 120583 (119879 119878) + 120583 (119878 cap 119879)

= 120583 (119878) + 120583 (119879)

(6)

Replacing le with ge we obtain the second assertion

Definition 11 (Choquet integral [19]) Given a fuzzy measure120583 2

119873rarr [0 1] and a nonnegative function 119891 on 119873 the

Choquet integral of 119891 with respect to 120583 is defined by

(119862) int119891119889120583 = int

+infin

0

120583 (119909119891 (119909) gt 119903) 119889119903 (7)

In the case that119873 is a finite set (7) can be rewritten as

(119862) int119891119889120583 =

119899

sum

119894=1

(119886119894minus 119886

119894minus1) 120583 (119878

119894) (8)

where 119886119894= 119891(119894) 119886

0= 0 119886

1le 119886

2le sdot sdot sdot le 119886

119899 and

119878119894= 119894 119899 (cf Figure 1) Note that this computation will

include a permutation reordering of the 119886119894to ensure that they

are in nondecreasing orderThe Choquet integral can be generalized to general

functions which are not necessarily nonnegative in severalways Here we show the following two generalizations theasymmetric Choquet integral and the symmetric Choquetintegral

Definition 12 (asymmetric Choquet integral [18]) Given afuzzy measure 120583 2119873 rarr [0 1] and a function 119891 on 119873 the

an

anminus1

a3

a2

a1

a0

Sn

S2

S1

Figure 1 Choquet integral

asymmetric Choquet integral of119891with respect to 120583 is definedby

(ASC) int119891119889120583 = int

0

minusinfin

(120583 (119909 | 119891 (119909) gt 119903)

minus119898 (119873)) 119889119903

+int

+infin

0

120583 (119909 | 119891 (119909) gt 119903) 119889119903

(9)

Definition 13 (symmetric Choquet integral [20]) Given afuzzy measure 120583 2119873 rarr [0 1] and a function 119891 on 119873 thesymmetric Choquet integral of 119891 with respect to 120583 is definedby

(SC) int119891119889120583 = minusint+infin

0

120583 (119909 | 119891minus(119909) gt 119903) 119889119903

+ int

+infin

0

120583 (119909 | 119891+(119909) gt 119903) 119889119903

(10)

where 119891+= 119891 or 0 119891

minus= minus(119891 and 0)

Proposition 14 Consider

(ASC) int119891119889120583

= minusint

0

minusinfin

120583119889(119909 | 119891

minus(119909) gt 119903) 119889119903

+int

+infin

0

120583 (119909 | 119891+(119909) gt 119903) 119889119903

(11)

6 Journal of Applied Mathematics

Proof Because

int

0

minusinfin

(120583 (119909 | 119891 (119909) gt 119903) minus 120583 (119873)) 119889119903

= int (120583 (119909 | minus119891minus(119909) gt 119903) minus 120583 (119873)) 119889119903

= minusint (120583 (119909 | 119891minus(119909) le 119903) minus 120583 (119873)) 119889119903

= minusint120583119889(119909 | 119891

minus(119909) gt 119903) 119889119903

(12)

by this proposition we can say that the difference betweenthe asymmetricChoquet integral and the symmetricChoquetintegral is based on which measure are used for integratingthe negative part of 119891

3 Description of Fairness bythe Choquet Integral

In this section we discuss the generalized Choquet integralsin terms of describing fairness by the Choquet integral Let 120583be a fuzzy measure on 2119873 and 119891 = (119909

1 119909

2 119909

119899) a function

on 119873 For example 119873 corresponds to a set consisting of119899 players and 119891 corresponds to a distribution of resourceswhere 119891 can take negative values The more players aresatisfied the fairer the distribution is

Example 15 Let 119899 = 3 and there are 6 resources Then1198911= (2 2 2) is fairer than 119891

2= (4 2 0) and 119891

2is fairer

than 1198913= (6 0 0) To describe this we use the Choquet

integral with respect to a submodular function Suppose that120583(119878) takes the same values according to |119878| (note that in thiscase the Choquet integral becomes an OWA operator andproperties like submodularity depend on the choice of OWAweights) If 120583 is submodular then we have

(119862) int1198911119889120583 ge (119862)int1198912119889120583 ge (119862)int1198913119889120583 (13)

Next we consider the negative distribution

Example 16 Let 119899 = 3 and there are minus6 resources Then 1198911=

(minus2 minus2 minus2) is fairer than1198912= (minus4 minus2 0) and119891

2is fairer than

1198913= (minus6 0 0) Let 120583 be a submodular function If we use the

symmetric Choquet integral then we have

(SC) int1198911119889120583 le (SC) int1198912119889120583 le (SC) int1198913119889120583 (14)

where all integrals take negative values And if 1205831015840 would beanother but superadditive fuzzy measure then we obtain

(SC) int11989111198891205831015840ge (SC) int1198912119889120583

1015840ge (SC) int1198913119889120583

1015840 (15)

which is not what we wanted In other words we cannever describe both positive fairness and negative fairness

by one fuzzy measure using the symmetric Choquet inte-gral On the other hand using the asymmetric Choquetintegral with respect to 120583 where 120583119889 is superadditive weobtain

(ASC) int1198911119889120583 ge (ASC) int1198912119889120583 ge (ASC) int1198913119889120583 (16)

If 120583 is a supermodular fuzzy measure then 120583 is superadditiveand 120583119889 is subadditive Therefore using the asymmetric Cho-quet integral with respect to a supermodular fuzzy measurewe can describe both positive fairness and negative fairnessby one fuzzy measure

According to these examples the asymmetric Choquetintegral with respect to a superadditive fuzzy measure isbetter suited for describing fairness The Mobius transformenables us to construct fuzzy measure easily

Definition 17 (Mobius transform) The Mobius transform of120583 denoted by119898120583

2119873rarr [minus1 1] is defined by

119898120583(119878) = sum

119879sube119878

(minus1)|119878119879|

120583 (119879) (17)

for any 119878 isin 2119873 Conversely we obtain

120583 (119878) = sum

119879sube119878

119898120583(119879) (18)

by the inverse Mobius transform for any 119878 isin 2119873 and there isa one-to-one correspondence between 120583 and119898120583

Note that if the sum of all Mobius masses is 1 then thecorresponding fuzzy measure of the whole set is 1

Proposition 18 If 119898120583(119878) ge 0 for any 119878 isin 2

119873 then 120583 issupermodular

Proof Assume for any 119878 isin 2119873 119898120583(119878) ge 0 Then denoting

that 119903 = 119878 119879 119904 = 119878 cap 119879 119905 = 119879 119878 for 119878 119879 isin 2119873 we have

120583 (119878 cup 119879) + 120583 (119878 cap 119879)

= sum

119862sube119878cup119879

119898120583(119880) + sum

119880sube119878cap119879

119898120583(119880)

= (sum

119880sube119903

+ sum

119880sube119904

+ sum

119880sube119905

+ sum

119880sube119903cup119904119880cap119903 = 0

119880cap119904 = 0

Journal of Applied Mathematics 7

+ sum

119880sube119904cup119905119880cap119904 = 0

119880cap119905 = 0

+ sum

119880sube119903cup119905

119880cap119903 = 0119880cap119905 = 0

+ sum

119880sube119903cup119904cup119905119880cap119903 = 0

119880cap119904 = 0119880cap119905 = 0

)119898120583(119880) + sum

119880sube119904

119898120583(119880)

120583 (119878) + 120583 (119879) = sum

119880sube119878

119898120583(119880) + sum

119880sube119879

119898120583(119880)

= (sum

119880sube119903

+ sum

119880sube119904

+ sum

119880sube119903cup119904119880cap119903 = 0

119880cap119904 = 0

)119898120583(119880)

+ (sum

119880sube119904

+ sum

119880sube119905

+ sum

119880sube119904cup119905119880cap119904 = 0

119880cap119905 = 0

)119898120583(119880)

(19)

so that we obtain

(120583 (119878 cup 119879) + 120583 (119904)) minus (120583 (119878) + 120583 (119879))

= ( sum

119880sube119903cup119905

119880cap119903 = 0119880cap119905 = 0

+ sum

119880sube119903cup119904cup119905119880cap119903 = 0

119880cap119904 = 0119880cap119905 = 0

)119898120583(119880) ge 0

(20)

Remark 19 The converse of Proposition 18 is not necessarilytrue In fact define119898120583 on119873 = 1 2 3 by

119898120583(119878) =

0 119878 = 0

1 119878 = 0 1 2 3

minus1

2 119878 = 1 2 3

(21)

Then we obtain a supermodular fuzzy measure

120583 (0) = 0 120583 (1) = 120583 (2) = 120583 (3) = 1

120583 (1 2) = 120583 (1 3) = 120583 (2 3) = 3

120583 (1 2 3) =11

2

(22)

4 Fairness Relation Using the AsymmetricChoquet Integral

Definition 20 (Choquet Integral relation) Let 119909 = (1199091

1199092 119909

119899) let 119910 = (119910

1 119910

2 119910

119899) and let 120583 be a fuzzy

measure 119909 is said to CI-dominate 119910 denoted by 119909ge120583119910 if and

only if

(ASC) int (119909 minus 119910) 119889120583 ge 120579 (23)

Assume119873 is a set of all players 119909 and 119910 are distributionsof resources to all players Then 119909 minus 119910 means each degreeof satisfaction of the distribution 119909 compared with thedistribution 119910 Since the more players are satisfied the fairerthe distribution is 119909ge

120583119910 means 119909 is at least as fair as 119910 By

119909gt120583119910 we denote the corresponding strict relation

Proposition 21 Let120583 be a superadditive fuzzymeasure on 2119873Then it is an antisymmetric relation that is 119909ge

120583119910 and 119910ge

120583119909

implies 119909 = 119910 for any functions 119909 119910 on119873

Proposition 22 If 120583 is a supermodular measure on 2119873 thengt120583is a transitive relation that is if 119909ge

120583119910 and 119910ge

120583119911 then 119909ge

120583119911

for any functions 119909 119910 119911 on119873

By Propositions 21 and 22 using the asymmetric Choquetintegral with respect to supermodular functions we obtain afair ordering relation

The proofs for Propositions 21 and 22 will be given in theAppendix of this paper

5 Decision Making by CI-Fairness ANumerical Example

In this section we want to demonstrate how CI-fairnessrelation can be used for decisionmaking Consider a situationwhere resources have to be allocated to 3 agents 119860 119861 and 119862but sets of agents can utilize resources differently (eg givinga commodity to one agent how many items can she producewithin a time unit) Such resource utilization can be assessedby a fuzzy measure

subset 119878 120583 (119878)

0 0

119860 01

119861 03

119862 05

119860 119861 05

119860 119862 05

119861 119862 08

119860 119861 119862 1

(24)

Here for example 119862 appears to be the best in resourceutilization but 119860 is good in collaboration The measure issuperadditive We can also see that this is a supermodularmeasure There are 3 cases to check the other follow fromsubsethood or superadditivity

(1) 120583(119860 119861cup119860 119862)+120583(119860 119861cap119860 119862) = 120583(119860 119861 119862)+120583(119860) = 11 ge 120583(119860 119861) + 120583(119860 119862) = 10

(2) 120583(119860 119861cup 119861 119862)+120583(119860 119861cap 119861 119862) = 120583(119860 119861 119862)+120583(119861) = 13 ge 120583(119860 119861) + 120583(119861 119862) = 13

8 Journal of Applied Mathematics

(1 2 3)

(1 3 2)

(2 3 1) (3 1 2)(2 1 3)

(3 2 1)

Figure 2 CI-Fairness relation among six possible allocations Theconnections include transitivity

(3) 120583(119860 119862cup119861 119862)+120583(119860 119862cap119861 119862) = 120583(119860 119861 119862)+120583(119862) = 15 ge 120583(119860 119862) + 120583(119861 119862) = 13

(1 2 3) ge120583 (1 3 2)

(1 2 3) ge120583 (2 1 3)

(1 2 3) ge120583 (2 3 1)

(1 2 3) ge120583 (3 1 2)

(1 2 3) ge120583 (3 2 1)

(1 3 2) ge120583 (2 3 1)

(1 3 2) ge120583 (3 1 2)

(1 3 2) ge120583 (3 2 1)

(2 1 3) ge120583 (2 3 1)

(2 1 3) ge120583 (3 1 2)

(2 1 3) ge120583 (3 2 1)

(2 3 1) ge120583 (3 2 1)

(3 1 2) ge120583 (3 2 1)

(25)

We find the allocation (1 2 3) in relation to all other alloca-tions but no allocation is in relation to (1 2 3)This is naturalgiven the order of resource utilization weights for the threeagents By rank we can now also order the other 5 allocations

Now we want to allocate three shares of a resource of6 units (1 2 3) among 119860 119861 119862 and compare the 6 possiblepermutations of (1 2 3) by the Choquet integral relationwith120579 = 0 For example to see if allocation 119909 = (1 2 3) is inrelation to the allocation 119910 = (2 3 1) we have to test

(119862) int [119909 minus 119910]+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (26)

In this case [119909 minus 119910]+ = (0 0 2) and [119910 minus 119909]+ = (1 1 0) Sothe first integral becomes 2 times 120583(119862) = 10 and the secondintegral 1 times 120583119889(119860 119861) = (1 minus 120583(119862)) = 05 From 10 ge 05then 119909ge

120583119910 follows

Among all possible 30 pairs (119909 119910) with 119909 = 119910 we can find13 cases in total where the Choquet integral relation holds

Figure 2 shows the Hasse diagram of the relation for the 6possible allocations We can see that as an effect of the bettercollaboration weight of 119860 allocation (2 1 3) is promotedand appears on the same rank as the strong allocation(1 3 2) (strong with regard to single agent utilization ofa resource) and also of better rank than the allocation(2 3 1)mdashexpressing the preference for 119862 as partner of 119860when comparing possible wins and losses in the allocationFinally as expected there is no case to allocate 3 shares to 119860

6 Experiments

In this section we will show how the CI-fairness can beapplied in a practical situation We study wireless channelallocation in the network layer a base station BS that isconnected to the backbone can command over a number119898 of wireless transmission channels and allocates them to anumber 119899 of users (or subscriber stations mobile stationsrelay stations etcmdashdepending on the specific network archi-tecture) for uplink or downlink traffic In the most abstractnotation the BSmakes the allocation based on ameasured orestimated channel state information if it allocates a channel 119894to user 119895 there will be a channel coefficient 119862

119894119895from [0 1] that

represents (in a simplified form) to what degree the user canemploy that channel For example remote users are likely tohave smaller channel coefficients than users that are close tothe BS Here an allocation is seen as feasible if each channelis allocated to one user and to each user at least one channelis assigned

The sum of all channel coefficients of allocated channelsfor a specific user is seen as the performance of the allocationfor that user Then the optimality task is related to theperformance vector for all users This is a direct reference toa relation between performance vectors and various fairnessrelations can be easily and conveniently studied in thiscontext If the number of channels and users is not too largeexhaustive search for maximum sets is possible For examplefor 4 users and 6 channels there are 1560 feasible allocationswhich means about 24 Mio pairwise comparisons

In this model channel allocation ignores low channelinterferences between users and specific channels by therestriction that each channel is allocated to exactly one userHowever if implementing a wireless access where some userscan use the same channel for transmission it might be hardto decide the exact schedule of such multiple assignmentsor to decide if installing such a scheme at all Thereforewe follow a more simplified approach channel interferencewill be represented by a conflict matrix between userswhere low elements for pairs of users indicate the optionto use multichannel assignment Then we use the converseelements of the conflict matrix as masses in the Mobiustransform More specifically the Mobius masses for singleelement sets are set to equal values the masses for two-element are set to 1 minus the corresponding element of theconflictmatrix and all other are set to 0Then themassvectoris normalized to a total of all masses of 1 to ensure that

Journal of Applied Mathematics 9

Table 1 Comparison of total performances for capacity propor-tional and maxmin fairness

Fairness relation Average performanceConflict matrix range 0 to 1

CI 42882Proportional 454661maxmin 414688

Conflict matrix range 09 to 1CI 456051Proportional 451135maxmin 422561

Conflict matrix range 0 to 02CI 409365Proportional 441848maxmin 404879

for the fuzzy measure the measure for the whole set is 1The inverse Mobius transform of such a mass vector gives asupermodularmeasure that can be used to specify a CI-basedfairness relation according to Definition 20

Now this relation is applied to all feasible channelallocationsmdashit means we are not considering a specific use ofmultichannel assignment but we are looking for the influencethat such amultichannel allocation could have on the specificselection of maximal elements Then the focus is on thedisadvantage that some pairs of users can experience if theirfavoured situation (eg to be distant enough to use the samechannel) is not taken into account The similar approach canbe considered for cognitive radio opportunistic networkingP2P vehicular networks and so forth

We might especially look for the price of fairness [1]compared to a standard proportional fair or maxmin fairallocation

Some example results are shown in Table 1 There for thecase of 4 users and 6 cells the maximum sets over all feasibleallocations were computed for CI-fairness using a measure asdescribed above proportional fairness andmaxmin fairnessThe conflictmatrix elements for the CI-fairness were uniformrandomly set within specific ranges All results are averagedover 30 repetitions Note that average sizes of maximum setswere found to be 42 formaxmin fairness 61 for proportionalfairness and 419 for CI-fairness so the CI-fairness produceslarger maximum sets an issue that should be addressed infutureworksThe table now shows average performances overall maximal elements

We see with regard to efficiency the well-known fact thatproportional-fair allocations are in average more efficientthan maxmin fair allocations The CI-fairness appears toselect maximal relations with a total performance betweenproportional fairness and maxmin fairness In case of stronginterferences (where a multichannel assignment is not rea-sonable) it is very close to proportional fairness while in caseof low interference it appears to be more close to maxminfairness that is the fairness relation favouring least elementsThus such an empirical result can be understood in the sensethat the neglecting of the potential multichannel assignment

corresponds with the neglection of strong users This is alsoparticularly appearing for the general case where all conflictmatrix elements were randomly selected between 0 and 1 alsohere the average performance appearsmore close tomaxminfairness than proportional fairness

7 Conclusion

A fairness relation among vectors based on the asymmetricChoquet integral was studied It formally follows the Nashstandard of comparison where the relative losses and winsare replaced by absolute losses and wins but instead takingweights for groups of vector components into account Thusit can represent for example pairs of users that are ina favourable situation regarding resource allocation Theappealing points of this CI-relation are as follows (1) Incase a supermodular measure is used for the integrationthe relation will be transitive which suits faster search formaximal elements (2) it can be parametrized by givinga weight to each subset of vector components (while inpractice this might be relevant only for smaller subsets) Therepresentation of the fuzzy measure by its Mobius transformappears to be a convenient way to yield a supermodularfuzzy measure the masses just need to be all nonnegativeThus the relation can be conveniently applied in many prac-tical applications to specify optimality Slight disadvantagesare with the complexity of the involved calculations andsome empirical evidence for specifying larger maximum setsthan other fairness relations like proportional fairness andmaxmin fairness

In case of wireless channel allocation how CI-fairnesscan be used to help deciding whether implementations ofmore complex allocation schemes are indeed worth the effortwas suggested This method can be applied to many otherproblems of higher efficiency of network resource utilizationIn future work we will consider the use of advanced fuzzyintegrals like balancing and fusion Choquet integral andthe adjustment of related CI-fairness parameters to specificsituations and what can be concluded from such parametersWe will also more intensively study the performance issueswith regard to computational effort and search for maximalelements and study conditions where also greatest elementsexist

Appendix

Proofs of Theorems

In the following we provide the proofs for Propositions 21and 22 For formal convenience of the proofs the notationwill be a little bit changing therefore we repeat the basicdefinitions

We define the Choquet-integral-based (fairness) relationas follows be 119909 and 119910 two vectors from 119877

119899 By [119909]+ weindicate the positive support of a vector 119909 that is the 119894thcomponent of [119909]+ is 119909

119894if 119909

119894ge 0 or 0 if 119909

119894lt 0 The Choquet

integral is denoted by (119862) int119909 119889120583 There 120583 is a fuzzy measure

10 Journal of Applied Mathematics

Also 120583119889 denotes its dual measure (ie 120583119889(119860) = 120583119883minus 120583(119860

119862))

Then 119909 is said to CI-dominate 119910 if and only if

119909ge120583119910 larrrarr (119862)int [119909 minus 119910]

+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (A1)

We can alternatively rewrite this as the asymmetric Choquetintegral

119909ge120583119910 larrrarr (ASC) int (119909 minus 119910) 119889120583 ge 0 (A2)

since the right-hand side condition of (A1) exactly cor-responds with the definition of the asymmetric Choquetintegral

First we can show the following

Lemma 23 If 120583 is a superadditive fuzzy measure where foreach 119860 119861 = 0 and 119860 cap 119861 = 0 120583

119860+ 120583

119861lt 120583

119860cup119861 then for

119860 = 0 120583119889

119860gt 120583

119860

Proof Since 120583 is superadditive for119860 = 0 120583119860+120583

119860119862 lt 120583

119860cup119860119862 =

120583119883 Then 120583

119860lt 120583

119883minus 120583

119860119862 = 120583

119889

119860follows directly

Theorem 24 If 120583 is superadditive then from 119909ge120583119910 and

119910ge120583119909119909 = 119910 follows

Proof 119909ge120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889 If

119909 = 119910 we can use Lemma 23 to obtain

(119862) int [119909 minus 119910]+119889120583

119889gt (119862)int [119909 minus 119910]

+119889120583

ge (119862)int [119910 minus 119909]+119889120583

119889gt (119862)int [119910 minus 119909]

+119889120583

(A3)

that is (119862) int [119910 minus 119909]+119889120583 ge (119862) int [119909 minus 119910]+119889120583

