1

An Axiomatic Theory of Fairnessin Resource Allocation

Tian Lan1 and Mung Chiang2

1 Department of Electrical and Computer Engineering, George Washington University2 Department of Electrical Engineering, Princeton University

Abstract—We develop a set of five axioms for fairness measuresin resource allocation: the axiom of continuity, of homogeneity, ofsaturation, of partition, and of starvation. We prove that there isa unique family of fairness measures satisfying the axioms, whichis constructed and then shown to include Atkinson’s index, α-fairness, Jain’s index, entropy, and other “decomposable” globalmeasures of fairness as special cases. We prove properties offairness measures satisfying the axioms, including symmetryand Schur-concavity. Among the engineering implications is ageneralized Jain’s index that tunes the resolution of fairnessmeasure, a decomposition of α-fair utility functions into fairnessand efficiency components, and an interpretation of “larger α ismore fair” and of Rawl’s difference principle.

The axiomatic theory is further extended in three directions.First, we quantify fairness of continuous-dimension inputs, whereresource allocations vary over time or domain. Second, bystarting with both a vector of resource allocation and a vector ofuser-specific weights, and modifying the axiom of partition, wederive a new family of fairness measures that are asymmetricamong users. Finally, we develop a set of four axioms by removingthe axiom of homogeneity to capture a fairness-efficiency tradeoff.

We present illustrative examples in congestion control, routing,power control, and spectrum management problems in commu-nication networks, with the potential of a fairness evaluation toolexplored. We also compare with other work of axiomatization ininformation, computer science, economics, sociology, and politicalphilosophy.

I. INTRODUCTION

Given a vector x ∈ Rn+, where xi is the resource allocated

to user i, how fair is it? Consider two allocations among threeusers: x = [1, 2, 3] and y = [1, 10, 100]. Among the largevariety of choices we have in quantifying fairness, we can getfairness values such as 0.33 or 0.86 for x and 0.01 or 0.41for y: x is viewed as 33 times more fair than y, or just twiceas fair as y. How many such “viewpoints” are there? Whatwould disqualify a quantitative metric of fairness? Can theyall be constructed from a set of simple statements taken astrue for the sake of subsequent inference?

One existing approach to quantify fairness of x is througha fairness measure, which is a function f that maps x intoa real number. These measures are sometimes referred to asdiversity indices in statistics. Various fairness measures havebeen proposed throughout the years, e.g., in [1]–[6] in thefield of networking and many more in other fields. Theserange from simple ones, e.g., the ratio between the smallestand the largest entries of x, to more sophisticated functions,

This work has been supported in part by NSF CNS-0905086 and CNS-0519880.

e.g., Jain’s index and the entropy function. Some of thesefairness measures map x to a normalized range between 0and 1, where 0 denotes the minimum fairness, 1 denotes themaximum fairness (often corresponding to an x where all xi

are the same), and a larger value indicates more fairness. Forexample, min-max ratio [1] is given by the maximum ratioof any two user’s resource allocation, while Jain’s index [3]computes a normalized square mean. How are these fairnessmeasures related? Is one measure “better” than any other?What other measures of fairness might be useful?

An alternative approach that has gained attention in the net-working research community since [7], [8] is the optimization-theoretic approach of α-fairness and the associated utilitymaximization problem. Given a set of feasible allocations, amaximizer of the α-fair utility function satisfies the definitionof α-fairness. Two well-known examples are as follows: amaximizer of the log utility function (α = 1) is proportionallyfair, and a maximizer of the α-fair utility function as α →∞is max-min fair. More recently, α-fair utility functions havebeen connected to divergence measures [9]. In [10], [11], theparameter α was viewed as a fairness measure in the sensethat a fairer allocation is one that is the maximizer of an α-fair utility function with larger α – although the exact roleof α in trading-off fairness and throughput can sometimes besurprising [12]. While it is often believed that α →∞ is morefair than α = 1, which is in turn more fair than α = 0, itremains unclear what it means to say, for example, that α = 3is more fair than α = 2.

Clearly, these two approaches for quantifying fairness aredifferent. One difference is the treatment of efficiency, or mag-nitude, of resources. On the one hand, α-fair utility functionsare continuous and strictly increasing in each entry of x, thusits maximization results in Pareto optimal resource allocations.On the other hand, scale-invariant fairness measures (ones thatmap x to the same value as a normalized x) are unaffectedby the magnitude of x, and [1 1] is as fair as [100 100]. Canthe two approaches be unified?