119889 This wouldmean that 119910ge

120583119909 would not hold Therefore it must be 119909 =

119910

For a supermodular measure and nonnegative vectors 119909and 119910 we have the known property (119862) int 119909 119889120583+ (119862) int 119910 119889120583 le(119862) int(119909 + 119910)119889120583 Since the dual measure of a supermodularmeasure is submodular it also holds that (119862) int 119909 119889120583 +(119862) int 119910 119889120583 ge (119862) int(119909 + 119910)119889120583 Using this the following canbe shown

Theorem25 If120583 is supermodular then the relationgt120583is cycle-

free

Proof We will only show the case of nonexistence of a 3-cycle here the concept generalizes directly to any other caseAssume that 119909gt

120583119910 and 119910gt

120583119911 First we note that

[119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+

= [119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+

(A4)

and therefore

(119862) int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

= (119862)int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

(A5)

Using Lemma 23 we yield

(119862) int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

119889

gt (119862)int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

(A6)

and from supermodularity of 120583 and submodularity of 120583119889

(119862) int [119910 minus 119909]+119889120583

119889+ (119862)int [119911 minus 119910]

+119889120583

119889+ (119862)int [119909 minus 119911]

+119889120583

119889

gt (119862)int [119909 minus 119910]+119889120583 + (119862)int [119910 minus 119911]

+119889120583

+ (119862)int [119911 minus 119909]+119889120583

(A7)

However 119909gt120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889

and 119910gt120583119911 means (119862) int [119910 minus 119911]+119889120583 ge (119862) int [119911 minus 119910]

+119889120583

119889Therefore above inequality can only hold if

(119862) int [119911 minus 119909]+119889120583 lt (119862)int [119909 minus 119911]

+119889120583

119889 (A8)

which means 119911 gt120583119909

If the measure 120583 is supermodular also transitivity ofthe relation can be shown Before we can show this weneed to introduce a notation and a corresponding lemmaThe notation refers to computing the (asymmetric) Choquetintegral of 119909 by measure 120583 in a ldquodifferent orderrdquo If 119909

119909(119894)

indicates the 119894th largest element of 119909 then the Choquetintegral is calculated as

(119862) int 119909119889120583 = 119909119909(1)120583119909(1)

+

119899

sum

119894=2

119909119909(119894)[120583

119909(1)119909(2)sdotsdotsdot119909(119894)minus 120583

119909(1)sdotsdotsdot119909(119894minus1)]

(A9)

For any other119910 we can compute the same expression justfollowing the order of the elements of 119910 instead of 119909 Then119909119910(119894)

indicates the element of119909with the index of the 119894th largestelement of 119910 and we define (for convenience we still use thesymbol for the Choquet integral keeping in mind that it isnot a real fuzzy integral anymore)

(119862) int(119910)

119909119889120583 = 119909119910(1)120583119910(1)

+

119899

sum

119894=2

119909119910(119894)[120583

119910(1)119910(2)sdotsdotsdot119910(119894)minus 120583

119910(1)sdotsdotsdot119910(119894minus1)]

(A10)

Journal of Applied Mathematics 11

Consequently we could also write (119862) int(119909)119909119889120583 for the

ldquooriginalrdquo Choquet integral We can do the same for theasymmetric Choquet integral by applying the resorting to thenegative components of the vector 119909 as well and will use thenotation (ASC) int

(119910)119909 119889120583 here Now we have the following

Lemma 26 If 120583 is a supermodular measure then for any119910 ( ASC ) int

(119910)119909 119889120583 ge ( ASC ) int

(119909)119909119889120583

Proof A moment of reasoning gives that any sorting of theindizes can be achieved by suitable application of a sequenceof three kinds of ldquoneighbour swapsrdquo starting from the order119909(1)119909(2) sdot sdot sdot 119909(119899))

Swap 1 In the computation of the integral a larger orequal nonnegative element 119889

119894at index 119894 is swapped with its

immediate nonnegative neighbour 119889119894+1

to the right Thenwhat was computed as 119889

119894(120583

119886119894minus120583

119886) +119889

119894+1(120583

119886119894119894+1minus120583

119886119894) before

the swap changes to 119889119894+1(120583

119886119894+1minus120583

119886) + 119889

119894(120583

119886119894119894+1minus120583

119886119894+1) after

the swap all other parts of the resorted asymmetric Choquetintegral expression remain the same (by 119886 we indicate theindex order before index 119894) Thus the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886119894)

minus 119889119894+1(120583

119886119894119894+1minus 120583

119886119894minus 120583

119886119894+1+ 120583

119886119894)

(A11)

and from 119889119894ge 119889

119894+1and 120583

119886119894119894+1+ 120583

119886ge 120583

119886119894+1+ 120583

119886119894

(supermodularity of 120583) it directly follows that the computedvalue will not decrease by this swap

Swap 2 In the computation of the integral a larger orequal negative element 119889

119894+1at index 119894 + 1 is swapped with

its immediate negative left neighbour 119889119894 From a similar

evaluation as for Swap 1 it can be seen that also here the valueof the computed expression will not decrease after the swap

Swap 3 A nonnegative element at index 119894 is swapped withits immediate negative neighbor to the right with index(119894 + 1) Then before the swap we compute 119889

119894(120583

119886119894minus 120583

119886) minus

119889119894+1(120583

119886119894119894+1minus 120583

119886119894) (119889

119894+1is the absolute value of the negative-

valued neighbour of 119889119894) and after the swap this part of the

expression changes to minus119889119894+1(120583

119886119894+1minus 120583

119886) + 119889

119894(120583

119886119894119894+1minus 120583

119886119894+1)

Then the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886)

minus 119889119894+1(120583

119886119894+1minus 120583

119886minus 120583

119886119894119894+1+ 120583

119886119894)

(A12)

Supermodularity of 120583 gives 120583119886119894+ 120583

119886119894+1le 120583

119886+ 120583

119886119894119894+1and

therefore the factor of 119889119894in former expression is nonnegative

and the factor of 119889119894+1

(which is the negation of the factor of119889119894) is negative or 0Therefore the total change is nonnegative

as wellThus with each of these swap operations we will never

decrease the value of the expression and starting from theasymmetric Choquet integral for 119909 upon reaching the finalorder the final value will be larger or equal

An example might be helpful Assume that the order in 119909is (1 2 3 minus 4 minus 5) (a negative index should indicate that

the corresponding element of 119909 is negative) and in 119910 we havethe order (minus5 3 1 2 minus4) (so 119910

5is the largest component of

119910 and 1199104the smallest) Then the order of swaps is as follows

Swap 1 applied as (1 2 3larr997888rarr

minus4 minus5) gives (1 3 2 minus4 minus5)Swap 1 applied as (1 3

larr997888rarr2 minus4 minus5) gives (3 1 2 minus4 minus5)

Next Swap 2 applied as (3 1 2 minus4 minus 5larr997888997888997888997888rarr

) gives (3 1 2 minus

5 minus 4) Swap 3 applied as (3 1 2 minus 5larr997888997888997888rarr

minus 4) gives (3 1 minus

5 2 minus 4) Swap 3 applied as (3 1 minus 5larr997888997888997888rarr

2 minus 4) gives (3 minus

5 1 2 minus4) and finally Swap 3 applied as (3 minus 5larr997888997888997888rarr

1 2 minus4)

gives the order (minus5 3 1 2 minus 4) of 119910 Note also that in casethe elements of 119909 are not in that order we can always relabelthe indizes and measures correspondingly

This means if we compute (using 119909 =

(1199091 119909

2 119909

3 minus119909

4 minus119909

5))

(ASC) int(119910)

119909119889120583

= minus11990951205835+ 119909

3(120583

35minus 120583

5) + 119909

1(120583

135minus 120583

35)

+ 1199092(120583

1235minus 120583

135) minus 119909

4(120583

12345minus 120583

1235)

(A13)

instead of

(ASC) int 119909 119889120583

= 11990911205831+ 119909

2(120583

12minus 120583

1) + 119909

3(120583

123minus 120583

12)

minus 1199094(120583

1234minus 120583

123) minus 119909

5(120583

12345minus 120583

1234)

(A14)

then (ASC) int(119910)119909 119889120583 ge (ASC) int 119909 119889120583

Using this Lemma we can now easily show the following

Theorem 27 If 120583 is a supermodular measure then ge120583is a

transitive relation

Proof For transitivity we have to show that for any 119909 119910 and119911 such that 119909ge

120583119910 and 119910ge

120583119911 it follows that 119909ge

120583119911 Now 119909ge

120583119910

means (ASC) int(119909 minus 119910)119889120583 ge 0 and 119910ge120583119911 means (ASC) int(119910 minus

119911)119889120583 ge 0 From supermodularity of 120583 and Lemma 26 then italso follows that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 ge (ASC) int (119909 minus 119910) 119889120583 ge 0

(ASC) int(119909minus119911)

(119910 minus 119911) 119889120583 ge (ASC) int (119910 minus 119911) 119889120583 ge 0(A15)

However by using the fact that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 + (ASC) int(119909minus119911)

(119910 minus 119911) 119889120583

= (ASC) int (119909 minus 119911) 119889120583(A16)

we see that also

(ASC) int (119909 minus 119911) 119889120583 ge 0 (A17)

which means 119909ge120583119911

12 Journal of Applied Mathematics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Bertsimas V F Farias and N Trichakis ldquoThe price offairnessrdquo Operations Research vol 59 no 1 pp 17ndash31 2011

[2] J F Nash Jr ldquoThe bargaining problemrdquo Econometrica vol 18pp 155ndash162 1950

[3] E Kalai and M Smorodinsky ldquoOther solutions to Nashrsquosbargaining problemrdquo Econometrica vol 43 pp 513ndash518 1975

[4] J Rawls ATheory of Justice Harvard University Press 1999[5] F Kelly ldquoCharging and rate control for elastic trafficrdquo European

Transactions onTelecommunications vol 8 no 1 pp 33ndash37 1997[6] D P Bertsekas R G Gallager and P Humblet Data Networks

vol 2 Prentice-Hall International 1992[7] H P Young Equity In Theory and Practice Princeton Univer-

sity Press Princeton NJ USA 1995[8] J Mo and J Walrand ldquoFair end-to-end window-based conges-

tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[9] M Alicherry R Bhatia and L E Li ldquoJoint channel assignmentand routing for throughput optimization inmulti-radiowirelessmesh networksrdquo in Proceedings of the 11th Annual InternationalConference on Mobile Computing and Networking (MobiCom05) pp 58ndash72 ACM Cologne Germany August-September2005

[10] A Iyer C Rosenberg and A Karnik ldquoWhat is the rightmodel for wireless channel interferencerdquo IEEE Transactions onWireless Communications vol 8 no 5 pp 2662ndash2671 2009

[11] A Mishra V Shrivastava S Banerjee and W ArbaughldquoPartially overlapped channels not considered harmfulrdquo ACMSIGMETRICS Performance Evaluation Review vol 34 no 1 pp63ndash74 2006

[12] N Nie and C Comaniciu ldquoAdaptive channel allocation spec-trum etiquette for cognitive radio networksrdquo Mobile Networksand Applications vol 11 no 6 pp 779ndash797 2006

[13] C Peng H Zheng and B Y Zhao ldquoUtilization and fairnessin spectrum assignment for opportunistic spectrum accessrdquoMobile Networks and Applications vol 11 no 4 pp 555ndash5762006

[14] X Yang andA P Petropulu ldquoCo-channel interferencemodelingand analysis in a Poisson field of interferers in wireless commu-nicationsrdquo IEEE Transactions on Signal Processing vol 51 no 1pp 64ndash76 2003

[15] M Koppen ldquoRelational optimization and its application frombottleneck flow control to wireless channel allocationrdquo Infor-matica vol 24 no 3 pp 413ndash433 2013

[16] M Sugeno Theory of fuzzy integrals and its applications [PhDthesis] 1974

[17] S Michio ldquoFuzzymeasures and fuzzy integrals a surveyrdquo FuzzyAutomata and Decision Processes vol 78 no 33 pp 89ndash1021977

[18] D Denneberg Non-Additive Measure and Integral vol 27Springer 1994

[19] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[20] J Sipos ldquoIntegral with respect to a pre-measurerdquo MathematicaSlovaca vol 29 no 2 pp 141ndash155 1979

[21] M Grabisch and C Labreuche ldquoThe symmetric and asymmet-ric Choquet integrals on finite spaces for decision makingrdquoStatistical Papers vol 43 no 1 pp 37ndash52 2002

[22] M Grabisch C Labreuche and J-C Vansnick ldquoOn theextension of pseudo-Boolean functions for the aggregation ofinteracting criteriardquo European Journal of Operational Researchvol 148 no 1 pp 28ndash47 2003

[23] R Mesiar A Mesiarova-Zemankova and K Ahmad ldquoDiscreteChoquet integral and some of its symmetric extensionsrdquo FuzzySets and Systems vol 184 pp 148ndash155 2011

[24] A Mesiarova-Zemankova R Mesiar and K Ahmad ldquoThebalancing Choquet integralrdquo Fuzzy Sets and Systems vol 161no 17 pp 2243ndash2255 2010

[25] H Fouchal X Gandibleux and F Lehuede ldquoA lower bound ofthe Choquet integral integrated within Martinsalgorithmrdquo inNewState ofMCDM in the 21st Century vol 648 ofLectureNotesin Economics and Mathematical Systems pp 79ndash89 SpringerHeidelberg Germany 2011

[26] L Galand L Julien and P Patrice ldquoDominance rules forthe choquet integral in multiobjective dynamic programmingrdquoin Proceedings of the 23rd International Joint Conference onArtificial Intelligence pp 538ndash544 AAAI Press 2013

[27] A W Marshall I Olkin and B C Arnold Inequalities TheoryofMajorization and Its Applications Springer 2nd edition 2010

[28] W Ogryczak AWierzbicki andMMilewski ldquoAmulti-criteriaapproach to fair and efficient bandwidth allocationrdquoOmega vol36 no 3 pp 451ndash463 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

Proof Because

int

0

minusinfin

(120583 (119909 | 119891 (119909) gt 119903) minus 120583 (119873)) 119889119903

= int (120583 (119909 | minus119891minus(119909) gt 119903) minus 120583 (119873)) 119889119903

= minusint (120583 (119909 | 119891minus(119909) le 119903) minus 120583 (119873)) 119889119903

= minusint120583119889(119909 | 119891

minus(119909) gt 119903) 119889119903

(12)

by this proposition we can say that the difference betweenthe asymmetricChoquet integral and the symmetricChoquetintegral is based on which measure are used for integratingthe negative part of 119891

3 Description of Fairness bythe Choquet Integral

In this section we discuss the generalized Choquet integralsin terms of describing fairness by the Choquet integral Let 120583be a fuzzy measure on 2119873 and 119891 = (119909

1 119909

2 119909

119899) a function

on 119873 For example 119873 corresponds to a set consisting of119899 players and 119891 corresponds to a distribution of resourceswhere 119891 can take negative values The more players aresatisfied the fairer the distribution is

Example 15 Let 119899 = 3 and there are 6 resources Then1198911= (2 2 2) is fairer than 119891

2= (4 2 0) and 119891

2is fairer

than 1198913= (6 0 0) To describe this we use the Choquet

integral with respect to a submodular function Suppose that120583(119878) takes the same values according to |119878| (note that in thiscase the Choquet integral becomes an OWA operator andproperties like submodularity depend on the choice of OWAweights) If 120583 is submodular then we have

(119862) int1198911119889120583 ge (119862)int1198912119889120583 ge (119862)int1198913119889120583 (13)

Next we consider the negative distribution

Example 16 Let 119899 = 3 and there are minus6 resources Then 1198911=

(minus2 minus2 minus2) is fairer than1198912= (minus4 minus2 0) and119891

2is fairer than

1198913= (minus6 0 0) Let 120583 be a submodular function If we use the

symmetric Choquet integral then we have

(SC) int1198911119889120583 le (SC) int1198912119889120583 le (SC) int1198913119889120583 (14)

where all integrals take negative values And if 1205831015840 would beanother but superadditive fuzzy measure then we obtain

(SC) int11989111198891205831015840ge (SC) int1198912119889120583

1015840ge (SC) int1198913119889120583

1015840 (15)

which is not what we wanted In other words we cannever describe both positive fairness and negative fairness

by one fuzzy measure using the symmetric Choquet inte-gral On the other hand using the asymmetric Choquetintegral with respect to 120583 where 120583119889 is superadditive weobtain

(ASC) int1198911119889120583 ge (ASC) int1198912119889120583 ge (ASC) int1198913119889120583 (16)

If 120583 is a supermodular fuzzy measure then 120583 is superadditiveand 120583119889 is subadditive Therefore using the asymmetric Cho-quet integral with respect to a supermodular fuzzy measurewe can describe both positive fairness and negative fairnessby one fuzzy measure

According to these examples the asymmetric Choquetintegral with respect to a superadditive fuzzy measure isbetter suited for describing fairness The Mobius transformenables us to construct fuzzy measure easily

Definition 17 (Mobius transform) The Mobius transform of120583 denoted by119898120583

2119873rarr [minus1 1] is defined by

119898120583(119878) = sum

119879sube119878

(minus1)|119878119879|

120583 (119879) (17)

for any 119878 isin 2119873 Conversely we obtain

120583 (119878) = sum

119879sube119878

119898120583(119879) (18)

by the inverse Mobius transform for any 119878 isin 2119873 and there isa one-to-one correspondence between 120583 and119898120583

Note that if the sum of all Mobius masses is 1 then thecorresponding fuzzy measure of the whole set is 1

Proposition 18 If 119898120583(119878) ge 0 for any 119878 isin 2

119873 then 120583 issupermodular

Proof Assume for any 119878 isin 2119873 119898120583(119878) ge 0 Then denoting

that 119903 = 119878 119879 119904 = 119878 cap 119879 119905 = 119879 119878 for 119878 119879 isin 2119873 we have

120583 (119878 cup 119879) + 120583 (119878 cap 119879)

= sum

119862sube119878cup119879

119898120583(119880) + sum

119880sube119878cap119879

119898120583(119880)

= (sum

119880sube119903

+ sum

119880sube119904

+ sum

119880sube119905

+ sum

119880sube119903cup119904119880cap119903 = 0

119880cap119904 = 0

Journal of Applied Mathematics 7

+ sum

119880sube119904cup119905119880cap119904 = 0

119880cap119905 = 0

+ sum

119880sube119903cup119905

119880cap119903 = 0119880cap119905 = 0

+ sum

119880sube119903cup119904cup119905119880cap119903 = 0

119880cap119904 = 0119880cap119905 = 0

)119898120583(119880) + sum

119880sube119904

119898120583(119880)

120583 (119878) + 120583 (119879) = sum

119880sube119878

119898120583(119880) + sum

119880sube119879

119898120583(119880)

= (sum

119880sube119903

+ sum

119880sube119904

+ sum

119880sube119903cup119904119880cap119903 = 0

119880cap119904 = 0

)119898120583(119880)

+ (sum

119880sube119904

+ sum

119880sube119905

+ sum

119880sube119904cup119905119880cap119904 = 0

119880cap119905 = 0

)119898120583(119880)

(19)

so that we obtain

(120583 (119878 cup 119879) + 120583 (119904)) minus (120583 (119878) + 120583 (119879))

= ( sum

119880sube119903cup119905

119880cap119903 = 0119880cap119905 = 0

+ sum

119880sube119903cup119904cup119905119880cap119903 = 0

119880cap119904 = 0119880cap119905 = 0

)119898120583(119880) ge 0

(20)

Remark 19 The converse of Proposition 18 is not necessarilytrue In fact define119898120583 on119873 = 1 2 3 by

119898120583(119878) =

0 119878 = 0

1 119878 = 0 1 2 3

minus1

2 119878 = 1 2 3

(21)

Then we obtain a supermodular fuzzy measure

120583 (0) = 0 120583 (1) = 120583 (2) = 120583 (3) = 1

120583 (1 2) = 120583 (1 3) = 120583 (2 3) = 3

120583 (1 2 3) =11

2

(22)

4 Fairness Relation Using the AsymmetricChoquet Integral

Definition 20 (Choquet Integral relation) Let 119909 = (1199091

1199092 119909

119899) let 119910 = (119910

1 119910

2 119910

119899) and let 120583 be a fuzzy

measure 119909 is said to CI-dominate 119910 denoted by 119909ge120583119910 if and

only if

(ASC) int (119909 minus 119910) 119889120583 ge 120579 (23)

Assume119873 is a set of all players 119909 and 119910 are distributionsof resources to all players Then 119909 minus 119910 means each degreeof satisfaction of the distribution 119909 compared with thedistribution 119910 Since the more players are satisfied the fairerthe distribution is 119909ge

120583119910 means 119909 is at least as fair as 119910 By

119909gt120583119910 we denote the corresponding strict relation

Proposition 21 Let120583 be a superadditive fuzzymeasure on 2119873Then it is an antisymmetric relation that is 119909ge

120583119910 and 119910ge

120583119909

implies 119909 = 119910 for any functions 119909 119910 on119873

Proposition 22 If 120583 is a supermodular measure on 2119873 thengt120583is a transitive relation that is if 119909ge

120583119910 and 119910ge

120583119911 then 119909ge

120583119911

for any functions 119909 119910 119911 on119873

By Propositions 21 and 22 using the asymmetric Choquetintegral with respect to supermodular functions we obtain afair ordering relation

The proofs for Propositions 21 and 22 will be given in theAppendix of this paper

5 Decision Making by CI-Fairness ANumerical Example

In this section we want to demonstrate how CI-fairnessrelation can be used for decisionmaking Consider a situationwhere resources have to be allocated to 3 agents 119860 119861 and 119862but sets of agents can utilize resources differently (eg givinga commodity to one agent how many items can she producewithin a time unit) Such resource utilization can be assessedby a fuzzy measure

subset 119878 120583 (119878)

0 0

119860 01

119861 03

119862 05

119860 119861 05

119860 119862 05

119861 119862 08

119860 119861 119862 1

(24)

Here for example 119862 appears to be the best in resourceutilization but 119860 is good in collaboration The measure issuperadditive We can also see that this is a supermodularmeasure There are 3 cases to check the other follow fromsubsethood or superadditivity

(1) 120583(119860 119861cup119860 119862)+120583(119860 119861cap119860 119862) = 120583(119860 119861 119862)+120583(119860) = 11 ge 120583(119860 119861) + 120583(119860 119862) = 10

(2) 120583(119860 119861cup 119861 119862)+120583(119860 119861cap 119861 119862) = 120583(119860 119861 119862)+120583(119861) = 13 ge 120583(119860 119861) + 120583(119861 119862) = 13

8 Journal of Applied Mathematics

(1 2 3)

(1 3 2)

(2 3 1) (3 1 2)(2 1 3)

(3 2 1)

Figure 2 CI-Fairness relation among six possible allocations Theconnections include transitivity

(3) 120583(119860 119862cup119861 119862)+120583(119860 119862cap119861 119862) = 120583(119860 119861 119862)+120583(119862) = 15 ge 120583(119860 119862) + 120583(119861 119862) = 13

(1 2 3) ge120583 (1 3 2)

(1 2 3) ge120583 (2 1 3)

(1 2 3) ge120583 (2 3 1)

(1 2 3) ge120583 (3 1 2)

(1 2 3) ge120583 (3 2 1)

(1 3 2) ge120583 (2 3 1)

(1 3 2) ge120583 (3 1 2)

(1 3 2) ge120583 (3 2 1)

(2 1 3) ge120583 (2 3 1)

(2 1 3) ge120583 (3 1 2)

(2 1 3) ge120583 (3 2 1)

(2 3 1) ge120583 (3 2 1)

(3 1 2) ge120583 (3 2 1)

(25)

We find the allocation (1 2 3) in relation to all other alloca-tions but no allocation is in relation to (1 2 3)This is naturalgiven the order of resource utilization weights for the threeagents By rank we can now also order the other 5 allocations

Now we want to allocate three shares of a resource of6 units (1 2 3) among 119860 119861 119862 and compare the 6 possiblepermutations of (1 2 3) by the Choquet integral relationwith120579 = 0 For example to see if allocation 119909 = (1 2 3) is inrelation to the allocation 119910 = (2 3 1) we have to test

(119862) int [119909 minus 119910]+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (26)

In this case [119909 minus 119910]+ = (0 0 2) and [119910 minus 119909]+ = (1 1 0) Sothe first integral becomes 2 times 120583(119862) = 10 and the secondintegral 1 times 120583119889(119860 119861) = (1 minus 120583(119862)) = 05 From 10 ge 05then 119909ge

120583119910 follows

Among all possible 30 pairs (119909 119910) with 119909 = 119910 we can find13 cases in total where the Choquet integral relation holds

Figure 2 shows the Hasse diagram of the relation for the 6possible allocations We can see that as an effect of the bettercollaboration weight of 119860 allocation (2 1 3) is promotedand appears on the same rank as the strong allocation(1 3 2) (strong with regard to single agent utilization ofa resource) and also of better rank than the allocation(2 3 1)mdashexpressing the preference for 119862 as partner of 119860when comparing possible wins and losses in the allocationFinally as expected there is no case to allocate 3 shares to 119860

6 Experiments

In this section we will show how the CI-fairness can beapplied in a practical situation We study wireless channelallocation in the network layer a base station BS that isconnected to the backbone can command over a number119898 of wireless transmission channels and allocates them to anumber 119899 of users (or subscriber stations mobile stationsrelay stations etcmdashdepending on the specific network archi-tecture) for uplink or downlink traffic In the most abstractnotation the BSmakes the allocation based on ameasured orestimated channel state information if it allocates a channel 119894to user 119895 there will be a channel coefficient 119862

119894119895from [0 1] that

represents (in a simplified form) to what degree the user canemploy that channel For example remote users are likely tohave smaller channel coefficients than users that are close tothe BS Here an allocation is seen as feasible if each channelis allocated to one user and to each user at least one channelis assigned

The sum of all channel coefficients of allocated channelsfor a specific user is seen as the performance of the allocationfor that user Then the optimality task is related to theperformance vector for all users This is a direct reference toa relation between performance vectors and various fairnessrelations can be easily and conveniently studied in thiscontext If the number of channels and users is not too largeexhaustive search for maximum sets is possible For examplefor 4 users and 6 channels there are 1560 feasible allocationswhich means about 24 Mio pairwise comparisons

In this model channel allocation ignores low channelinterferences between users and specific channels by therestriction that each channel is allocated to exactly one userHowever if implementing a wireless access where some userscan use the same channel for transmission it might be hardto decide the exact schedule of such multiple assignmentsor to decide if installing such a scheme at all Thereforewe follow a more simplified approach channel interferencewill be represented by a conflict matrix between userswhere low elements for pairs of users indicate the optionto use multichannel assignment Then we use the converseelements of the conflict matrix as masses in the Mobiustransform More specifically the Mobius masses for singleelement sets are set to equal values the masses for two-element are set to 1 minus the corresponding element of theconflictmatrix and all other are set to 0Then themassvectoris normalized to a total of all masses of 1 to ensure that

Journal of Applied Mathematics 9

Table 1 Comparison of total performances for capacity propor-tional and maxmin fairness

Fairness relation Average performanceConflict matrix range 0 to 1

CI 42882Proportional 454661maxmin 414688

Conflict matrix range 09 to 1CI 456051Proportional 451135maxmin 422561

Conflict matrix range 0 to 02CI 409365Proportional 441848maxmin 404879

for the fuzzy measure the measure for the whole set is 1The inverse Mobius transform of such a mass vector gives asupermodularmeasure that can be used to specify a CI-basedfairness relation according to Definition 20

Now this relation is applied to all feasible channelallocationsmdashit means we are not considering a specific use ofmultichannel assignment but we are looking for the influencethat such amultichannel allocation could have on the specificselection of maximal elements Then the focus is on thedisadvantage that some pairs of users can experience if theirfavoured situation (eg to be distant enough to use the samechannel) is not taken into account The similar approach canbe considered for cognitive radio opportunistic networkingP2P vehicular networks and so forth

We might especially look for the price of fairness [1]compared to a standard proportional fair or maxmin fairallocation

Some example results are shown in Table 1 There for thecase of 4 users and 6 cells the maximum sets over all feasibleallocations were computed for CI-fairness using a measure asdescribed above proportional fairness andmaxmin fairnessThe conflictmatrix elements for the CI-fairness were uniformrandomly set within specific ranges All results are averagedover 30 repetitions Note that average sizes of maximum setswere found to be 42 formaxmin fairness 61 for proportionalfairness and 419 for CI-fairness so the CI-fairness produceslarger maximum sets an issue that should be addressed infutureworksThe table now shows average performances overall maximal elements

We see with regard to efficiency the well-known fact thatproportional-fair allocations are in average more efficientthan maxmin fair allocations The CI-fairness appears toselect maximal relations with a total performance betweenproportional fairness and maxmin fairness In case of stronginterferences (where a multichannel assignment is not rea-sonable) it is very close to proportional fairness while in caseof low interference it appears to be more close to maxminfairness that is the fairness relation favouring least elementsThus such an empirical result can be understood in the sensethat the neglecting of the potential multichannel assignment

corresponds with the neglection of strong users This is alsoparticularly appearing for the general case where all conflictmatrix elements were randomly selected between 0 and 1 alsohere the average performance appearsmore close tomaxminfairness than proportional fairness

7 Conclusion

A fairness relation among vectors based on the asymmetricChoquet integral was studied It formally follows the Nashstandard of comparison where the relative losses and winsare replaced by absolute losses and wins but instead takingweights for groups of vector components into account Thusit can represent for example pairs of users that are ina favourable situation regarding resource allocation Theappealing points of this CI-relation are as follows (1) Incase a supermodular measure is used for the integrationthe relation will be transitive which suits faster search formaximal elements (2) it can be parametrized by givinga weight to each subset of vector components (while inpractice this might be relevant only for smaller subsets) Therepresentation of the fuzzy measure by its Mobius transformappears to be a convenient way to yield a supermodularfuzzy measure the masses just need to be all nonnegativeThus the relation can be conveniently applied in many prac-tical applications to specify optimality Slight disadvantagesare with the complexity of the involved calculations andsome empirical evidence for specifying larger maximum setsthan other fairness relations like proportional fairness andmaxmin fairness

In case of wireless channel allocation how CI-fairnesscan be used to help deciding whether implementations ofmore complex allocation schemes are indeed worth the effortwas suggested This method can be applied to many otherproblems of higher efficiency of network resource utilizationIn future work we will consider the use of advanced fuzzyintegrals like balancing and fusion Choquet integral andthe adjustment of related CI-fairness parameters to specificsituations and what can be concluded from such parametersWe will also more intensively study the performance issueswith regard to computational effort and search for maximalelements and study conditions where also greatest elementsexist

Appendix

Proofs of Theorems

In the following we provide the proofs for Propositions 21and 22 For formal convenience of the proofs the notationwill be a little bit changing therefore we repeat the basicdefinitions

We define the Choquet-integral-based (fairness) relationas follows be 119909 and 119910 two vectors from 119877

119899 By [119909]+ weindicate the positive support of a vector 119909 that is the 119894thcomponent of [119909]+ is 119909

119894if 119909

119894ge 0 or 0 if 119909

119894lt 0 The Choquet

integral is denoted by (119862) int119909 119889120583 There 120583 is a fuzzy measure

10 Journal of Applied Mathematics

Also 120583119889 denotes its dual measure (ie 120583119889(119860) = 120583119883minus 120583(119860

119862))

Then 119909 is said to CI-dominate 119910 if and only if

119909ge120583119910 larrrarr (119862)int [119909 minus 119910]

+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (A1)

We can alternatively rewrite this as the asymmetric Choquetintegral

119909ge120583119910 larrrarr (ASC) int (119909 minus 119910) 119889120583 ge 0 (A2)

since the right-hand side condition of (A1) exactly cor-responds with the definition of the asymmetric Choquetintegral

First we can show the following

Lemma 23 If 120583 is a superadditive fuzzy measure where foreach 119860 119861 = 0 and 119860 cap 119861 = 0 120583

119860+ 120583

119861lt 120583

119860cup119861 then for

119860 = 0 120583119889

119860gt 120583

119860

Proof Since 120583 is superadditive for119860 = 0 120583119860+120583

119860119862 lt 120583

119860cup119860119862 =

120583119883 Then 120583

119860lt 120583

119883minus 120583

119860119862 = 120583

119889

119860follows directly

Theorem 24 If 120583 is superadditive then from 119909ge120583119910 and

119910ge120583119909119909 = 119910 follows

Proof 119909ge120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889 If

119909 = 119910 we can use Lemma 23 to obtain

(119862) int [119909 minus 119910]+119889120583

119889gt (119862)int [119909 minus 119910]

+119889120583

ge (119862)int [119910 minus 119909]+119889120583

119889gt (119862)int [119910 minus 119909]

+119889120583

(A3)

that is (119862) int [119910 minus 119909]+119889120583 ge (119862) int [119909 minus 119910]+119889120583

119889 This wouldmean that 119910ge

120583119909 would not hold Therefore it must be 119909 =

119910

For a supermodular measure and nonnegative vectors 119909and 119910 we have the known property (119862) int 119909 119889120583+ (119862) int 119910 119889120583 le(119862) int(119909 + 119910)119889120583 Since the dual measure of a supermodularmeasure is submodular it also holds that (119862) int 119909 119889120583 +(119862) int 119910 119889120583 ge (119862) int(119909 + 119910)119889120583 Using this the following canbe shown

Theorem25 If120583 is supermodular then the relationgt120583is cycle-

free

Proof We will only show the case of nonexistence of a 3-cycle here the concept generalizes directly to any other caseAssume that 119909gt

120583119910 and 119910gt

120583119911 First we note that

[119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+

= [119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+

(A4)

and therefore

(119862) int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

= (119862)int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

(A5)

Using Lemma 23 we yield

(119862) int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

119889

gt (119862)int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

(A6)

and from supermodularity of 120583 and submodularity of 120583119889

(119862) int [119910 minus 119909]+119889120583

119889+ (119862)int [119911 minus 119910]

+119889120583

119889+ (119862)int [119909 minus 119911]

+119889120583

119889

gt (119862)int [119909 minus 119910]+119889120583 + (119862)int [119910 minus 119911]

+119889120583

+ (119862)int [119911 minus 119909]+119889120583

(A7)

However 119909gt120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889

and 119910gt120583119911 means (119862) int [119910 minus 119911]+119889120583 ge (119862) int [119911 minus 119910]

+119889120583

119889Therefore above inequality can only hold if

(119862) int [119911 minus 119909]+119889120583 lt (119862)int [119909 minus 119911]

+119889120583

119889 (A8)

which means 119911 gt120583119909

If the measure 120583 is supermodular also transitivity ofthe relation can be shown Before we can show this weneed to introduce a notation and a corresponding lemmaThe notation refers to computing the (asymmetric) Choquetintegral of 119909 by measure 120583 in a ldquodifferent orderrdquo If 119909

119909(119894)

indicates the 119894th largest element of 119909 then the Choquetintegral is calculated as

(119862) int 119909119889120583 = 119909119909(1)120583119909(1)

+

119899

sum

119894=2

119909119909(119894)[120583

119909(1)119909(2)sdotsdotsdot119909(119894)minus 120583

119909(1)sdotsdotsdot119909(119894minus1)]

(A9)

For any other119910 we can compute the same expression justfollowing the order of the elements of 119910 instead of 119909 Then119909119910(119894)

indicates the element of119909with the index of the 119894th largestelement of 119910 and we define (for convenience we still use thesymbol for the Choquet integral keeping in mind that it isnot a real fuzzy integral anymore)

(119862) int(119910)

119909119889120583 = 119909119910(1)120583119910(1)

+

119899

sum

119894=2

119909119910(119894)[120583

119910(1)119910(2)sdotsdotsdot119910(119894)minus 120583

119910(1)sdotsdotsdot119910(119894minus1)]

(A10)

Journal of Applied Mathematics 11

Consequently we could also write (119862) int(119909)119909119889120583 for the

ldquooriginalrdquo Choquet integral We can do the same for theasymmetric Choquet integral by applying the resorting to thenegative components of the vector 119909 as well and will use thenotation (ASC) int

(119910)119909 119889120583 here Now we have the following

Lemma 26 If 120583 is a supermodular measure then for any119910 ( ASC ) int

(119910)119909 119889120583 ge ( ASC ) int

(119909)119909119889120583

Proof A moment of reasoning gives that any sorting of theindizes can be achieved by suitable application of a sequenceof three kinds of ldquoneighbour swapsrdquo starting from the order119909(1)119909(2) sdot sdot sdot 119909(119899))

Swap 1 In the computation of the integral a larger orequal nonnegative element 119889

119894at index 119894 is swapped with its

immediate nonnegative neighbour 119889119894+1

to the right Thenwhat was computed as 119889

119894(120583

119886119894minus120583

119886) +119889

119894+1(120583

119886119894119894+1minus120583

119886119894) before

the swap changes to 119889119894+1(120583

119886119894+1minus120583

119886) + 119889

119894(120583

119886119894119894+1minus120583

119886119894+1) after

the swap all other parts of the resorted asymmetric Choquetintegral expression remain the same (by 119886 we indicate theindex order before index 119894) Thus the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886119894)

minus 119889119894+1(120583

119886119894119894+1minus 120583

119886119894minus 120583

119886119894+1+ 120583

119886119894)

(A11)

and from 119889119894ge 119889

119894+1and 120583

119886119894119894+1+ 120583

119886ge 120583

119886119894+1+ 120583

119886119894

(supermodularity of 120583) it directly follows that the computedvalue will not decrease by this swap

Swap 2 In the computation of the integral a larger orequal negative element 119889

119894+1at index 119894 + 1 is swapped with

its immediate negative left neighbour 119889119894 From a similar

evaluation as for Swap 1 it can be seen that also here the valueof the computed expression will not decrease after the swap

Swap 3 A nonnegative element at index 119894 is swapped withits immediate negative neighbor to the right with index(119894 + 1) Then before the swap we compute 119889

119894(120583

119886119894minus 120583

119886) minus

119889119894+1(120583

119886119894119894+1minus 120583

119886119894) (119889

119894+1is the absolute value of the negative-

valued neighbour of 119889119894) and after the swap this part of the

expression changes to minus119889119894+1(120583

119886119894+1minus 120583

119886) + 119889

119894(120583

119886119894119894+1minus 120583

119886119894+1)

Then the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886)

minus 119889119894+1(120583

119886119894+1minus 120583

119886minus 120583

119886119894119894+1+ 120583

119886119894)

(A12)

Supermodularity of 120583 gives 120583119886119894+ 120583

119886119894+1le 120583

119886+ 120583

119886119894119894+1and

therefore the factor of 119889119894in former expression is nonnegative

and the factor of 119889119894+1

(which is the negation of the factor of119889119894) is negative or 0Therefore the total change is nonnegative

as wellThus with each of these swap operations we will never

decrease the value of the expression and starting from theasymmetric Choquet integral for 119909 upon reaching the finalorder the final value will be larger or equal

An example might be helpful Assume that the order in 119909is (1 2 3 minus 4 minus 5) (a negative index should indicate that

the corresponding element of 119909 is negative) and in 119910 we havethe order (minus5 3 1 2 minus4) (so 119910

5is the largest component of

119910 and 1199104the smallest) Then the order of swaps is as follows

Swap 1 applied as (1 2 3larr997888rarr

minus4 minus5) gives (1 3 2 minus4 minus5)Swap 1 applied as (1 3

larr997888rarr2 minus4 minus5) gives (3 1 2 minus4 minus5)

Next Swap 2 applied as (3 1 2 minus4 minus 5larr997888997888997888997888rarr

) gives (3 1 2 minus

5 minus 4) Swap 3 applied as (3 1 2 minus 5larr997888997888997888rarr

minus 4) gives (3 1 minus

5 2 minus 4) Swap 3 applied as (3 1 minus 5larr997888997888997888rarr

2 minus 4) gives (3 minus

5 1 2 minus4) and finally Swap 3 applied as (3 minus 5larr997888997888997888rarr

1 2 minus4)

gives the order (minus5 3 1 2 minus 4) of 119910 Note also that in casethe elements of 119909 are not in that order we can always relabelthe indizes and measures correspondingly

This means if we compute (using 119909 =

(1199091 119909

2 119909

3 minus119909

4 minus119909

5))

(ASC) int(119910)

119909119889120583

= minus11990951205835+ 119909

3(120583

35minus 120583

5) + 119909

1(120583

135minus 120583

35)

+ 1199092(120583

1235minus 120583

135) minus 119909

4(120583

12345minus 120583

1235)

(A13)

instead of

(ASC) int 119909 119889120583

= 11990911205831+ 119909

2(120583

12minus 120583

1) + 119909

3(120583

123minus 120583

12)

minus 1199094(120583

1234minus 120583

123) minus 119909

5(120583

12345minus 120583

1234)

(A14)

then (ASC) int(119910)119909 119889120583 ge (ASC) int 119909 119889120583

Using this Lemma we can now easily show the following

Theorem 27 If 120583 is a supermodular measure then ge120583is a

transitive relation

Proof For transitivity we have to show that for any 119909 119910 and119911 such that 119909ge

120583119910 and 119910ge

120583119911 it follows that 119909ge

120583119911 Now 119909ge

120583119910

means (ASC) int(119909 minus 119910)119889120583 ge 0 and 119910ge120583119911 means (ASC) int(119910 minus

119911)119889120583 ge 0 From supermodularity of 120583 and Lemma 26 then italso follows that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 ge (ASC) int (119909 minus 119910) 119889120583 ge 0

(ASC) int(119909minus119911)

(119910 minus 119911) 119889120583 ge (ASC) int (119910 minus 119911) 119889120583 ge 0(A15)

However by using the fact that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 + (ASC) int(119909minus119911)

(119910 minus 119911) 119889120583

= (ASC) int (119909 minus 119911) 119889120583(A16)

we see that also

(ASC) int (119909 minus 119911) 119889120583 ge 0 (A17)

which means 119909ge120583119911

12 Journal of Applied Mathematics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Bertsimas V F Farias and N Trichakis ldquoThe price offairnessrdquo Operations Research vol 59 no 1 pp 17ndash31 2011

[2] J F Nash Jr ldquoThe bargaining problemrdquo Econometrica vol 18pp 155ndash162 1950

[3] E Kalai and M Smorodinsky ldquoOther solutions to Nashrsquosbargaining problemrdquo Econometrica vol 43 pp 513ndash518 1975

[4] J Rawls ATheory of Justice Harvard University Press 1999[5] F Kelly ldquoCharging and rate control for elastic trafficrdquo European

Transactions onTelecommunications vol 8 no 1 pp 33ndash37 1997[6] D P Bertsekas R G Gallager and P Humblet Data Networks

vol 2 Prentice-Hall International 1992[7] H P Young Equity In Theory and Practice Princeton Univer-

sity Press Princeton NJ USA 1995[8] J Mo and J Walrand ldquoFair end-to-end window-based conges-

tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[9] M Alicherry R Bhatia and L E Li ldquoJoint channel assignmentand routing for throughput optimization inmulti-radiowirelessmesh networksrdquo in Proceedings of the 11th Annual InternationalConference on Mobile Computing and Networking (MobiCom05) pp 58ndash72 ACM Cologne Germany August-September2005