To address the above questions, we develop an axiomaticapproach to fairness measures. We show that a set of fiveaxioms, each of which simple and intuitive, thus accepted astrue for the sake of subsequent inference, can lead to a usefulfamily of fairness measures. The axioms are: the axiom ofcontinuity, of homogeneity, of saturation, of partition, and ofstarvation. Starting with these five axioms, we can generatea family of fairness measures from generator functions g:any increasing and continuous functions that lead to a well-

defined “mean” function (i.e., from any Kolmogorov-Nagumofunction [13], [14]). For example, using power functions withexponent β as the generator function, we derive a uniquefamily of fairness measures fβ that include all of the followingas special cases, depending on the choice of β: Jain’s index,maximum or minimum ratio, entropy, and α-fair utility, andreveals new fairness measures corresponding to other rangesof β. While we start with the approach of fairness measurerather than optimization objective function, it turns out thatthe latter approach can also be recovered from fβ .

We also prove the uniqueness of fairness measures. First,we show that for each generator function g, its correspondingfairness measure f satisfying the five axioms is unique. In amuch stronger result, generator g must belong to the followingclass of functions: logarithm functions g(y) = log y and powergenerator functions g(y) = yβ . This result is sufficient andnecessary, in the sense that no other generator function ispossible, and for each of the generator function within thisclass, we can construct a unique fairness measure fβ satisfyingthe five axioms.

In particular, for β ≤ 0, several well-known fairnessmeasures (e.g., Jain’s index and entropy) are special cases ofour construction, and we generalize Jain’s index to providea flexible tradeoff between “resolution” and “strictness” ofthe fairness measure. For β ≥ 0, α-fair utility functions canbe factorized as the product of two components: our fairnessmeasure with β = α and a function of the total throughputthat captures the scale, or efficiency, of x. Such a factorizationquantifies a tradeoff between fairness and efficiency: what isthe maximum weight that can be given to fairness while stillmaintaining Pareto efficiency. It also facilitates an unambigu-ous understanding of what it means to say that a larger α is“more fair” for general α ∈ [0,∞).

Any fairness measure satisfying the five axioms can beproven to have many properties quantifying common beliefsabout fairness, including Schur-concavity [15]. Our fairnessmeasures presents a new ordering of Lorenz curves [6], [16],different from the Gini coefficient.

We extend this axiomatic theory in three directions. First,utilizing fairness measure f that satisfies the five axioms,we derive a new class of fairness measures to quantify thedegree of fairness associated with a time varying resourceallocation. In Section III, we tackle the following question:Given a continuous-dimension resource allocation over time[0, T ], represented by {Xt ∈ R, ∀t ∈ [0, T ]}. How do wequantify its degree of fairness using a function F? Our resultshows that fairness of continuous-dimension resource alloca-tion can be characterized by the limit of fairness measuresfor n-dimensional resource allocations. Fairness F can beequivalently computed from resource allocation over time,cumulative distribution, or the Lorenz curve. We prove a set ofinteresting properties and implications, including a generalizednotion of Schur-concavity.

Given a set of useful axioms, it is important to ask ifother useful axiomatic systems are possible. By removingor modifying some of the axioms, what kind of fairnessmeasures and properties would result? One limitation of theaxiom of partition is that the resulting fairness measures

are symmetric: labeling of users can be permuted withoutchanging fairness value. In practice, fairness measures shouldincorporate asymmetric function. For example, user i has aweight wi, which may depend on her needs or roles in the sizeof feasible region of allocations. In Section IV we proposean alternative set of five axioms and replace the Axiom ofPartition by a new version. Starting with this new set ofaxioms, we can generate asymmetric fairness measures, whichincorporates all special cases for symmetric fairness measure,as well as weighted p-norms and weighted geometric mean.Many properties previously proven for symmetric fairnessmeasures are revised by the user-specific weights. In particular,equal allocation is no longer fairness-maximizing.

This family of asymmetric fairness measures lead to ourdiscussion of a fairness evaluation tool, which can be usedto perform a consistency check of fairness evaluation, modeluser-specific weights for different applications, and tune thetradeoff between fairness and efficiency in design objectives.In Section IV.D, we present a human-subject experiment forfβ reverse engineering and fairness evaluation’s consistencycheck.