[10] A Iyer C Rosenberg and A Karnik ldquoWhat is the rightmodel for wireless channel interferencerdquo IEEE Transactions onWireless Communications vol 8 no 5 pp 2662ndash2671 2009

[11] A Mishra V Shrivastava S Banerjee and W ArbaughldquoPartially overlapped channels not considered harmfulrdquo ACMSIGMETRICS Performance Evaluation Review vol 34 no 1 pp63ndash74 2006

[12] N Nie and C Comaniciu ldquoAdaptive channel allocation spec-trum etiquette for cognitive radio networksrdquo Mobile Networksand Applications vol 11 no 6 pp 779ndash797 2006

[13] C Peng H Zheng and B Y Zhao ldquoUtilization and fairnessin spectrum assignment for opportunistic spectrum accessrdquoMobile Networks and Applications vol 11 no 4 pp 555ndash5762006

[14] X Yang andA P Petropulu ldquoCo-channel interferencemodelingand analysis in a Poisson field of interferers in wireless commu-nicationsrdquo IEEE Transactions on Signal Processing vol 51 no 1pp 64ndash76 2003

[15] M Koppen ldquoRelational optimization and its application frombottleneck flow control to wireless channel allocationrdquo Infor-matica vol 24 no 3 pp 413ndash433 2013

[16] M Sugeno Theory of fuzzy integrals and its applications [PhDthesis] 1974

[17] S Michio ldquoFuzzymeasures and fuzzy integrals a surveyrdquo FuzzyAutomata and Decision Processes vol 78 no 33 pp 89ndash1021977

[18] D Denneberg Non-Additive Measure and Integral vol 27Springer 1994

[19] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[20] J Sipos ldquoIntegral with respect to a pre-measurerdquo MathematicaSlovaca vol 29 no 2 pp 141ndash155 1979

[21] M Grabisch and C Labreuche ldquoThe symmetric and asymmet-ric Choquet integrals on finite spaces for decision makingrdquoStatistical Papers vol 43 no 1 pp 37ndash52 2002

[22] M Grabisch C Labreuche and J-C Vansnick ldquoOn theextension of pseudo-Boolean functions for the aggregation ofinteracting criteriardquo European Journal of Operational Researchvol 148 no 1 pp 28ndash47 2003

[23] R Mesiar A Mesiarova-Zemankova and K Ahmad ldquoDiscreteChoquet integral and some of its symmetric extensionsrdquo FuzzySets and Systems vol 184 pp 148ndash155 2011

[24] A Mesiarova-Zemankova R Mesiar and K Ahmad ldquoThebalancing Choquet integralrdquo Fuzzy Sets and Systems vol 161no 17 pp 2243ndash2255 2010

[25] H Fouchal X Gandibleux and F Lehuede ldquoA lower bound ofthe Choquet integral integrated within Martinsalgorithmrdquo inNewState ofMCDM in the 21st Century vol 648 ofLectureNotesin Economics and Mathematical Systems pp 79ndash89 SpringerHeidelberg Germany 2011

[26] L Galand L Julien and P Patrice ldquoDominance rules forthe choquet integral in multiobjective dynamic programmingrdquoin Proceedings of the 23rd International Joint Conference onArtificial Intelligence pp 538ndash544 AAAI Press 2013

[27] A W Marshall I Olkin and B C Arnold Inequalities TheoryofMajorization and Its Applications Springer 2nd edition 2010

[28] W Ogryczak AWierzbicki andMMilewski ldquoAmulti-criteriaapproach to fair and efficient bandwidth allocationrdquoOmega vol36 no 3 pp 451ndash463 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

+ sum

119880sube119904cup119905119880cap119904 = 0

119880cap119905 = 0

+ sum

119880sube119903cup119905

119880cap119903 = 0119880cap119905 = 0

+ sum

119880sube119903cup119904cup119905119880cap119903 = 0

119880cap119904 = 0119880cap119905 = 0

)119898120583(119880) + sum

119880sube119904

119898120583(119880)

120583 (119878) + 120583 (119879) = sum

119880sube119878

119898120583(119880) + sum

119880sube119879

119898120583(119880)

= (sum

119880sube119903

+ sum

119880sube119904

+ sum

119880sube119903cup119904119880cap119903 = 0

119880cap119904 = 0

)119898120583(119880)

+ (sum

119880sube119904

+ sum

119880sube119905

+ sum

119880sube119904cup119905119880cap119904 = 0

119880cap119905 = 0

)119898120583(119880)

(19)

so that we obtain

(120583 (119878 cup 119879) + 120583 (119904)) minus (120583 (119878) + 120583 (119879))

= ( sum

119880sube119903cup119905

119880cap119903 = 0119880cap119905 = 0

+ sum

119880sube119903cup119904cup119905119880cap119903 = 0

119880cap119904 = 0119880cap119905 = 0

)119898120583(119880) ge 0

(20)

Remark 19 The converse of Proposition 18 is not necessarilytrue In fact define119898120583 on119873 = 1 2 3 by

119898120583(119878) =

0 119878 = 0

1 119878 = 0 1 2 3

minus1

2 119878 = 1 2 3

(21)

Then we obtain a supermodular fuzzy measure

120583 (0) = 0 120583 (1) = 120583 (2) = 120583 (3) = 1

120583 (1 2) = 120583 (1 3) = 120583 (2 3) = 3

120583 (1 2 3) =11

2

(22)

4 Fairness Relation Using the AsymmetricChoquet Integral

Definition 20 (Choquet Integral relation) Let 119909 = (1199091

1199092 119909

119899) let 119910 = (119910

1 119910

2 119910

119899) and let 120583 be a fuzzy

measure 119909 is said to CI-dominate 119910 denoted by 119909ge120583119910 if and

only if

(ASC) int (119909 minus 119910) 119889120583 ge 120579 (23)

Assume119873 is a set of all players 119909 and 119910 are distributionsof resources to all players Then 119909 minus 119910 means each degreeof satisfaction of the distribution 119909 compared with thedistribution 119910 Since the more players are satisfied the fairerthe distribution is 119909ge

120583119910 means 119909 is at least as fair as 119910 By

119909gt120583119910 we denote the corresponding strict relation

Proposition 21 Let120583 be a superadditive fuzzymeasure on 2119873Then it is an antisymmetric relation that is 119909ge

120583119910 and 119910ge

120583119909

implies 119909 = 119910 for any functions 119909 119910 on119873

Proposition 22 If 120583 is a supermodular measure on 2119873 thengt120583is a transitive relation that is if 119909ge

120583119910 and 119910ge

120583119911 then 119909ge

120583119911

for any functions 119909 119910 119911 on119873

By Propositions 21 and 22 using the asymmetric Choquetintegral with respect to supermodular functions we obtain afair ordering relation

The proofs for Propositions 21 and 22 will be given in theAppendix of this paper

5 Decision Making by CI-Fairness ANumerical Example

In this section we want to demonstrate how CI-fairnessrelation can be used for decisionmaking Consider a situationwhere resources have to be allocated to 3 agents 119860 119861 and 119862but sets of agents can utilize resources differently (eg givinga commodity to one agent how many items can she producewithin a time unit) Such resource utilization can be assessedby a fuzzy measure

subset 119878 120583 (119878)

0 0

119860 01

119861 03

119862 05

119860 119861 05

119860 119862 05

119861 119862 08

119860 119861 119862 1

(24)

Here for example 119862 appears to be the best in resourceutilization but 119860 is good in collaboration The measure issuperadditive We can also see that this is a supermodularmeasure There are 3 cases to check the other follow fromsubsethood or superadditivity

(1) 120583(119860 119861cup119860 119862)+120583(119860 119861cap119860 119862) = 120583(119860 119861 119862)+120583(119860) = 11 ge 120583(119860 119861) + 120583(119860 119862) = 10

(2) 120583(119860 119861cup 119861 119862)+120583(119860 119861cap 119861 119862) = 120583(119860 119861 119862)+120583(119861) = 13 ge 120583(119860 119861) + 120583(119861 119862) = 13

8 Journal of Applied Mathematics

(1 2 3)

(1 3 2)

(2 3 1) (3 1 2)(2 1 3)

(3 2 1)

Figure 2 CI-Fairness relation among six possible allocations Theconnections include transitivity

(3) 120583(119860 119862cup119861 119862)+120583(119860 119862cap119861 119862) = 120583(119860 119861 119862)+120583(119862) = 15 ge 120583(119860 119862) + 120583(119861 119862) = 13

(1 2 3) ge120583 (1 3 2)

(1 2 3) ge120583 (2 1 3)

(1 2 3) ge120583 (2 3 1)

(1 2 3) ge120583 (3 1 2)

(1 2 3) ge120583 (3 2 1)

(1 3 2) ge120583 (2 3 1)

(1 3 2) ge120583 (3 1 2)

(1 3 2) ge120583 (3 2 1)

(2 1 3) ge120583 (2 3 1)

(2 1 3) ge120583 (3 1 2)

(2 1 3) ge120583 (3 2 1)

(2 3 1) ge120583 (3 2 1)

(3 1 2) ge120583 (3 2 1)

(25)

We find the allocation (1 2 3) in relation to all other alloca-tions but no allocation is in relation to (1 2 3)This is naturalgiven the order of resource utilization weights for the threeagents By rank we can now also order the other 5 allocations

Now we want to allocate three shares of a resource of6 units (1 2 3) among 119860 119861 119862 and compare the 6 possiblepermutations of (1 2 3) by the Choquet integral relationwith120579 = 0 For example to see if allocation 119909 = (1 2 3) is inrelation to the allocation 119910 = (2 3 1) we have to test

(119862) int [119909 minus 119910]+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (26)

In this case [119909 minus 119910]+ = (0 0 2) and [119910 minus 119909]+ = (1 1 0) Sothe first integral becomes 2 times 120583(119862) = 10 and the secondintegral 1 times 120583119889(119860 119861) = (1 minus 120583(119862)) = 05 From 10 ge 05then 119909ge

120583119910 follows

Among all possible 30 pairs (119909 119910) with 119909 = 119910 we can find13 cases in total where the Choquet integral relation holds

Figure 2 shows the Hasse diagram of the relation for the 6possible allocations We can see that as an effect of the bettercollaboration weight of 119860 allocation (2 1 3) is promotedand appears on the same rank as the strong allocation(1 3 2) (strong with regard to single agent utilization ofa resource) and also of better rank than the allocation(2 3 1)mdashexpressing the preference for 119862 as partner of 119860when comparing possible wins and losses in the allocationFinally as expected there is no case to allocate 3 shares to 119860

6 Experiments

In this section we will show how the CI-fairness can beapplied in a practical situation We study wireless channelallocation in the network layer a base station BS that isconnected to the backbone can command over a number119898 of wireless transmission channels and allocates them to anumber 119899 of users (or subscriber stations mobile stationsrelay stations etcmdashdepending on the specific network archi-tecture) for uplink or downlink traffic In the most abstractnotation the BSmakes the allocation based on ameasured orestimated channel state information if it allocates a channel 119894to user 119895 there will be a channel coefficient 119862

119894119895from [0 1] that

represents (in a simplified form) to what degree the user canemploy that channel For example remote users are likely tohave smaller channel coefficients than users that are close tothe BS Here an allocation is seen as feasible if each channelis allocated to one user and to each user at least one channelis assigned

The sum of all channel coefficients of allocated channelsfor a specific user is seen as the performance of the allocationfor that user Then the optimality task is related to theperformance vector for all users This is a direct reference toa relation between performance vectors and various fairnessrelations can be easily and conveniently studied in thiscontext If the number of channels and users is not too largeexhaustive search for maximum sets is possible For examplefor 4 users and 6 channels there are 1560 feasible allocationswhich means about 24 Mio pairwise comparisons

In this model channel allocation ignores low channelinterferences between users and specific channels by therestriction that each channel is allocated to exactly one userHowever if implementing a wireless access where some userscan use the same channel for transmission it might be hardto decide the exact schedule of such multiple assignmentsor to decide if installing such a scheme at all Thereforewe follow a more simplified approach channel interferencewill be represented by a conflict matrix between userswhere low elements for pairs of users indicate the optionto use multichannel assignment Then we use the converseelements of the conflict matrix as masses in the Mobiustransform More specifically the Mobius masses for singleelement sets are set to equal values the masses for two-element are set to 1 minus the corresponding element of theconflictmatrix and all other are set to 0Then themassvectoris normalized to a total of all masses of 1 to ensure that

Journal of Applied Mathematics 9

Table 1 Comparison of total performances for capacity propor-tional and maxmin fairness

Fairness relation Average performanceConflict matrix range 0 to 1

CI 42882Proportional 454661maxmin 414688

Conflict matrix range 09 to 1CI 456051Proportional 451135maxmin 422561

Conflict matrix range 0 to 02CI 409365Proportional 441848maxmin 404879

for the fuzzy measure the measure for the whole set is 1The inverse Mobius transform of such a mass vector gives asupermodularmeasure that can be used to specify a CI-basedfairness relation according to Definition 20

Now this relation is applied to all feasible channelallocationsmdashit means we are not considering a specific use ofmultichannel assignment but we are looking for the influencethat such amultichannel allocation could have on the specificselection of maximal elements Then the focus is on thedisadvantage that some pairs of users can experience if theirfavoured situation (eg to be distant enough to use the samechannel) is not taken into account The similar approach canbe considered for cognitive radio opportunistic networkingP2P vehicular networks and so forth

We might especially look for the price of fairness [1]compared to a standard proportional fair or maxmin fairallocation

Some example results are shown in Table 1 There for thecase of 4 users and 6 cells the maximum sets over all feasibleallocations were computed for CI-fairness using a measure asdescribed above proportional fairness andmaxmin fairnessThe conflictmatrix elements for the CI-fairness were uniformrandomly set within specific ranges All results are averagedover 30 repetitions Note that average sizes of maximum setswere found to be 42 formaxmin fairness 61 for proportionalfairness and 419 for CI-fairness so the CI-fairness produceslarger maximum sets an issue that should be addressed infutureworksThe table now shows average performances overall maximal elements

We see with regard to efficiency the well-known fact thatproportional-fair allocations are in average more efficientthan maxmin fair allocations The CI-fairness appears toselect maximal relations with a total performance betweenproportional fairness and maxmin fairness In case of stronginterferences (where a multichannel assignment is not rea-sonable) it is very close to proportional fairness while in caseof low interference it appears to be more close to maxminfairness that is the fairness relation favouring least elementsThus such an empirical result can be understood in the sensethat the neglecting of the potential multichannel assignment

corresponds with the neglection of strong users This is alsoparticularly appearing for the general case where all conflictmatrix elements were randomly selected between 0 and 1 alsohere the average performance appearsmore close tomaxminfairness than proportional fairness

7 Conclusion

A fairness relation among vectors based on the asymmetricChoquet integral was studied It formally follows the Nashstandard of comparison where the relative losses and winsare replaced by absolute losses and wins but instead takingweights for groups of vector components into account Thusit can represent for example pairs of users that are ina favourable situation regarding resource allocation Theappealing points of this CI-relation are as follows (1) Incase a supermodular measure is used for the integrationthe relation will be transitive which suits faster search formaximal elements (2) it can be parametrized by givinga weight to each subset of vector components (while inpractice this might be relevant only for smaller subsets) Therepresentation of the fuzzy measure by its Mobius transformappears to be a convenient way to yield a supermodularfuzzy measure the masses just need to be all nonnegativeThus the relation can be conveniently applied in many prac-tical applications to specify optimality Slight disadvantagesare with the complexity of the involved calculations andsome empirical evidence for specifying larger maximum setsthan other fairness relations like proportional fairness andmaxmin fairness

In case of wireless channel allocation how CI-fairnesscan be used to help deciding whether implementations ofmore complex allocation schemes are indeed worth the effortwas suggested This method can be applied to many otherproblems of higher efficiency of network resource utilizationIn future work we will consider the use of advanced fuzzyintegrals like balancing and fusion Choquet integral andthe adjustment of related CI-fairness parameters to specificsituations and what can be concluded from such parametersWe will also more intensively study the performance issueswith regard to computational effort and search for maximalelements and study conditions where also greatest elementsexist

Appendix

Proofs of Theorems

In the following we provide the proofs for Propositions 21and 22 For formal convenience of the proofs the notationwill be a little bit changing therefore we repeat the basicdefinitions

We define the Choquet-integral-based (fairness) relationas follows be 119909 and 119910 two vectors from 119877

119899 By [119909]+ weindicate the positive support of a vector 119909 that is the 119894thcomponent of [119909]+ is 119909

119894if 119909

119894ge 0 or 0 if 119909

119894lt 0 The Choquet

integral is denoted by (119862) int119909 119889120583 There 120583 is a fuzzy measure

10 Journal of Applied Mathematics

Also 120583119889 denotes its dual measure (ie 120583119889(119860) = 120583119883minus 120583(119860

119862))

Then 119909 is said to CI-dominate 119910 if and only if

119909ge120583119910 larrrarr (119862)int [119909 minus 119910]

+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (A1)

We can alternatively rewrite this as the asymmetric Choquetintegral

119909ge120583119910 larrrarr (ASC) int (119909 minus 119910) 119889120583 ge 0 (A2)

since the right-hand side condition of (A1) exactly cor-responds with the definition of the asymmetric Choquetintegral

First we can show the following

Lemma 23 If 120583 is a superadditive fuzzy measure where foreach 119860 119861 = 0 and 119860 cap 119861 = 0 120583

119860+ 120583

119861lt 120583

119860cup119861 then for

119860 = 0 120583119889

119860gt 120583

119860

Proof Since 120583 is superadditive for119860 = 0 120583119860+120583

119860119862 lt 120583

119860cup119860119862 =

120583119883 Then 120583

119860lt 120583

119883minus 120583

119860119862 = 120583

119889

119860follows directly

Theorem 24 If 120583 is superadditive then from 119909ge120583119910 and

119910ge120583119909119909 = 119910 follows

Proof 119909ge120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889 If

119909 = 119910 we can use Lemma 23 to obtain

(119862) int [119909 minus 119910]+119889120583

119889gt (119862)int [119909 minus 119910]

+119889120583

ge (119862)int [119910 minus 119909]+119889120583

119889gt (119862)int [119910 minus 119909]

+119889120583

(A3)

that is (119862) int [119910 minus 119909]+119889120583 ge (119862) int [119909 minus 119910]+119889120583

119889 This wouldmean that 119910ge

120583119909 would not hold Therefore it must be 119909 =

119910

For a supermodular measure and nonnegative vectors 119909and 119910 we have the known property (119862) int 119909 119889120583+ (119862) int 119910 119889120583 le(119862) int(119909 + 119910)119889120583 Since the dual measure of a supermodularmeasure is submodular it also holds that (119862) int 119909 119889120583 +(119862) int 119910 119889120583 ge (119862) int(119909 + 119910)119889120583 Using this the following canbe shown

Theorem25 If120583 is supermodular then the relationgt120583is cycle-

free

Proof We will only show the case of nonexistence of a 3-cycle here the concept generalizes directly to any other caseAssume that 119909gt

120583119910 and 119910gt

120583119911 First we note that

[119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+

= [119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+

(A4)

and therefore

(119862) int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

= (119862)int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

(A5)

Using Lemma 23 we yield

(119862) int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

119889

gt (119862)int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

(A6)

and from supermodularity of 120583 and submodularity of 120583119889

(119862) int [119910 minus 119909]+119889120583

119889+ (119862)int [119911 minus 119910]

+119889120583

119889+ (119862)int [119909 minus 119911]

+119889120583

119889

gt (119862)int [119909 minus 119910]+119889120583 + (119862)int [119910 minus 119911]

+119889120583

+ (119862)int [119911 minus 119909]+119889120583

(A7)

However 119909gt120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889

and 119910gt120583119911 means (119862) int [119910 minus 119911]+119889120583 ge (119862) int [119911 minus 119910]

+119889120583

119889Therefore above inequality can only hold if

(119862) int [119911 minus 119909]+119889120583 lt (119862)int [119909 minus 119911]

+119889120583

119889 (A8)

which means 119911 gt120583119909

If the measure 120583 is supermodular also transitivity ofthe relation can be shown Before we can show this weneed to introduce a notation and a corresponding lemmaThe notation refers to computing the (asymmetric) Choquetintegral of 119909 by measure 120583 in a ldquodifferent orderrdquo If 119909

119909(119894)

indicates the 119894th largest element of 119909 then the Choquetintegral is calculated as

(119862) int 119909119889120583 = 119909119909(1)120583119909(1)

+

119899

sum

119894=2

119909119909(119894)[120583

119909(1)119909(2)sdotsdotsdot119909(119894)minus 120583

119909(1)sdotsdotsdot119909(119894minus1)]

(A9)

For any other119910 we can compute the same expression justfollowing the order of the elements of 119910 instead of 119909 Then119909119910(119894)

indicates the element of119909with the index of the 119894th largestelement of 119910 and we define (for convenience we still use thesymbol for the Choquet integral keeping in mind that it isnot a real fuzzy integral anymore)

(119862) int(119910)

119909119889120583 = 119909119910(1)120583119910(1)

+

119899

sum

119894=2

119909119910(119894)[120583

119910(1)119910(2)sdotsdotsdot119910(119894)minus 120583

119910(1)sdotsdotsdot119910(119894minus1)]

(A10)

Journal of Applied Mathematics 11

Consequently we could also write (119862) int(119909)119909119889120583 for the

ldquooriginalrdquo Choquet integral We can do the same for theasymmetric Choquet integral by applying the resorting to thenegative components of the vector 119909 as well and will use thenotation (ASC) int

(119910)119909 119889120583 here Now we have the following

Lemma 26 If 120583 is a supermodular measure then for any119910 ( ASC ) int

(119910)119909 119889120583 ge ( ASC ) int

(119909)119909119889120583

Proof A moment of reasoning gives that any sorting of theindizes can be achieved by suitable application of a sequenceof three kinds of ldquoneighbour swapsrdquo starting from the order119909(1)119909(2) sdot sdot sdot 119909(119899))

Swap 1 In the computation of the integral a larger orequal nonnegative element 119889

119894at index 119894 is swapped with its

immediate nonnegative neighbour 119889119894+1

to the right Thenwhat was computed as 119889

119894(120583

119886119894minus120583

119886) +119889

119894+1(120583

119886119894119894+1minus120583

119886119894) before

the swap changes to 119889119894+1(120583

119886119894+1minus120583

119886) + 119889

119894(120583

119886119894119894+1minus120583

119886119894+1) after

the swap all other parts of the resorted asymmetric Choquetintegral expression remain the same (by 119886 we indicate theindex order before index 119894) Thus the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886119894)

minus 119889119894+1(120583

119886119894119894+1minus 120583

119886119894minus 120583

119886119894+1+ 120583

119886119894)

(A11)

and from 119889119894ge 119889

119894+1and 120583

119886119894119894+1+ 120583

119886ge 120583

119886119894+1+ 120583

119886119894

(supermodularity of 120583) it directly follows that the computedvalue will not decrease by this swap

Swap 2 In the computation of the integral a larger orequal negative element 119889

119894+1at index 119894 + 1 is swapped with

its immediate negative left neighbour 119889119894 From a similar

evaluation as for Swap 1 it can be seen that also here the valueof the computed expression will not decrease after the swap

Swap 3 A nonnegative element at index 119894 is swapped withits immediate negative neighbor to the right with index(119894 + 1) Then before the swap we compute 119889

119894(120583

119886119894minus 120583

119886) minus

119889119894+1(120583

119886119894119894+1minus 120583

119886119894) (119889

119894+1is the absolute value of the negative-

valued neighbour of 119889119894) and after the swap this part of the

expression changes to minus119889119894+1(120583

119886119894+1minus 120583

119886) + 119889

119894(120583

119886119894119894+1minus 120583

119886119894+1)

Then the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886)

minus 119889119894+1(120583

119886119894+1minus 120583

119886minus 120583

119886119894119894+1+ 120583

119886119894)

(A12)

Supermodularity of 120583 gives 120583119886119894+ 120583

119886119894+1le 120583

119886+ 120583

119886119894119894+1and

therefore the factor of 119889119894in former expression is nonnegative

and the factor of 119889119894+1

(which is the negation of the factor of119889119894) is negative or 0Therefore the total change is nonnegative

as wellThus with each of these swap operations we will never

decrease the value of the expression and starting from theasymmetric Choquet integral for 119909 upon reaching the finalorder the final value will be larger or equal

An example might be helpful Assume that the order in 119909is (1 2 3 minus 4 minus 5) (a negative index should indicate that

the corresponding element of 119909 is negative) and in 119910 we havethe order (minus5 3 1 2 minus4) (so 119910

5is the largest component of

119910 and 1199104the smallest) Then the order of swaps is as follows

Swap 1 applied as (1 2 3larr997888rarr

minus4 minus5) gives (1 3 2 minus4 minus5)Swap 1 applied as (1 3

larr997888rarr2 minus4 minus5) gives (3 1 2 minus4 minus5)

Next Swap 2 applied as (3 1 2 minus4 minus 5larr997888997888997888997888rarr

) gives (3 1 2 minus

5 minus 4) Swap 3 applied as (3 1 2 minus 5larr997888997888997888rarr

minus 4) gives (3 1 minus

5 2 minus 4) Swap 3 applied as (3 1 minus 5larr997888997888997888rarr

2 minus 4) gives (3 minus

5 1 2 minus4) and finally Swap 3 applied as (3 minus 5larr997888997888997888rarr

1 2 minus4)

gives the order (minus5 3 1 2 minus 4) of 119910 Note also that in casethe elements of 119909 are not in that order we can always relabelthe indizes and measures correspondingly

This means if we compute (using 119909 =

(1199091 119909

2 119909

3 minus119909

4 minus119909

5))

(ASC) int(119910)

119909119889120583

= minus11990951205835+ 119909

3(120583

35minus 120583

5) + 119909

1(120583

135minus 120583

35)

+ 1199092(120583

1235minus 120583

135) minus 119909

4(120583

12345minus 120583

1235)

(A13)

instead of

(ASC) int 119909 119889120583

= 11990911205831+ 119909

2(120583

12minus 120583

1) + 119909

3(120583

123minus 120583

12)

minus 1199094(120583

1234minus 120583

123) minus 119909

5(120583

12345minus 120583

1234)

(A14)

then (ASC) int(119910)119909 119889120583 ge (ASC) int 119909 119889120583

Using this Lemma we can now easily show the following

Theorem 27 If 120583 is a supermodular measure then ge120583is a

transitive relation

Proof For transitivity we have to show that for any 119909 119910 and119911 such that 119909ge

120583119910 and 119910ge

120583119911 it follows that 119909ge

120583119911 Now 119909ge

120583119910

means (ASC) int(119909 minus 119910)119889120583 ge 0 and 119910ge120583119911 means (ASC) int(119910 minus

119911)119889120583 ge 0 From supermodularity of 120583 and Lemma 26 then italso follows that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 ge (ASC) int (119909 minus 119910) 119889120583 ge 0

(ASC) int(119909minus119911)

(119910 minus 119911) 119889120583 ge (ASC) int (119910 minus 119911) 119889120583 ge 0(A15)

However by using the fact that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 + (ASC) int(119909minus119911)

(119910 minus 119911) 119889120583

= (ASC) int (119909 minus 119911) 119889120583(A16)

we see that also

(ASC) int (119909 minus 119911) 119889120583 ge 0 (A17)

which means 119909ge120583119911

12 Journal of Applied Mathematics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Bertsimas V F Farias and N Trichakis ldquoThe price offairnessrdquo Operations Research vol 59 no 1 pp 17ndash31 2011

[2] J F Nash Jr ldquoThe bargaining problemrdquo Econometrica vol 18pp 155ndash162 1950

[3] E Kalai and M Smorodinsky ldquoOther solutions to Nashrsquosbargaining problemrdquo Econometrica vol 43 pp 513ndash518 1975

[4] J Rawls ATheory of Justice Harvard University Press 1999[5] F Kelly ldquoCharging and rate control for elastic trafficrdquo European

Transactions onTelecommunications vol 8 no 1 pp 33ndash37 1997[6] D P Bertsekas R G Gallager and P Humblet Data Networks

vol 2 Prentice-Hall International 1992[7] H P Young Equity In Theory and Practice Princeton Univer-

sity Press Princeton NJ USA 1995[8] J Mo and J Walrand ldquoFair end-to-end window-based conges-

tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[9] M Alicherry R Bhatia and L E Li ldquoJoint channel assignmentand routing for throughput optimization inmulti-radiowirelessmesh networksrdquo in Proceedings of the 11th Annual InternationalConference on Mobile Computing and Networking (MobiCom05) pp 58ndash72 ACM Cologne Germany August-September2005

[10] A Iyer C Rosenberg and A Karnik ldquoWhat is the rightmodel for wireless channel interferencerdquo IEEE Transactions onWireless Communications vol 8 no 5 pp 2662ndash2671 2009

[11] A Mishra V Shrivastava S Banerjee and W ArbaughldquoPartially overlapped channels not considered harmfulrdquo ACMSIGMETRICS Performance Evaluation Review vol 34 no 1 pp63ndash74 2006

[12] N Nie and C Comaniciu ldquoAdaptive channel allocation spec-trum etiquette for cognitive radio networksrdquo Mobile Networksand Applications vol 11 no 6 pp 779ndash797 2006

[13] C Peng H Zheng and B Y Zhao ldquoUtilization and fairnessin spectrum assignment for opportunistic spectrum accessrdquoMobile Networks and Applications vol 11 no 4 pp 555ndash5762006

[14] X Yang andA P Petropulu ldquoCo-channel interferencemodelingand analysis in a Poisson field of interferers in wireless commu-nicationsrdquo IEEE Transactions on Signal Processing vol 51 no 1pp 64ndash76 2003

[15] M Koppen ldquoRelational optimization and its application frombottleneck flow control to wireless channel allocationrdquo Infor-matica vol 24 no 3 pp 413ndash433 2013

[16] M Sugeno Theory of fuzzy integrals and its applications [PhDthesis] 1974

[17] S Michio ldquoFuzzymeasures and fuzzy integrals a surveyrdquo FuzzyAutomata and Decision Processes vol 78 no 33 pp 89ndash1021977

[18] D Denneberg Non-Additive Measure and Integral vol 27Springer 1994

[19] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[20] J Sipos ldquoIntegral with respect to a pre-measurerdquo MathematicaSlovaca vol 29 no 2 pp 141ndash155 1979

[21] M Grabisch and C Labreuche ldquoThe symmetric and asymmet-ric Choquet integrals on finite spaces for decision makingrdquoStatistical Papers vol 43 no 1 pp 37ndash52 2002

[22] M Grabisch C Labreuche and J-C Vansnick ldquoOn theextension of pseudo-Boolean functions for the aggregation ofinteracting criteriardquo European Journal of Operational Researchvol 148 no 1 pp 28ndash47 2003

[23] R Mesiar A Mesiarova-Zemankova and K Ahmad ldquoDiscreteChoquet integral and some of its symmetric extensionsrdquo FuzzySets and Systems vol 184 pp 148ndash155 2011

[24] A Mesiarova-Zemankova R Mesiar and K Ahmad ldquoThebalancing Choquet integralrdquo Fuzzy Sets and Systems vol 161no 17 pp 2243ndash2255 2010

[25] H Fouchal X Gandibleux and F Lehuede ldquoA lower bound ofthe Choquet integral integrated within Martinsalgorithmrdquo inNewState ofMCDM in the 21st Century vol 648 ofLectureNotesin Economics and Mathematical Systems pp 79ndash89 SpringerHeidelberg Germany 2011

[26] L Galand L Julien and P Patrice ldquoDominance rules forthe choquet integral in multiobjective dynamic programmingrdquoin Proceedings of the 23rd International Joint Conference onArtificial Intelligence pp 538ndash544 AAAI Press 2013

[27] A W Marshall I Olkin and B C Arnold Inequalities TheoryofMajorization and Its Applications Springer 2nd edition 2010

[28] W Ogryczak AWierzbicki andMMilewski ldquoAmulti-criteriaapproach to fair and efficient bandwidth allocationrdquoOmega vol36 no 3 pp 451ndash463 2008

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

(1 2 3)

(1 3 2)

(2 3 1) (3 1 2)(2 1 3)

(3 2 1)

Figure 2 CI-Fairness relation among six possible allocations Theconnections include transitivity

(3) 120583(119860 119862cup119861 119862)+120583(119860 119862cap119861 119862) = 120583(119860 119861 119862)+120583(119862) = 15 ge 120583(119860 119862) + 120583(119861 119862) = 13

(1 2 3) ge120583 (1 3 2)

(1 2 3) ge120583 (2 1 3)

(1 2 3) ge120583 (2 3 1)

(1 2 3) ge120583 (3 1 2)

(1 2 3) ge120583 (3 2 1)

(1 3 2) ge120583 (2 3 1)

(1 3 2) ge120583 (3 1 2)

(1 3 2) ge120583 (3 2 1)

(2 1 3) ge120583 (2 3 1)

(2 1 3) ge120583 (3 1 2)

(2 1 3) ge120583 (3 2 1)

(2 3 1) ge120583 (3 2 1)

(3 1 2) ge120583 (3 2 1)

(25)

We find the allocation (1 2 3) in relation to all other alloca-tions but no allocation is in relation to (1 2 3)This is naturalgiven the order of resource utilization weights for the threeagents By rank we can now also order the other 5 allocations

Now we want to allocate three shares of a resource of6 units (1 2 3) among 119860 119861 119862 and compare the 6 possiblepermutations of (1 2 3) by the Choquet integral relationwith120579 = 0 For example to see if allocation 119909 = (1 2 3) is inrelation to the allocation 119910 = (2 3 1) we have to test

(119862) int [119909 minus 119910]+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (26)

In this case [119909 minus 119910]+ = (0 0 2) and [119910 minus 119909]+ = (1 1 0) Sothe first integral becomes 2 times 120583(119862) = 10 and the secondintegral 1 times 120583119889(119860 119861) = (1 minus 120583(119862)) = 05 From 10 ge 05then 119909ge

120583119910 follows

Among all possible 30 pairs (119909 119910) with 119909 = 119910 we can find13 cases in total where the Choquet integral relation holds

Figure 2 shows the Hasse diagram of the relation for the 6possible allocations We can see that as an effect of the bettercollaboration weight of 119860 allocation (2 1 3) is promotedand appears on the same rank as the strong allocation(1 3 2) (strong with regard to single agent utilization ofa resource) and also of better rank than the allocation(2 3 1)mdashexpressing the preference for 119862 as partner of 119860when comparing possible wins and losses in the allocationFinally as expected there is no case to allocate 3 shares to 119860

6 Experiments

In this section we will show how the CI-fairness can beapplied in a practical situation We study wireless channelallocation in the network layer a base station BS that isconnected to the backbone can command over a number119898 of wireless transmission channels and allocates them to anumber 119899 of users (or subscriber stations mobile stationsrelay stations etcmdashdepending on the specific network archi-tecture) for uplink or downlink traffic In the most abstractnotation the BSmakes the allocation based on ameasured orestimated channel state information if it allocates a channel 119894to user 119895 there will be a channel coefficient 119862

119894119895from [0 1] that

represents (in a simplified form) to what degree the user canemploy that channel For example remote users are likely tohave smaller channel coefficients than users that are close tothe BS Here an allocation is seen as feasible if each channelis allocated to one user and to each user at least one channelis assigned

The sum of all channel coefficients of allocated channelsfor a specific user is seen as the performance of the allocationfor that user Then the optimality task is related to theperformance vector for all users This is a direct reference toa relation between performance vectors and various fairnessrelations can be easily and conveniently studied in thiscontext If the number of channels and users is not too largeexhaustive search for maximum sets is possible For examplefor 4 users and 6 channels there are 1560 feasible allocationswhich means about 24 Mio pairwise comparisons

In this model channel allocation ignores low channelinterferences between users and specific channels by therestriction that each channel is allocated to exactly one userHowever if implementing a wireless access where some userscan use the same channel for transmission it might be hardto decide the exact schedule of such multiple assignmentsor to decide if installing such a scheme at all Thereforewe follow a more simplified approach channel interferencewill be represented by a conflict matrix between userswhere low elements for pairs of users indicate the optionto use multichannel assignment Then we use the converseelements of the conflict matrix as masses in the Mobiustransform More specifically the Mobius masses for singleelement sets are set to equal values the masses for two-element are set to 1 minus the corresponding element of theconflictmatrix and all other are set to 0Then themassvectoris normalized to a total of all masses of 1 to ensure that

Journal of Applied Mathematics 9

Table 1 Comparison of total performances for capacity propor-tional and maxmin fairness

Fairness relation Average performanceConflict matrix range 0 to 1

CI 42882Proportional 454661maxmin 414688

Conflict matrix range 09 to 1CI 456051Proportional 451135maxmin 422561

Conflict matrix range 0 to 02CI 409365Proportional 441848maxmin 404879

for the fuzzy measure the measure for the whole set is 1The inverse Mobius transform of such a mass vector gives asupermodularmeasure that can be used to specify a CI-basedfairness relation according to Definition 20

Now this relation is applied to all feasible channelallocationsmdashit means we are not considering a specific use ofmultichannel assignment but we are looking for the influencethat such amultichannel allocation could have on the specificselection of maximal elements Then the focus is on thedisadvantage that some pairs of users can experience if theirfavoured situation (eg to be distant enough to use the samechannel) is not taken into account The similar approach canbe considered for cognitive radio opportunistic networkingP2P vehicular networks and so forth

We might especially look for the price of fairness [1]compared to a standard proportional fair or maxmin fairallocation

Some example results are shown in Table 1 There for thecase of 4 users and 6 cells the maximum sets over all feasibleallocations were computed for CI-fairness using a measure asdescribed above proportional fairness andmaxmin fairnessThe conflictmatrix elements for the CI-fairness were uniformrandomly set within specific ranges All results are averagedover 30 repetitions Note that average sizes of maximum setswere found to be 42 formaxmin fairness 61 for proportionalfairness and 419 for CI-fairness so the CI-fairness produceslarger maximum sets an issue that should be addressed infutureworksThe table now shows average performances overall maximal elements

We see with regard to efficiency the well-known fact thatproportional-fair allocations are in average more efficientthan maxmin fair allocations The CI-fairness appears toselect maximal relations with a total performance betweenproportional fairness and maxmin fairness In case of stronginterferences (where a multichannel assignment is not rea-sonable) it is very close to proportional fairness while in caseof low interference it appears to be more close to maxminfairness that is the fairness relation favouring least elementsThus such an empirical result can be understood in the sensethat the neglecting of the potential multichannel assignment

corresponds with the neglection of strong users This is alsoparticularly appearing for the general case where all conflictmatrix elements were randomly selected between 0 and 1 alsohere the average performance appearsmore close tomaxminfairness than proportional fairness

7 Conclusion

A fairness relation among vectors based on the asymmetricChoquet integral was studied It formally follows the Nashstandard of comparison where the relative losses and winsare replaced by absolute losses and wins but instead takingweights for groups of vector components into account Thusit can represent for example pairs of users that are ina favourable situation regarding resource allocation Theappealing points of this CI-relation are as follows (1) Incase a supermodular measure is used for the integrationthe relation will be transitive which suits faster search formaximal elements (2) it can be parametrized by givinga weight to each subset of vector components (while inpractice this might be relevant only for smaller subsets) Therepresentation of the fuzzy measure by its Mobius transformappears to be a convenient way to yield a supermodularfuzzy measure the masses just need to be all nonnegativeThus the relation can be conveniently applied in many prac-tical applications to specify optimality Slight disadvantagesare with the complexity of the involved calculations andsome empirical evidence for specifying larger maximum setsthan other fairness relations like proportional fairness andmaxmin fairness

In case of wireless channel allocation how CI-fairnesscan be used to help deciding whether implementations ofmore complex allocation schemes are indeed worth the effortwas suggested This method can be applied to many otherproblems of higher efficiency of network resource utilizationIn future work we will consider the use of advanced fuzzyintegrals like balancing and fusion Choquet integral andthe adjustment of related CI-fairness parameters to specificsituations and what can be concluded from such parametersWe will also more intensively study the performance issueswith regard to computational effort and search for maximalelements and study conditions where also greatest elementsexist

Appendix

Proofs of Theorems

In the following we provide the proofs for Propositions 21and 22 For formal convenience of the proofs the notationwill be a little bit changing therefore we repeat the basicdefinitions

We define the Choquet-integral-based (fairness) relationas follows be 119909 and 119910 two vectors from 119877

119899 By [119909]+ weindicate the positive support of a vector 119909 that is the 119894thcomponent of [119909]+ is 119909

119894if 119909

119894ge 0 or 0 if 119909

119894lt 0 The Choquet

integral is denoted by (119862) int119909 119889120583 There 120583 is a fuzzy measure

10 Journal of Applied Mathematics

Also 120583119889 denotes its dual measure (ie 120583119889(119860) = 120583119883minus 120583(119860

119862))

Then 119909 is said to CI-dominate 119910 if and only if

119909ge120583119910 larrrarr (119862)int [119909 minus 119910]

+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (A1)

We can alternatively rewrite this as the asymmetric Choquetintegral

119909ge120583119910 larrrarr (ASC) int (119909 minus 119910) 119889120583 ge 0 (A2)

since the right-hand side condition of (A1) exactly cor-responds with the definition of the asymmetric Choquetintegral

First we can show the following

Lemma 23 If 120583 is a superadditive fuzzy measure where foreach 119860 119861 = 0 and 119860 cap 119861 = 0 120583

119860+ 120583

119861lt 120583

119860cup119861 then for

119860 = 0 120583119889

119860gt 120583

119860

Proof Since 120583 is superadditive for119860 = 0 120583119860+120583

119860119862 lt 120583

119860cup119860119862 =

120583119883 Then 120583

119860lt 120583

119883minus 120583

119860119862 = 120583

119889

119860follows directly

Theorem 24 If 120583 is superadditive then from 119909ge120583119910 and

119910ge120583119909119909 = 119910 follows

Proof 119909ge120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889 If

119909 = 119910 we can use Lemma 23 to obtain

(119862) int [119909 minus 119910]+119889120583

119889gt (119862)int [119909 minus 119910]

+119889120583

ge (119862)int [119910 minus 119909]+119889120583

119889gt (119862)int [119910 minus 119909]

+119889120583

(A3)

that is (119862) int [119910 minus 119909]+119889120583 ge (119862) int [119909 minus 119910]+119889120583

119889 This wouldmean that 119910ge

120583119909 would not hold Therefore it must be 119909 =

119910

For a supermodular measure and nonnegative vectors 119909and 119910 we have the known property (119862) int 119909 119889120583+ (119862) int 119910 119889120583 le(119862) int(119909 + 119910)119889120583 Since the dual measure of a supermodularmeasure is submodular it also holds that (119862) int 119909 119889120583 +(119862) int 119910 119889120583 ge (119862) int(119909 + 119910)119889120583 Using this the following canbe shown

Theorem25 If120583 is supermodular then the relationgt120583is cycle-

free

Proof We will only show the case of nonexistence of a 3-cycle here the concept generalizes directly to any other caseAssume that 119909gt

120583119910 and 119910gt

120583119911 First we note that

[119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+

= [119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+

(A4)

and therefore

(119862) int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

= (119862)int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

(A5)

Using Lemma 23 we yield

(119862) int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

119889

gt (119862)int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

(A6)

and from supermodularity of 120583 and submodularity of 120583119889

(119862) int [119910 minus 119909]+119889120583

119889+ (119862)int [119911 minus 119910]

+119889120583

119889+ (119862)int [119909 minus 119911]

+119889120583

119889

gt (119862)int [119909 minus 119910]+119889120583 + (119862)int [119910 minus 119911]

+119889120583

+ (119862)int [119911 minus 119909]+119889120583

(A7)

However 119909gt120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889

and 119910gt120583119911 means (119862) int [119910 minus 119911]+119889120583 ge (119862) int [119911 minus 119910]

+119889120583

119889Therefore above inequality can only hold if

(119862) int [119911 minus 119909]+119889120583 lt (119862)int [119909 minus 119911]

+119889120583

119889 (A8)

which means 119911 gt120583119909

If the measure 120583 is supermodular also transitivity ofthe relation can be shown Before we can show this weneed to introduce a notation and a corresponding lemmaThe notation refers to computing the (asymmetric) Choquetintegral of 119909 by measure 120583 in a ldquodifferent orderrdquo If 119909

119909(119894)

indicates the 119894th largest element of 119909 then the Choquetintegral is calculated as

(119862) int 119909119889120583 = 119909119909(1)120583119909(1)

+

119899

sum

119894=2

119909119909(119894)[120583

119909(1)119909(2)sdotsdotsdot119909(119894)minus 120583

119909(1)sdotsdotsdot119909(119894minus1)]

(A9)

For any other119910 we can compute the same expression justfollowing the order of the elements of 119910 instead of 119909 Then119909119910(119894)

indicates the element of119909with the index of the 119894th largestelement of 119910 and we define (for convenience we still use thesymbol for the Choquet integral keeping in mind that it isnot a real fuzzy integral anymore)

(119862) int(119910)

119909119889120583 = 119909119910(1)120583119910(1)

+

119899

sum

119894=2

119909119910(119894)[120583

119910(1)119910(2)sdotsdotsdot119910(119894)minus 120583

119910(1)sdotsdotsdot119910(119894minus1)]

(A10)

Journal of Applied Mathematics 11

Consequently we could also write (119862) int(119909)119909119889120583 for the

ldquooriginalrdquo Choquet integral We can do the same for theasymmetric Choquet integral by applying the resorting to thenegative components of the vector 119909 as well and will use thenotation (ASC) int

(119910)119909 119889120583 here Now we have the following

Lemma 26 If 120583 is a supermodular measure then for any119910 ( ASC ) int

(119910)119909 119889120583 ge ( ASC ) int

(119909)119909119889120583

Proof A moment of reasoning gives that any sorting of theindizes can be achieved by suitable application of a sequenceof three kinds of ldquoneighbour swapsrdquo starting from the order119909(1)119909(2) sdot sdot sdot 119909(119899))

Swap 1 In the computation of the integral a larger orequal nonnegative element 119889

119894at index 119894 is swapped with its

immediate nonnegative neighbour 119889119894+1

to the right Thenwhat was computed as 119889

119894(120583

119886119894minus120583

119886) +119889

119894+1(120583

119886119894119894+1minus120583

119886119894) before

the swap changes to 119889119894+1(120583

119886119894+1minus120583

119886) + 119889

119894(120583

119886119894119894+1minus120583

119886119894+1) after

the swap all other parts of the resorted asymmetric Choquetintegral expression remain the same (by 119886 we indicate theindex order before index 119894) Thus the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886119894)

minus 119889119894+1(120583

119886119894119894+1minus 120583

119886119894minus 120583

119886119894+1+ 120583

119886119894)

(A11)

and from 119889119894ge 119889

119894+1and 120583

119886119894119894+1+ 120583

119886ge 120583

119886119894+1+ 120583

119886119894

(supermodularity of 120583) it directly follows that the computedvalue will not decrease by this swap

Swap 2 In the computation of the integral a larger orequal negative element 119889

119894+1at index 119894 + 1 is swapped with

its immediate negative left neighbour 119889119894 From a similar

evaluation as for Swap 1 it can be seen that also here the valueof the computed expression will not decrease after the swap

Swap 3 A nonnegative element at index 119894 is swapped withits immediate negative neighbor to the right with index(119894 + 1) Then before the swap we compute 119889

119894(120583

119886119894minus 120583

119886) minus

119889119894+1(120583

119886119894119894+1minus 120583

119886119894) (119889

119894+1is the absolute value of the negative-

valued neighbour of 119889119894) and after the swap this part of the

expression changes to minus119889119894+1(120583

119886119894+1minus 120583

119886) + 119889

119894(120583

119886119894119894+1minus 120583

119886119894+1)

Then the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886)

minus 119889119894+1(120583

119886119894+1minus 120583

119886minus 120583

119886119894119894+1+ 120583

119886119894)

(A12)

Supermodularity of 120583 gives 120583119886119894+ 120583

119886119894+1le 120583

119886+ 120583

119886119894119894+1and

therefore the factor of 119889119894in former expression is nonnegative

and the factor of 119889119894+1

(which is the negation of the factor of119889119894) is negative or 0Therefore the total change is nonnegative

as wellThus with each of these swap operations we will never

decrease the value of the expression and starting from theasymmetric Choquet integral for 119909 upon reaching the finalorder the final value will be larger or equal

An example might be helpful Assume that the order in 119909is (1 2 3 minus 4 minus 5) (a negative index should indicate that

the corresponding element of 119909 is negative) and in 119910 we havethe order (minus5 3 1 2 minus4) (so 119910

5is the largest component of

119910 and 1199104the smallest) Then the order of swaps is as follows

Swap 1 applied as (1 2 3larr997888rarr

minus4 minus5) gives (1 3 2 minus4 minus5)Swap 1 applied as (1 3

larr997888rarr2 minus4 minus5) gives (3 1 2 minus4 minus5)

Next Swap 2 applied as (3 1 2 minus4 minus 5larr997888997888997888997888rarr

) gives (3 1 2 minus

5 minus 4) Swap 3 applied as (3 1 2 minus 5larr997888997888997888rarr

minus 4) gives (3 1 minus

5 2 minus 4) Swap 3 applied as (3 1 minus 5larr997888997888997888rarr

2 minus 4) gives (3 minus

5 1 2 minus4) and finally Swap 3 applied as (3 minus 5larr997888997888997888rarr

1 2 minus4)

gives the order (minus5 3 1 2 minus 4) of 119910 Note also that in casethe elements of 119909 are not in that order we can always relabelthe indizes and measures correspondingly

This means if we compute (using 119909 =

(1199091 119909

2 119909

3 minus119909

4 minus119909

5))

(ASC) int(119910)

119909119889120583

= minus11990951205835+ 119909

3(120583

35minus 120583

5) + 119909

1(120583

135minus 120583

35)

+ 1199092(120583

1235minus 120583

135) minus 119909

4(120583

12345minus 120583

1235)

(A13)

instead of

(ASC) int 119909 119889120583

= 11990911205831+ 119909

2(120583

12minus 120583

1) + 119909

3(120583

123minus 120583

12)

minus 1199094(120583

1234minus 120583

123) minus 119909

5(120583

12345minus 120583

1234)

(A14)

then (ASC) int(119910)119909 119889120583 ge (ASC) int 119909 119889120583

Using this Lemma we can now easily show the following

Theorem 27 If 120583 is a supermodular measure then ge120583is a

transitive relation

Proof For transitivity we have to show that for any 119909 119910 and119911 such that 119909ge

120583119910 and 119910ge

120583119911 it follows that 119909ge

120583119911 Now 119909ge

120583119910

means (ASC) int(119909 minus 119910)119889120583 ge 0 and 119910ge120583119911 means (ASC) int(119910 minus

119911)119889120583 ge 0 From supermodularity of 120583 and Lemma 26 then italso follows that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 ge (ASC) int (119909 minus 119910) 119889120583 ge 0

(ASC) int(119909minus119911)

(119910 minus 119911) 119889120583 ge (ASC) int (119910 minus 119911) 119889120583 ge 0(A15)

However by using the fact that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 + (ASC) int(119909minus119911)

(119910 minus 119911) 119889120583

= (ASC) int (119909 minus 119911) 119889120583(A16)

we see that also

(ASC) int (119909 minus 119911) 119889120583 ge 0 (A17)

which means 119909ge120583119911

12 Journal of Applied Mathematics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Bertsimas V F Farias and N Trichakis ldquoThe price offairnessrdquo Operations Research vol 59 no 1 pp 17ndash31 2011

[2] J F Nash Jr ldquoThe bargaining problemrdquo Econometrica vol 18pp 155ndash162 1950

[3] E Kalai and M Smorodinsky ldquoOther solutions to Nashrsquosbargaining problemrdquo Econometrica vol 43 pp 513ndash518 1975

[4] J Rawls ATheory of Justice Harvard University Press 1999[5] F Kelly ldquoCharging and rate control for elastic trafficrdquo European

Transactions onTelecommunications vol 8 no 1 pp 33ndash37 1997[6] D P Bertsekas R G Gallager and P Humblet Data Networks

vol 2 Prentice-Hall International 1992[7] H P Young Equity In Theory and Practice Princeton Univer-

sity Press Princeton NJ USA 1995[8] J Mo and J Walrand ldquoFair end-to-end window-based conges-

tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[9] M Alicherry R Bhatia and L E Li ldquoJoint channel assignmentand routing for throughput optimization inmulti-radiowirelessmesh networksrdquo in Proceedings of the 11th Annual InternationalConference on Mobile Computing and Networking (MobiCom05) pp 58ndash72 ACM Cologne Germany August-September2005

[10] A Iyer C Rosenberg and A Karnik ldquoWhat is the rightmodel for wireless channel interferencerdquo IEEE Transactions onWireless Communications vol 8 no 5 pp 2662ndash2671 2009

[11] A Mishra V Shrivastava S Banerjee and W ArbaughldquoPartially overlapped channels not considered harmfulrdquo ACMSIGMETRICS Performance Evaluation Review vol 34 no 1 pp63ndash74 2006

[12] N Nie and C Comaniciu ldquoAdaptive channel allocation spec-trum etiquette for cognitive radio networksrdquo Mobile Networksand Applications vol 11 no 6 pp 779ndash797 2006

[13] C Peng H Zheng and B Y Zhao ldquoUtilization and fairnessin spectrum assignment for opportunistic spectrum accessrdquoMobile Networks and Applications vol 11 no 4 pp 555ndash5762006

[14] X Yang andA P Petropulu ldquoCo-channel interferencemodelingand analysis in a Poisson field of interferers in wireless commu-nicationsrdquo IEEE Transactions on Signal Processing vol 51 no 1pp 64ndash76 2003

[15] M Koppen ldquoRelational optimization and its application frombottleneck flow control to wireless channel allocationrdquo Infor-matica vol 24 no 3 pp 413ndash433 2013

[16] M Sugeno Theory of fuzzy integrals and its applications [PhDthesis] 1974

[17] S Michio ldquoFuzzymeasures and fuzzy integrals a surveyrdquo FuzzyAutomata and Decision Processes vol 78 no 33 pp 89ndash1021977

[18] D Denneberg Non-Additive Measure and Integral vol 27Springer 1994

[19] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[20] J Sipos ldquoIntegral with respect to a pre-measurerdquo MathematicaSlovaca vol 29 no 2 pp 141ndash155 1979

[21] M Grabisch and C Labreuche ldquoThe symmetric and asymmet-ric Choquet integrals on finite spaces for decision makingrdquoStatistical Papers vol 43 no 1 pp 37ndash52 2002

[22] M Grabisch C Labreuche and J-C Vansnick ldquoOn theextension of pseudo-Boolean functions for the aggregation ofinteracting criteriardquo European Journal of Operational Researchvol 148 no 1 pp 28ndash47 2003

[23] R Mesiar A Mesiarova-Zemankova and K Ahmad ldquoDiscreteChoquet integral and some of its symmetric extensionsrdquo FuzzySets and Systems vol 184 pp 148ndash155 2011

[24] A Mesiarova-Zemankova R Mesiar and K Ahmad ldquoThebalancing Choquet integralrdquo Fuzzy Sets and Systems vol 161no 17 pp 2243ndash2255 2010

[25] H Fouchal X Gandibleux and F Lehuede ldquoA lower bound ofthe Choquet integral integrated within Martinsalgorithmrdquo inNewState ofMCDM in the 21st Century vol 648 ofLectureNotesin Economics and Mathematical Systems pp 79ndash89 SpringerHeidelberg Germany 2011

[26] L Galand L Julien and P Patrice ldquoDominance rules forthe choquet integral in multiobjective dynamic programmingrdquoin Proceedings of the 23rd International Joint Conference onArtificial Intelligence pp 538ndash544 AAAI Press 2013

[27] A W Marshall I Olkin and B C Arnold Inequalities TheoryofMajorization and Its Applications Springer 2nd edition 2010

[28] W Ogryczak AWierzbicki andMMilewski ldquoAmulti-criteriaapproach to fair and efficient bandwidth allocationrdquoOmega vol36 no 3 pp 451ndash463 2008

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

Table 1 Comparison of total performances for capacity propor-tional and maxmin fairness

Fairness relation Average performanceConflict matrix range 0 to 1

CI 42882Proportional 454661maxmin 414688

Conflict matrix range 09 to 1CI 456051Proportional 451135maxmin 422561

Conflict matrix range 0 to 02CI 409365Proportional 441848maxmin 404879

for the fuzzy measure the measure for the whole set is 1The inverse Mobius transform of such a mass vector gives asupermodularmeasure that can be used to specify a CI-basedfairness relation according to Definition 20

Now this relation is applied to all feasible channelallocationsmdashit means we are not considering a specific use ofmultichannel assignment but we are looking for the influencethat such amultichannel allocation could have on the specificselection of maximal elements Then the focus is on thedisadvantage that some pairs of users can experience if theirfavoured situation (eg to be distant enough to use the samechannel) is not taken into account The similar approach canbe considered for cognitive radio opportunistic networkingP2P vehicular networks and so forth

We might especially look for the price of fairness [1]compared to a standard proportional fair or maxmin fairallocation

Some example results are shown in Table 1 There for thecase of 4 users and 6 cells the maximum sets over all feasibleallocations were computed for CI-fairness using a measure asdescribed above proportional fairness andmaxmin fairnessThe conflictmatrix elements for the CI-fairness were uniformrandomly set within specific ranges All results are averagedover 30 repetitions Note that average sizes of maximum setswere found to be 42 formaxmin fairness 61 for proportionalfairness and 419 for CI-fairness so the CI-fairness produceslarger maximum sets an issue that should be addressed infutureworksThe table now shows average performances overall maximal elements

We see with regard to efficiency the well-known fact thatproportional-fair allocations are in average more efficientthan maxmin fair allocations The CI-fairness appears toselect maximal relations with a total performance betweenproportional fairness and maxmin fairness In case of stronginterferences (where a multichannel assignment is not rea-sonable) it is very close to proportional fairness while in caseof low interference it appears to be more close to maxminfairness that is the fairness relation favouring least elementsThus such an empirical result can be understood in the sensethat the neglecting of the potential multichannel assignment

corresponds with the neglection of strong users This is alsoparticularly appearing for the general case where all conflictmatrix elements were randomly selected between 0 and 1 alsohere the average performance appearsmore close tomaxminfairness than proportional fairness

7 Conclusion

A fairness relation among vectors based on the asymmetricChoquet integral was studied It formally follows the Nashstandard of comparison where the relative losses and winsare replaced by absolute losses and wins but instead takingweights for groups of vector components into account Thusit can represent for example pairs of users that are ina favourable situation regarding resource allocation Theappealing points of this CI-relation are as follows (1) Incase a supermodular measure is used for the integrationthe relation will be transitive which suits faster search formaximal elements (2) it can be parametrized by givinga weight to each subset of vector components (while inpractice this might be relevant only for smaller subsets) Therepresentation of the fuzzy measure by its Mobius transformappears to be a convenient way to yield a supermodularfuzzy measure the masses just need to be all nonnegativeThus the relation can be conveniently applied in many prac-tical applications to specify optimality Slight disadvantagesare with the complexity of the involved calculations andsome empirical evidence for specifying larger maximum setsthan other fairness relations like proportional fairness andmaxmin fairness

In case of wireless channel allocation how CI-fairnesscan be used to help deciding whether implementations ofmore complex allocation schemes are indeed worth the effortwas suggested This method can be applied to many otherproblems of higher efficiency of network resource utilizationIn future work we will consider the use of advanced fuzzyintegrals like balancing and fusion Choquet integral andthe adjustment of related CI-fairness parameters to specificsituations and what can be concluded from such parametersWe will also more intensively study the performance issueswith regard to computational effort and search for maximalelements and study conditions where also greatest elementsexist

Appendix

Proofs of Theorems

In the following we provide the proofs for Propositions 21and 22 For formal convenience of the proofs the notationwill be a little bit changing therefore we repeat the basicdefinitions

We define the Choquet-integral-based (fairness) relationas follows be 119909 and 119910 two vectors from 119877

119899 By [119909]+ weindicate the positive support of a vector 119909 that is the 119894thcomponent of [119909]+ is 119909

119894if 119909

119894ge 0 or 0 if 119909

119894lt 0 The Choquet

integral is denoted by (119862) int119909 119889120583 There 120583 is a fuzzy measure

10 Journal of Applied Mathematics

Also 120583119889 denotes its dual measure (ie 120583119889(119860) = 120583119883minus 120583(119860

119862))

Then 119909 is said to CI-dominate 119910 if and only if

119909ge120583119910 larrrarr (119862)int [119909 minus 119910]

+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (A1)

We can alternatively rewrite this as the asymmetric Choquetintegral

119909ge120583119910 larrrarr (ASC) int (119909 minus 119910) 119889120583 ge 0 (A2)

since the right-hand side condition of (A1) exactly cor-responds with the definition of the asymmetric Choquetintegral

First we can show the following

Lemma 23 If 120583 is a superadditive fuzzy measure where foreach 119860 119861 = 0 and 119860 cap 119861 = 0 120583

119860+ 120583

119861lt 120583

119860cup119861 then for

119860 = 0 120583119889

119860gt 120583

119860

Proof Since 120583 is superadditive for119860 = 0 120583119860+120583

119860119862 lt 120583

119860cup119860119862 =

120583119883 Then 120583

119860lt 120583

119883minus 120583

119860119862 = 120583

119889

119860follows directly

Theorem 24 If 120583 is superadditive then from 119909ge120583119910 and

119910ge120583119909119909 = 119910 follows

Proof 119909ge120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889 If

119909 = 119910 we can use Lemma 23 to obtain

(119862) int [119909 minus 119910]+119889120583

119889gt (119862)int [119909 minus 119910]

+119889120583

ge (119862)int [119910 minus 119909]+119889120583

119889gt (119862)int [119910 minus 119909]

+119889120583

(A3)

that is (119862) int [119910 minus 119909]+119889120583 ge (119862) int [119909 minus 119910]+119889120583

119889 This wouldmean that 119910ge

120583119909 would not hold Therefore it must be 119909 =

119910

For a supermodular measure and nonnegative vectors 119909and 119910 we have the known property (119862) int 119909 119889120583+ (119862) int 119910 119889120583 le(119862) int(119909 + 119910)119889120583 Since the dual measure of a supermodularmeasure is submodular it also holds that (119862) int 119909 119889120583 +(119862) int 119910 119889120583 ge (119862) int(119909 + 119910)119889120583 Using this the following canbe shown

Theorem25 If120583 is supermodular then the relationgt120583is cycle-

free

Proof We will only show the case of nonexistence of a 3-cycle here the concept generalizes directly to any other caseAssume that 119909gt

120583119910 and 119910gt

120583119911 First we note that

[119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+

= [119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+

(A4)

and therefore

(119862) int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

= (119862)int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

(A5)

Using Lemma 23 we yield

(119862) int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

119889

gt (119862)int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

(A6)

and from supermodularity of 120583 and submodularity of 120583119889

(119862) int [119910 minus 119909]+119889120583

119889+ (119862)int [119911 minus 119910]

+119889120583

119889+ (119862)int [119909 minus 119911]

+119889120583

119889

gt (119862)int [119909 minus 119910]+119889120583 + (119862)int [119910 minus 119911]

+119889120583

+ (119862)int [119911 minus 119909]+119889120583

(A7)

However 119909gt120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889

and 119910gt120583119911 means (119862) int [119910 minus 119911]+119889120583 ge (119862) int [119911 minus 119910]

+119889120583

119889Therefore above inequality can only hold if

(119862) int [119911 minus 119909]+119889120583 lt (119862)int [119909 minus 119911]

+119889120583

119889 (A8)

which means 119911 gt120583119909

If the measure 120583 is supermodular also transitivity ofthe relation can be shown Before we can show this weneed to introduce a notation and a corresponding lemmaThe notation refers to computing the (asymmetric) Choquetintegral of 119909 by measure 120583 in a ldquodifferent orderrdquo If 119909

119909(119894)

indicates the 119894th largest element of 119909 then the Choquetintegral is calculated as

(119862) int 119909119889120583 = 119909119909(1)120583119909(1)

+

119899

sum

119894=2

119909119909(119894)[120583

119909(1)119909(2)sdotsdotsdot119909(119894)minus 120583

119909(1)sdotsdotsdot119909(119894minus1)]

(A9)

For any other119910 we can compute the same expression justfollowing the order of the elements of 119910 instead of 119909 Then119909119910(119894)

indicates the element of119909with the index of the 119894th largestelement of 119910 and we define (for convenience we still use thesymbol for the Choquet integral keeping in mind that it isnot a real fuzzy integral anymore)

(119862) int(119910)

119909119889120583 = 119909119910(1)120583119910(1)

+

119899

sum

119894=2

119909119910(119894)[120583

119910(1)119910(2)sdotsdotsdot119910(119894)minus 120583

119910(1)sdotsdotsdot119910(119894minus1)]

(A10)

Journal of Applied Mathematics 11

Consequently we could also write (119862) int(119909)119909119889120583 for the

ldquooriginalrdquo Choquet integral We can do the same for theasymmetric Choquet integral by applying the resorting to thenegative components of the vector 119909 as well and will use thenotation (ASC) int

(119910)119909 119889120583 here Now we have the following

Lemma 26 If 120583 is a supermodular measure then for any119910 ( ASC ) int

(119910)119909 119889120583 ge ( ASC ) int

(119909)119909119889120583

Proof A moment of reasoning gives that any sorting of theindizes can be achieved by suitable application of a sequenceof three kinds of ldquoneighbour swapsrdquo starting from the order119909(1)119909(2) sdot sdot sdot 119909(119899))

Swap 1 In the computation of the integral a larger orequal nonnegative element 119889

119894at index 119894 is swapped with its

immediate nonnegative neighbour 119889119894+1

to the right Thenwhat was computed as 119889

119894(120583

119886119894minus120583

119886) +119889

119894+1(120583

119886119894119894+1minus120583

119886119894) before

the swap changes to 119889119894+1(120583

119886119894+1minus120583

119886) + 119889

119894(120583

119886119894119894+1minus120583

119886119894+1) after

the swap all other parts of the resorted asymmetric Choquetintegral expression remain the same (by 119886 we indicate theindex order before index 119894) Thus the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886119894)

minus 119889119894+1(120583

119886119894119894+1minus 120583

119886119894minus 120583

119886119894+1+ 120583

119886119894)

(A11)

and from 119889119894ge 119889

119894+1and 120583

119886119894119894+1+ 120583

119886ge 120583

119886119894+1+ 120583

119886119894

(supermodularity of 120583) it directly follows that the computedvalue will not decrease by this swap

Swap 2 In the computation of the integral a larger orequal negative element 119889

119894+1at index 119894 + 1 is swapped with

its immediate negative left neighbour 119889119894 From a similar

evaluation as for Swap 1 it can be seen that also here the valueof the computed expression will not decrease after the swap

Swap 3 A nonnegative element at index 119894 is swapped withits immediate negative neighbor to the right with index(119894 + 1) Then before the swap we compute 119889

119894(120583

119886119894minus 120583

119886) minus

119889119894+1(120583

119886119894119894+1minus 120583

119886119894) (119889

119894+1is the absolute value of the negative-

valued neighbour of 119889119894) and after the swap this part of the

expression changes to minus119889119894+1(120583

119886119894+1minus 120583

119886) + 119889

119894(120583

119886119894119894+1minus 120583

119886119894+1)

Then the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886)

minus 119889119894+1(120583

119886119894+1minus 120583

119886minus 120583

119886119894119894+1+ 120583

119886119894)

(A12)

Supermodularity of 120583 gives 120583119886119894+ 120583

119886119894+1le 120583

119886+ 120583

119886119894119894+1and

therefore the factor of 119889119894in former expression is nonnegative

and the factor of 119889119894+1

(which is the negation of the factor of119889119894) is negative or 0Therefore the total change is nonnegative

as wellThus with each of these swap operations we will never

decrease the value of the expression and starting from theasymmetric Choquet integral for 119909 upon reaching the finalorder the final value will be larger or equal

An example might be helpful Assume that the order in 119909is (1 2 3 minus 4 minus 5) (a negative index should indicate that

the corresponding element of 119909 is negative) and in 119910 we havethe order (minus5 3 1 2 minus4) (so 119910

5is the largest component of

119910 and 1199104the smallest) Then the order of swaps is as follows

Swap 1 applied as (1 2 3larr997888rarr

minus4 minus5) gives (1 3 2 minus4 minus5)Swap 1 applied as (1 3

larr997888rarr2 minus4 minus5) gives (3 1 2 minus4 minus5)

Next Swap 2 applied as (3 1 2 minus4 minus 5larr997888997888997888997888rarr

) gives (3 1 2 minus

5 minus 4) Swap 3 applied as (3 1 2 minus 5larr997888997888997888rarr

minus 4) gives (3 1 minus

5 2 minus 4) Swap 3 applied as (3 1 minus 5larr997888997888997888rarr

2 minus 4) gives (3 minus

5 1 2 minus4) and finally Swap 3 applied as (3 minus 5larr997888997888997888rarr

1 2 minus4)

gives the order (minus5 3 1 2 minus 4) of 119910 Note also that in casethe elements of 119909 are not in that order we can always relabelthe indizes and measures correspondingly

This means if we compute (using 119909 =

(1199091 119909

2 119909

3 minus119909

4 minus119909

5))

(ASC) int(119910)

119909119889120583

= minus11990951205835+ 119909

3(120583

35minus 120583

5) + 119909

1(120583

135minus 120583

35)

+ 1199092(120583

1235minus 120583

135) minus 119909

4(120583

12345minus 120583

1235)

(A13)

instead of

(ASC) int 119909 119889120583

= 11990911205831+ 119909

2(120583

12minus 120583

1) + 119909

3(120583

123minus 120583

12)

minus 1199094(120583

1234minus 120583

123) minus 119909

5(120583

12345minus 120583

1234)

(A14)

then (ASC) int(119910)119909 119889120583 ge (ASC) int 119909 119889120583

Using this Lemma we can now easily show the following

Theorem 27 If 120583 is a supermodular measure then ge120583is a

transitive relation

Proof For transitivity we have to show that for any 119909 119910 and119911 such that 119909ge

120583119910 and 119910ge

120583119911 it follows that 119909ge

120583119911 Now 119909ge

120583119910

means (ASC) int(119909 minus 119910)119889120583 ge 0 and 119910ge120583119911 means (ASC) int(119910 minus

119911)119889120583 ge 0 From supermodularity of 120583 and Lemma 26 then italso follows that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 ge (ASC) int (119909 minus 119910) 119889120583 ge 0

(ASC) int(119909minus119911)

(119910 minus 119911) 119889120583 ge (ASC) int (119910 minus 119911) 119889120583 ge 0(A15)

However by using the fact that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 + (ASC) int(119909minus119911)

(119910 minus 119911) 119889120583

= (ASC) int (119909 minus 119911) 119889120583(A16)

we see that also

(ASC) int (119909 minus 119911) 119889120583 ge 0 (A17)

which means 119909ge120583119911

12 Journal of Applied Mathematics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Bertsimas V F Farias and N Trichakis ldquoThe price offairnessrdquo Operations Research vol 59 no 1 pp 17ndash31 2011

[2] J F Nash Jr ldquoThe bargaining problemrdquo Econometrica vol 18pp 155ndash162 1950

[3] E Kalai and M Smorodinsky ldquoOther solutions to Nashrsquosbargaining problemrdquo Econometrica vol 43 pp 513ndash518 1975

[4] J Rawls ATheory of Justice Harvard University Press 1999[5] F Kelly ldquoCharging and rate control for elastic trafficrdquo European

Transactions onTelecommunications vol 8 no 1 pp 33ndash37 1997[6] D P Bertsekas R G Gallager and P Humblet Data Networks

vol 2 Prentice-Hall International 1992[7] H P Young Equity In Theory and Practice Princeton Univer-

sity Press Princeton NJ USA 1995[8] J Mo and J Walrand ldquoFair end-to-end window-based conges-

tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[9] M Alicherry R Bhatia and L E Li ldquoJoint channel assignmentand routing for throughput optimization inmulti-radiowirelessmesh networksrdquo in Proceedings of the 11th Annual InternationalConference on Mobile Computing and Networking (MobiCom05) pp 58ndash72 ACM Cologne Germany August-September2005

[10] A Iyer C Rosenberg and A Karnik ldquoWhat is the rightmodel for wireless channel interferencerdquo IEEE Transactions onWireless Communications vol 8 no 5 pp 2662ndash2671 2009

[11] A Mishra V Shrivastava S Banerjee and W ArbaughldquoPartially overlapped channels not considered harmfulrdquo ACMSIGMETRICS Performance Evaluation Review vol 34 no 1 pp63ndash74 2006

[12] N Nie and C Comaniciu ldquoAdaptive channel allocation spec-trum etiquette for cognitive radio networksrdquo Mobile Networksand Applications vol 11 no 6 pp 779ndash797 2006

[13] C Peng H Zheng and B Y Zhao ldquoUtilization and fairnessin spectrum assignment for opportunistic spectrum accessrdquoMobile Networks and Applications vol 11 no 4 pp 555ndash5762006

[14] X Yang andA P Petropulu ldquoCo-channel interferencemodelingand analysis in a Poisson field of interferers in wireless commu-nicationsrdquo IEEE Transactions on Signal Processing vol 51 no 1pp 64ndash76 2003

[15] M Koppen ldquoRelational optimization and its application frombottleneck flow control to wireless channel allocationrdquo Infor-matica vol 24 no 3 pp 413ndash433 2013

[16] M Sugeno Theory of fuzzy integrals and its applications [PhDthesis] 1974

[17] S Michio ldquoFuzzymeasures and fuzzy integrals a surveyrdquo FuzzyAutomata and Decision Processes vol 78 no 33 pp 89ndash1021977

[18] D Denneberg Non-Additive Measure and Integral vol 27Springer 1994

[19] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[20] J Sipos ldquoIntegral with respect to a pre-measurerdquo MathematicaSlovaca vol 29 no 2 pp 141ndash155 1979

[21] M Grabisch and C Labreuche ldquoThe symmetric and asymmet-ric Choquet integrals on finite spaces for decision makingrdquoStatistical Papers vol 43 no 1 pp 37ndash52 2002

[22] M Grabisch C Labreuche and J-C Vansnick ldquoOn theextension of pseudo-Boolean functions for the aggregation ofinteracting criteriardquo European Journal of Operational Researchvol 148 no 1 pp 28ndash47 2003

[23] R Mesiar A Mesiarova-Zemankova and K Ahmad ldquoDiscreteChoquet integral and some of its symmetric extensionsrdquo FuzzySets and Systems vol 184 pp 148ndash155 2011

[24] A Mesiarova-Zemankova R Mesiar and K Ahmad ldquoThebalancing Choquet integralrdquo Fuzzy Sets and Systems vol 161no 17 pp 2243ndash2255 2010

[25] H Fouchal X Gandibleux and F Lehuede ldquoA lower bound ofthe Choquet integral integrated within Martinsalgorithmrdquo inNewState ofMCDM in the 21st Century vol 648 ofLectureNotesin Economics and Mathematical Systems pp 79ndash89 SpringerHeidelberg Germany 2011

[26] L Galand L Julien and P Patrice ldquoDominance rules forthe choquet integral in multiobjective dynamic programmingrdquoin Proceedings of the 23rd International Joint Conference onArtificial Intelligence pp 538ndash544 AAAI Press 2013

[27] A W Marshall I Olkin and B C Arnold Inequalities TheoryofMajorization and Its Applications Springer 2nd edition 2010

[28] W Ogryczak AWierzbicki andMMilewski ldquoAmulti-criteriaapproach to fair and efficient bandwidth allocationrdquoOmega vol36 no 3 pp 451ndash463 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

Also 120583119889 denotes its dual measure (ie 120583119889(119860) = 120583119883minus 120583(119860

119862))

Then 119909 is said to CI-dominate 119910 if and only if

119909ge120583119910 larrrarr (119862)int [119909 minus 119910]

+119889120583 ge (119862)int [119910 minus 119909]

+119889120583

119889 (A1)

We can alternatively rewrite this as the asymmetric Choquetintegral

119909ge120583119910 larrrarr (ASC) int (119909 minus 119910) 119889120583 ge 0 (A2)

since the right-hand side condition of (A1) exactly cor-responds with the definition of the asymmetric Choquetintegral

First we can show the following

Lemma 23 If 120583 is a superadditive fuzzy measure where foreach 119860 119861 = 0 and 119860 cap 119861 = 0 120583

119860+ 120583

119861lt 120583

119860cup119861 then for

119860 = 0 120583119889

119860gt 120583

119860

Proof Since 120583 is superadditive for119860 = 0 120583119860+120583

119860119862 lt 120583

119860cup119860119862 =

120583119883 Then 120583

119860lt 120583

119883minus 120583

119860119862 = 120583

119889

119860follows directly

Theorem 24 If 120583 is superadditive then from 119909ge120583119910 and

119910ge120583119909119909 = 119910 follows

Proof 119909ge120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889 If

119909 = 119910 we can use Lemma 23 to obtain

(119862) int [119909 minus 119910]+119889120583

119889gt (119862)int [119909 minus 119910]

+119889120583

ge (119862)int [119910 minus 119909]+119889120583

119889gt (119862)int [119910 minus 119909]

+119889120583

(A3)

that is (119862) int [119910 minus 119909]+119889120583 ge (119862) int [119909 minus 119910]+119889120583

119889 This wouldmean that 119910ge

120583119909 would not hold Therefore it must be 119909 =

119910

For a supermodular measure and nonnegative vectors 119909and 119910 we have the known property (119862) int 119909 119889120583+ (119862) int 119910 119889120583 le(119862) int(119909 + 119910)119889120583 Since the dual measure of a supermodularmeasure is submodular it also holds that (119862) int 119909 119889120583 +(119862) int 119910 119889120583 ge (119862) int(119909 + 119910)119889120583 Using this the following canbe shown

Theorem25 If120583 is supermodular then the relationgt120583is cycle-

free

Proof We will only show the case of nonexistence of a 3-cycle here the concept generalizes directly to any other caseAssume that 119909gt

120583119910 and 119910gt

120583119911 First we note that

[119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+

= [119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+

(A4)

and therefore

(119862) int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

= (119862)int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

(A5)

Using Lemma 23 we yield

(119862) int ([119910 minus 119909]++ [119911 minus 119910]

++ [119909 minus 119911]

+) 119889120583

119889

gt (119862)int ([119909 minus 119910]++ [119910 minus 119911]

++ [119911 minus 119909]

+) 119889120583

(A6)

and from supermodularity of 120583 and submodularity of 120583119889

(119862) int [119910 minus 119909]+119889120583

119889+ (119862)int [119911 minus 119910]

+119889120583

119889+ (119862)int [119909 minus 119911]

+119889120583

119889

gt (119862)int [119909 minus 119910]+119889120583 + (119862)int [119910 minus 119911]

+119889120583

+ (119862)int [119911 minus 119909]+119889120583

(A7)

However 119909gt120583119910means (119862) int [119909 minus 119910]+119889120583 ge (119862) int [119910 minus 119909]+119889120583119889

and 119910gt120583119911 means (119862) int [119910 minus 119911]+119889120583 ge (119862) int [119911 minus 119910]

+119889120583

119889Therefore above inequality can only hold if

(119862) int [119911 minus 119909]+119889120583 lt (119862)int [119909 minus 119911]

+119889120583

119889 (A8)

which means 119911 gt120583119909

If the measure 120583 is supermodular also transitivity ofthe relation can be shown Before we can show this weneed to introduce a notation and a corresponding lemmaThe notation refers to computing the (asymmetric) Choquetintegral of 119909 by measure 120583 in a ldquodifferent orderrdquo If 119909

119909(119894)

indicates the 119894th largest element of 119909 then the Choquetintegral is calculated as

(119862) int 119909119889120583 = 119909119909(1)120583119909(1)

+

119899

sum

119894=2

119909119909(119894)[120583

119909(1)119909(2)sdotsdotsdot119909(119894)minus 120583

119909(1)sdotsdotsdot119909(119894minus1)]

(A9)

For any other119910 we can compute the same expression justfollowing the order of the elements of 119910 instead of 119909 Then119909119910(119894)

indicates the element of119909with the index of the 119894th largestelement of 119910 and we define (for convenience we still use thesymbol for the Choquet integral keeping in mind that it isnot a real fuzzy integral anymore)

(119862) int(119910)

119909119889120583 = 119909119910(1)120583119910(1)

+

119899

sum

119894=2

119909119910(119894)[120583

119910(1)119910(2)sdotsdotsdot119910(119894)minus 120583

119910(1)sdotsdotsdot119910(119894minus1)]

(A10)

Journal of Applied Mathematics 11

Consequently we could also write (119862) int(119909)119909119889120583 for the

ldquooriginalrdquo Choquet integral We can do the same for theasymmetric Choquet integral by applying the resorting to thenegative components of the vector 119909 as well and will use thenotation (ASC) int

(119910)119909 119889120583 here Now we have the following

Lemma 26 If 120583 is a supermodular measure then for any119910 ( ASC ) int

(119910)119909 119889120583 ge ( ASC ) int

(119909)119909119889120583

Proof A moment of reasoning gives that any sorting of theindizes can be achieved by suitable application of a sequenceof three kinds of ldquoneighbour swapsrdquo starting from the order119909(1)119909(2) sdot sdot sdot 119909(119899))

Swap 1 In the computation of the integral a larger orequal nonnegative element 119889

119894at index 119894 is swapped with its

immediate nonnegative neighbour 119889119894+1

to the right Thenwhat was computed as 119889

119894(120583

119886119894minus120583

119886) +119889

119894+1(120583

119886119894119894+1minus120583

119886119894) before

the swap changes to 119889119894+1(120583

119886119894+1minus120583

119886) + 119889

119894(120583

119886119894119894+1minus120583

119886119894+1) after

the swap all other parts of the resorted asymmetric Choquetintegral expression remain the same (by 119886 we indicate theindex order before index 119894) Thus the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886119894)

minus 119889119894+1(120583

119886119894119894+1minus 120583

119886119894minus 120583

119886119894+1+ 120583

119886119894)

(A11)

and from 119889119894ge 119889

119894+1and 120583

119886119894119894+1+ 120583

119886ge 120583

119886119894+1+ 120583

119886119894

(supermodularity of 120583) it directly follows that the computedvalue will not decrease by this swap

Swap 2 In the computation of the integral a larger orequal negative element 119889

119894+1at index 119894 + 1 is swapped with

its immediate negative left neighbour 119889119894 From a similar

evaluation as for Swap 1 it can be seen that also here the valueof the computed expression will not decrease after the swap

Swap 3 A nonnegative element at index 119894 is swapped withits immediate negative neighbor to the right with index(119894 + 1) Then before the swap we compute 119889

119894(120583

119886119894minus 120583

119886) minus

119889119894+1(120583

119886119894119894+1minus 120583

119886119894) (119889

119894+1is the absolute value of the negative-

valued neighbour of 119889119894) and after the swap this part of the

expression changes to minus119889119894+1(120583

119886119894+1minus 120583

119886) + 119889

119894(120583

119886119894119894+1minus 120583

119886119894+1)

Then the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886)

minus 119889119894+1(120583

119886119894+1minus 120583

119886minus 120583

119886119894119894+1+ 120583

119886119894)

(A12)

Supermodularity of 120583 gives 120583119886119894+ 120583

119886119894+1le 120583

119886+ 120583

119886119894119894+1and

therefore the factor of 119889119894in former expression is nonnegative

and the factor of 119889119894+1

(which is the negation of the factor of119889119894) is negative or 0Therefore the total change is nonnegative

as wellThus with each of these swap operations we will never

decrease the value of the expression and starting from theasymmetric Choquet integral for 119909 upon reaching the finalorder the final value will be larger or equal

An example might be helpful Assume that the order in 119909is (1 2 3 minus 4 minus 5) (a negative index should indicate that

the corresponding element of 119909 is negative) and in 119910 we havethe order (minus5 3 1 2 minus4) (so 119910

5is the largest component of

119910 and 1199104the smallest) Then the order of swaps is as follows

Swap 1 applied as (1 2 3larr997888rarr

minus4 minus5) gives (1 3 2 minus4 minus5)Swap 1 applied as (1 3

larr997888rarr2 minus4 minus5) gives (3 1 2 minus4 minus5)

Next Swap 2 applied as (3 1 2 minus4 minus 5larr997888997888997888997888rarr

) gives (3 1 2 minus

5 minus 4) Swap 3 applied as (3 1 2 minus 5larr997888997888997888rarr

minus 4) gives (3 1 minus

5 2 minus 4) Swap 3 applied as (3 1 minus 5larr997888997888997888rarr

2 minus 4) gives (3 minus

5 1 2 minus4) and finally Swap 3 applied as (3 minus 5larr997888997888997888rarr

1 2 minus4)

gives the order (minus5 3 1 2 minus 4) of 119910 Note also that in casethe elements of 119909 are not in that order we can always relabelthe indizes and measures correspondingly

This means if we compute (using 119909 =

(1199091 119909

2 119909

3 minus119909

4 minus119909

5))

(ASC) int(119910)

119909119889120583

= minus11990951205835+ 119909

3(120583

35minus 120583

5) + 119909

1(120583

135minus 120583

35)

+ 1199092(120583

1235minus 120583

135) minus 119909

4(120583

12345minus 120583

1235)

(A13)

instead of

(ASC) int 119909 119889120583

= 11990911205831+ 119909

2(120583

12minus 120583

1) + 119909

3(120583

123minus 120583

12)

minus 1199094(120583

1234minus 120583

123) minus 119909

5(120583

12345minus 120583

1234)

(A14)

then (ASC) int(119910)119909 119889120583 ge (ASC) int 119909 119889120583

Using this Lemma we can now easily show the following

Theorem 27 If 120583 is a supermodular measure then ge120583is a

transitive relation

Proof For transitivity we have to show that for any 119909 119910 and119911 such that 119909ge

120583119910 and 119910ge

120583119911 it follows that 119909ge

120583119911 Now 119909ge

120583119910

means (ASC) int(119909 minus 119910)119889120583 ge 0 and 119910ge120583119911 means (ASC) int(119910 minus

119911)119889120583 ge 0 From supermodularity of 120583 and Lemma 26 then italso follows that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 ge (ASC) int (119909 minus 119910) 119889120583 ge 0

(ASC) int(119909minus119911)

(119910 minus 119911) 119889120583 ge (ASC) int (119910 minus 119911) 119889120583 ge 0(A15)

However by using the fact that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 + (ASC) int(119909minus119911)

(119910 minus 119911) 119889120583

= (ASC) int (119909 minus 119911) 119889120583(A16)

we see that also

(ASC) int (119909 minus 119911) 119889120583 ge 0 (A17)

which means 119909ge120583119911

12 Journal of Applied Mathematics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Bertsimas V F Farias and N Trichakis ldquoThe price offairnessrdquo Operations Research vol 59 no 1 pp 17ndash31 2011

[2] J F Nash Jr ldquoThe bargaining problemrdquo Econometrica vol 18pp 155ndash162 1950

[3] E Kalai and M Smorodinsky ldquoOther solutions to Nashrsquosbargaining problemrdquo Econometrica vol 43 pp 513ndash518 1975

[4] J Rawls ATheory of Justice Harvard University Press 1999[5] F Kelly ldquoCharging and rate control for elastic trafficrdquo European

Transactions onTelecommunications vol 8 no 1 pp 33ndash37 1997[6] D P Bertsekas R G Gallager and P Humblet Data Networks

vol 2 Prentice-Hall International 1992[7] H P Young Equity In Theory and Practice Princeton Univer-

sity Press Princeton NJ USA 1995[8] J Mo and J Walrand ldquoFair end-to-end window-based conges-

tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[9] M Alicherry R Bhatia and L E Li ldquoJoint channel assignmentand routing for throughput optimization inmulti-radiowirelessmesh networksrdquo in Proceedings of the 11th Annual InternationalConference on Mobile Computing and Networking (MobiCom05) pp 58ndash72 ACM Cologne Germany August-September2005

[10] A Iyer C Rosenberg and A Karnik ldquoWhat is the rightmodel for wireless channel interferencerdquo IEEE Transactions onWireless Communications vol 8 no 5 pp 2662ndash2671 2009

[11] A Mishra V Shrivastava S Banerjee and W ArbaughldquoPartially overlapped channels not considered harmfulrdquo ACMSIGMETRICS Performance Evaluation Review vol 34 no 1 pp63ndash74 2006

[12] N Nie and C Comaniciu ldquoAdaptive channel allocation spec-trum etiquette for cognitive radio networksrdquo Mobile Networksand Applications vol 11 no 6 pp 779ndash797 2006

[13] C Peng H Zheng and B Y Zhao ldquoUtilization and fairnessin spectrum assignment for opportunistic spectrum accessrdquoMobile Networks and Applications vol 11 no 4 pp 555ndash5762006

[14] X Yang andA P Petropulu ldquoCo-channel interferencemodelingand analysis in a Poisson field of interferers in wireless commu-nicationsrdquo IEEE Transactions on Signal Processing vol 51 no 1pp 64ndash76 2003

[15] M Koppen ldquoRelational optimization and its application frombottleneck flow control to wireless channel allocationrdquo Infor-matica vol 24 no 3 pp 413ndash433 2013

[16] M Sugeno Theory of fuzzy integrals and its applications [PhDthesis] 1974

[17] S Michio ldquoFuzzymeasures and fuzzy integrals a surveyrdquo FuzzyAutomata and Decision Processes vol 78 no 33 pp 89ndash1021977

[18] D Denneberg Non-Additive Measure and Integral vol 27Springer 1994

[19] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[20] J Sipos ldquoIntegral with respect to a pre-measurerdquo MathematicaSlovaca vol 29 no 2 pp 141ndash155 1979

[21] M Grabisch and C Labreuche ldquoThe symmetric and asymmet-ric Choquet integrals on finite spaces for decision makingrdquoStatistical Papers vol 43 no 1 pp 37ndash52 2002

[22] M Grabisch C Labreuche and J-C Vansnick ldquoOn theextension of pseudo-Boolean functions for the aggregation ofinteracting criteriardquo European Journal of Operational Researchvol 148 no 1 pp 28ndash47 2003

[23] R Mesiar A Mesiarova-Zemankova and K Ahmad ldquoDiscreteChoquet integral and some of its symmetric extensionsrdquo FuzzySets and Systems vol 184 pp 148ndash155 2011

[24] A Mesiarova-Zemankova R Mesiar and K Ahmad ldquoThebalancing Choquet integralrdquo Fuzzy Sets and Systems vol 161no 17 pp 2243ndash2255 2010

[25] H Fouchal X Gandibleux and F Lehuede ldquoA lower bound ofthe Choquet integral integrated within Martinsalgorithmrdquo inNewState ofMCDM in the 21st Century vol 648 ofLectureNotesin Economics and Mathematical Systems pp 79ndash89 SpringerHeidelberg Germany 2011

[26] L Galand L Julien and P Patrice ldquoDominance rules forthe choquet integral in multiobjective dynamic programmingrdquoin Proceedings of the 23rd International Joint Conference onArtificial Intelligence pp 538ndash544 AAAI Press 2013

[27] A W Marshall I Olkin and B C Arnold Inequalities TheoryofMajorization and Its Applications Springer 2nd edition 2010

[28] W Ogryczak AWierzbicki andMMilewski ldquoAmulti-criteriaapproach to fair and efficient bandwidth allocationrdquoOmega vol36 no 3 pp 451ndash463 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

Consequently we could also write (119862) int(119909)119909119889120583 for the

ldquooriginalrdquo Choquet integral We can do the same for theasymmetric Choquet integral by applying the resorting to thenegative components of the vector 119909 as well and will use thenotation (ASC) int

(119910)119909 119889120583 here Now we have the following

Lemma 26 If 120583 is a supermodular measure then for any119910 ( ASC ) int

(119910)119909 119889120583 ge ( ASC ) int

(119909)119909119889120583

Proof A moment of reasoning gives that any sorting of theindizes can be achieved by suitable application of a sequenceof three kinds of ldquoneighbour swapsrdquo starting from the order119909(1)119909(2) sdot sdot sdot 119909(119899))

Swap 1 In the computation of the integral a larger orequal nonnegative element 119889

119894at index 119894 is swapped with its

immediate nonnegative neighbour 119889119894+1

to the right Thenwhat was computed as 119889

119894(120583

119886119894minus120583

119886) +119889

119894+1(120583

119886119894119894+1minus120583

119886119894) before

the swap changes to 119889119894+1(120583

119886119894+1minus120583

119886) + 119889

119894(120583

119886119894119894+1minus120583

119886119894+1) after

the swap all other parts of the resorted asymmetric Choquetintegral expression remain the same (by 119886 we indicate theindex order before index 119894) Thus the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886119894)

minus 119889119894+1(120583

119886119894119894+1minus 120583

119886119894minus 120583

119886119894+1+ 120583

119886119894)

(A11)

and from 119889119894ge 119889

119894+1and 120583

119886119894119894+1+ 120583

119886ge 120583

119886119894+1+ 120583

119886119894

(supermodularity of 120583) it directly follows that the computedvalue will not decrease by this swap

Swap 2 In the computation of the integral a larger orequal negative element 119889

119894+1at index 119894 + 1 is swapped with

its immediate negative left neighbour 119889119894 From a similar

evaluation as for Swap 1 it can be seen that also here the valueof the computed expression will not decrease after the swap

Swap 3 A nonnegative element at index 119894 is swapped withits immediate negative neighbor to the right with index(119894 + 1) Then before the swap we compute 119889

119894(120583

119886119894minus 120583

119886) minus

119889119894+1(120583

119886119894119894+1minus 120583

119886119894) (119889

119894+1is the absolute value of the negative-

valued neighbour of 119889119894) and after the swap this part of the

expression changes to minus119889119894+1(120583

119886119894+1minus 120583

119886) + 119889

119894(120583

119886119894119894+1minus 120583

119886119894+1)

Then the total change is

119889119894(120583

119886119894119894+1minus 120583

119886119894+1minus 120583

119886119894+ 120583

119886)

minus 119889119894+1(120583

119886119894+1minus 120583

119886minus 120583

119886119894119894+1+ 120583

119886119894)

(A12)

Supermodularity of 120583 gives 120583119886119894+ 120583

119886119894+1le 120583

119886+ 120583

119886119894119894+1and

therefore the factor of 119889119894in former expression is nonnegative

and the factor of 119889119894+1

(which is the negation of the factor of119889119894) is negative or 0Therefore the total change is nonnegative

as wellThus with each of these swap operations we will never

decrease the value of the expression and starting from theasymmetric Choquet integral for 119909 upon reaching the finalorder the final value will be larger or equal

An example might be helpful Assume that the order in 119909is (1 2 3 minus 4 minus 5) (a negative index should indicate that

the corresponding element of 119909 is negative) and in 119910 we havethe order (minus5 3 1 2 minus4) (so 119910

5is the largest component of

119910 and 1199104the smallest) Then the order of swaps is as follows

Swap 1 applied as (1 2 3larr997888rarr

minus4 minus5) gives (1 3 2 minus4 minus5)Swap 1 applied as (1 3

larr997888rarr2 minus4 minus5) gives (3 1 2 minus4 minus5)

Next Swap 2 applied as (3 1 2 minus4 minus 5larr997888997888997888997888rarr

) gives (3 1 2 minus

5 minus 4) Swap 3 applied as (3 1 2 minus 5larr997888997888997888rarr

minus 4) gives (3 1 minus

5 2 minus 4) Swap 3 applied as (3 1 minus 5larr997888997888997888rarr

2 minus 4) gives (3 minus

5 1 2 minus4) and finally Swap 3 applied as (3 minus 5larr997888997888997888rarr

1 2 minus4)

gives the order (minus5 3 1 2 minus 4) of 119910 Note also that in casethe elements of 119909 are not in that order we can always relabelthe indizes and measures correspondingly

This means if we compute (using 119909 =

(1199091 119909

2 119909

3 minus119909

4 minus119909

5))

(ASC) int(119910)

119909119889120583

= minus11990951205835+ 119909

3(120583

35minus 120583

5) + 119909

1(120583

135minus 120583

35)

+ 1199092(120583

1235minus 120583

135) minus 119909

4(120583

12345minus 120583

1235)

(A13)

instead of

(ASC) int 119909 119889120583

= 11990911205831+ 119909

2(120583

12minus 120583

1) + 119909

3(120583

123minus 120583

12)

minus 1199094(120583

1234minus 120583

123) minus 119909

5(120583

12345minus 120583

1234)

(A14)

then (ASC) int(119910)119909 119889120583 ge (ASC) int 119909 119889120583

Using this Lemma we can now easily show the following

Theorem 27 If 120583 is a supermodular measure then ge120583is a

transitive relation

Proof For transitivity we have to show that for any 119909 119910 and119911 such that 119909ge

120583119910 and 119910ge

120583119911 it follows that 119909ge

120583119911 Now 119909ge

120583119910

means (ASC) int(119909 minus 119910)119889120583 ge 0 and 119910ge120583119911 means (ASC) int(119910 minus

119911)119889120583 ge 0 From supermodularity of 120583 and Lemma 26 then italso follows that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 ge (ASC) int (119909 minus 119910) 119889120583 ge 0

(ASC) int(119909minus119911)

(119910 minus 119911) 119889120583 ge (ASC) int (119910 minus 119911) 119889120583 ge 0(A15)

However by using the fact that

(ASC) int(119909minus119911)

(119909 minus 119910) 119889120583 + (ASC) int(119909minus119911)

(119910 minus 119911) 119889120583

= (ASC) int (119909 minus 119911) 119889120583(A16)

we see that also

(ASC) int (119909 minus 119911) 119889120583 ge 0 (A17)

which means 119909ge120583119911

12 Journal of Applied Mathematics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Bertsimas V F Farias and N Trichakis ldquoThe price offairnessrdquo Operations Research vol 59 no 1 pp 17ndash31 2011

[2] J F Nash Jr ldquoThe bargaining problemrdquo Econometrica vol 18pp 155ndash162 1950

[3] E Kalai and M Smorodinsky ldquoOther solutions to Nashrsquosbargaining problemrdquo Econometrica vol 43 pp 513ndash518 1975

[4] J Rawls ATheory of Justice Harvard University Press 1999[5] F Kelly ldquoCharging and rate control for elastic trafficrdquo European

Transactions onTelecommunications vol 8 no 1 pp 33ndash37 1997[6] D P Bertsekas R G Gallager and P Humblet Data Networks

vol 2 Prentice-Hall International 1992[7] H P Young Equity In Theory and Practice Princeton Univer-

sity Press Princeton NJ USA 1995[8] J Mo and J Walrand ldquoFair end-to-end window-based conges-

tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[9] M Alicherry R Bhatia and L E Li ldquoJoint channel assignmentand routing for throughput optimization inmulti-radiowirelessmesh networksrdquo in Proceedings of the 11th Annual InternationalConference on Mobile Computing and Networking (MobiCom05) pp 58ndash72 ACM Cologne Germany August-September2005

[10] A Iyer C Rosenberg and A Karnik ldquoWhat is the rightmodel for wireless channel interferencerdquo IEEE Transactions onWireless Communications vol 8 no 5 pp 2662ndash2671 2009

[11] A Mishra V Shrivastava S Banerjee and W ArbaughldquoPartially overlapped channels not considered harmfulrdquo ACMSIGMETRICS Performance Evaluation Review vol 34 no 1 pp63ndash74 2006

[12] N Nie and C Comaniciu ldquoAdaptive channel allocation spec-trum etiquette for cognitive radio networksrdquo Mobile Networksand Applications vol 11 no 6 pp 779ndash797 2006

[13] C Peng H Zheng and B Y Zhao ldquoUtilization and fairnessin spectrum assignment for opportunistic spectrum accessrdquoMobile Networks and Applications vol 11 no 4 pp 555ndash5762006

[14] X Yang andA P Petropulu ldquoCo-channel interferencemodelingand analysis in a Poisson field of interferers in wireless commu-nicationsrdquo IEEE Transactions on Signal Processing vol 51 no 1pp 64ndash76 2003

[15] M Koppen ldquoRelational optimization and its application frombottleneck flow control to wireless channel allocationrdquo Infor-matica vol 24 no 3 pp 413ndash433 2013

[16] M Sugeno Theory of fuzzy integrals and its applications [PhDthesis] 1974

[17] S Michio ldquoFuzzymeasures and fuzzy integrals a surveyrdquo FuzzyAutomata and Decision Processes vol 78 no 33 pp 89ndash1021977

[18] D Denneberg Non-Additive Measure and Integral vol 27Springer 1994

[19] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[20] J Sipos ldquoIntegral with respect to a pre-measurerdquo MathematicaSlovaca vol 29 no 2 pp 141ndash155 1979

[21] M Grabisch and C Labreuche ldquoThe symmetric and asymmet-ric Choquet integrals on finite spaces for decision makingrdquoStatistical Papers vol 43 no 1 pp 37ndash52 2002

[22] M Grabisch C Labreuche and J-C Vansnick ldquoOn theextension of pseudo-Boolean functions for the aggregation ofinteracting criteriardquo European Journal of Operational Researchvol 148 no 1 pp 28ndash47 2003

[23] R Mesiar A Mesiarova-Zemankova and K Ahmad ldquoDiscreteChoquet integral and some of its symmetric extensionsrdquo FuzzySets and Systems vol 184 pp 148ndash155 2011

[24] A Mesiarova-Zemankova R Mesiar and K Ahmad ldquoThebalancing Choquet integralrdquo Fuzzy Sets and Systems vol 161no 17 pp 2243ndash2255 2010

[25] H Fouchal X Gandibleux and F Lehuede ldquoA lower bound ofthe Choquet integral integrated within Martinsalgorithmrdquo inNewState ofMCDM in the 21st Century vol 648 ofLectureNotesin Economics and Mathematical Systems pp 79ndash89 SpringerHeidelberg Germany 2011

[26] L Galand L Julien and P Patrice ldquoDominance rules forthe choquet integral in multiobjective dynamic programmingrdquoin Proceedings of the 23rd International Joint Conference onArtificial Intelligence pp 538ndash544 AAAI Press 2013

[27] A W Marshall I Olkin and B C Arnold Inequalities TheoryofMajorization and Its Applications Springer 2nd edition 2010

[28] W Ogryczak AWierzbicki andMMilewski ldquoAmulti-criteriaapproach to fair and efficient bandwidth allocationrdquoOmega vol36 no 3 pp 451ndash463 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Journal of Applied Mathematics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Bertsimas V F Farias and N Trichakis ldquoThe price offairnessrdquo Operations Research vol 59 no 1 pp 17ndash31 2011

[2] J F Nash Jr ldquoThe bargaining problemrdquo Econometrica vol 18pp 155ndash162 1950

[3] E Kalai and M Smorodinsky ldquoOther solutions to Nashrsquosbargaining problemrdquo Econometrica vol 43 pp 513ndash518 1975

[4] J Rawls ATheory of Justice Harvard University Press 1999[5] F Kelly ldquoCharging and rate control for elastic trafficrdquo European

Transactions onTelecommunications vol 8 no 1 pp 33ndash37 1997[6] D P Bertsekas R G Gallager and P Humblet Data Networks

vol 2 Prentice-Hall International 1992[7] H P Young Equity In Theory and Practice Princeton Univer-

sity Press Princeton NJ USA 1995[8] J Mo and J Walrand ldquoFair end-to-end window-based conges-

tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[9] M Alicherry R Bhatia and L E Li ldquoJoint channel assignmentand routing for throughput optimization inmulti-radiowirelessmesh networksrdquo in Proceedings of the 11th Annual InternationalConference on Mobile Computing and Networking (MobiCom05) pp 58ndash72 ACM Cologne Germany August-September2005

[10] A Iyer C Rosenberg and A Karnik ldquoWhat is the rightmodel for wireless channel interferencerdquo IEEE Transactions onWireless Communications vol 8 no 5 pp 2662ndash2671 2009

[11] A Mishra V Shrivastava S Banerjee and W ArbaughldquoPartially overlapped channels not considered harmfulrdquo ACMSIGMETRICS Performance Evaluation Review vol 34 no 1 pp63ndash74 2006

[12] N Nie and C Comaniciu ldquoAdaptive channel allocation spec-trum etiquette for cognitive radio networksrdquo Mobile Networksand Applications vol 11 no 6 pp 779ndash797 2006

[13] C Peng H Zheng and B Y Zhao ldquoUtilization and fairnessin spectrum assignment for opportunistic spectrum accessrdquoMobile Networks and Applications vol 11 no 4 pp 555ndash5762006

[14] X Yang andA P Petropulu ldquoCo-channel interferencemodelingand analysis in a Poisson field of interferers in wireless commu-nicationsrdquo IEEE Transactions on Signal Processing vol 51 no 1pp 64ndash76 2003

[15] M Koppen ldquoRelational optimization and its application frombottleneck flow control to wireless channel allocationrdquo Infor-matica vol 24 no 3 pp 413ndash433 2013

[16] M Sugeno Theory of fuzzy integrals and its applications [PhDthesis] 1974

[17] S Michio ldquoFuzzymeasures and fuzzy integrals a surveyrdquo FuzzyAutomata and Decision Processes vol 78 no 33 pp 89ndash1021977

[18] D Denneberg Non-Additive Measure and Integral vol 27Springer 1994

[19] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[20] J Sipos ldquoIntegral with respect to a pre-measurerdquo MathematicaSlovaca vol 29 no 2 pp 141ndash155 1979

[21] M Grabisch and C Labreuche ldquoThe symmetric and asymmet-ric Choquet integrals on finite spaces for decision makingrdquoStatistical Papers vol 43 no 1 pp 37ndash52 2002

[22] M Grabisch C Labreuche and J-C Vansnick ldquoOn theextension of pseudo-Boolean functions for the aggregation ofinteracting criteriardquo European Journal of Operational Researchvol 148 no 1 pp 28ndash47 2003

[23] R Mesiar A Mesiarova-Zemankova and K Ahmad ldquoDiscreteChoquet integral and some of its symmetric extensionsrdquo FuzzySets and Systems vol 184 pp 148ndash155 2011

[24] A Mesiarova-Zemankova R Mesiar and K Ahmad ldquoThebalancing Choquet integralrdquo Fuzzy Sets and Systems vol 161no 17 pp 2243ndash2255 2010

[25] H Fouchal X Gandibleux and F Lehuede ldquoA lower bound ofthe Choquet integral integrated within Martinsalgorithmrdquo inNewState ofMCDM in the 21st Century vol 648 ofLectureNotesin Economics and Mathematical Systems pp 79ndash89 SpringerHeidelberg Germany 2011

[26] L Galand L Julien and P Patrice ldquoDominance rules forthe choquet integral in multiobjective dynamic programmingrdquoin Proceedings of the 23rd International Joint Conference onArtificial Intelligence pp 538ndash544 AAAI Press 2013

[27] A W Marshall I Olkin and B C Arnold Inequalities TheoryofMajorization and Its Applications Springer 2nd edition 2010

[28] W Ogryczak AWierzbicki andMMilewski ldquoAmulti-criteriaapproach to fair and efficient bandwidth allocationrdquoOmega vol36 no 3 pp 451ndash463 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of