The development of fairness measures takes another turntowards the end of the paper in Section V. By removingthe Axiom of Homogeneity, we propose another set of fouraxioms, which allows efficiency of resource allocation to bejointly captured in the fairness measure. We show how thisalternative system connects with optimization-based resourceallocation, where magnitude matters due to the feasibilityconstraint and an objective function that favors efficiency. Thefairness measures F constructed out of the reduced set ofaxioms is a generalization of that out of the previous twosets of axioms. In particular, the entire α-fair utility function,rather than just its fairness component, is a special case of Fnow.

Starting from axioms, this paper focuses on developing amodeling “language” in quantifying fairness. Fairness measurethus constructed are illustrated in Appendix A using numericalexamples, from typical networking functionalities: congestioncontrol, routing, power control, and spectrum management, inAppendix A.

Axiomatization of key notions in bargaining, collaboration,and sharing are studied in information, science, computerscience, economics, sociology, psychology, and political phi-losophy over the past 60 years. In Appendix B, we compareand contrast with several of them, including the well-knowntheories on Renyi Entropy, Lorenz Curve, Nash BargainingSolutions, Shapley Value, Fair Division, as well as the twoprinciples in Rawls’ Theory of Justice as Fairness.

Some open questions are outlined in that appendix too. Allproofs are collected in Appendix C.

II. THE FIRST AXIOMATIC THEORY

A. A Set of Five Axioms

Let x be a resource allocation vector with n non-negativeelements. A fairness measure is a sequence of mapping{fn(x),∀n ∈ Zn} from n-dimensional vectors x to real num-bers, called fairness values, i.e., {fn : Rn

+ → R, ∀n ∈ Z+}.

2

Symbol Meaning

x Resource allocation vector of length n

w(x) Sum of all elements of x

f(·), fβ(·) Fairness measure (of parameter β)

β Parameter for power function g(y) = yβ

g(·) Generator function

si Positive weights for weighted mean

1n Vector of all ones of length n

x↑ Sorted vector with smallest element being first

x º y Vector x majorizes vector y

Uα(·) α-fair utility with parameter α

H(·) Shannon entropy function

J(·) Jain’s index

Φλ(·) Our multicriteria (fairness and efficiency) utility function

Xt∈[0,T ] Resource allocation over time [0, T ]

F(·) Fairness measure for continuous-dimensional inputs

Px(·) pdf of resource allocation over [0, T ]

Lx(·) Lorenz curve of resource allocation over [0, T ]

qj Positive constant weight of user j

Ii Set of user indices in sub-system i

F (·), Fβ,λ(·) Fairness-efficiency measure

TABLE IMAIN NOTATION.

To simplify notations, we suppress n in {fn} and denote themsimply as f . We introduce the following set of axioms aboutf , whose explanations are provided after the statement of eachaxiom.

1) Axiom of Continuity. Fairness measure f(x) is contin-uous on Rn

+, for all n ∈ Z+.

Axioms 1 is intuitive: A slight change in resource allocationshows up as a slight change in the fairness measure.

2) Axiom of Homogeneity. Fairness measure f(x) is ahomogeneous function of degree 0:

f(x) = f(t · x), ∀ t > 0. (1)

Without loss of generality, for n = 1, we take |f(x1)| = 1for all x1 > 0, i.e., fairness is a constant for a one-user system.

Axiom 2 says that the fairness measure is independent of theunit of measurement or magnitude of the resource allocation.For an optimization formulation of resource allocation, thefairness measure f(x) alone cannot be used as the objectivefunction if efficiency (which depends on magnitude

∑i xi) is

to be captured. In Section II-E, we will connect this fairnessmeasure with an efficiency measure in α-fair utility function.In Section V, we will remove Axiom 2 and propose analternative set of axioms, which make measure f(x) dependent

on both the magnitude and distribution of x, thus capturingfairness and efficiency at the same time.

3) Axiom of Saturation. Equal allocation’s fairness value isindependent of number of users as the number of usersbecomes large, i.e.,

limn→∞

f(1n+1)f(1n)

= 1. (2)

This axiom is a technical condition used to help ensureuniqueness of the fairness measure. Note that it is not statingthat equal allocation is most fair.

A primary motivation for quantifying fairness is to allowa comparison of fairness values. Therefore, we must ensurewell-definedness of ratio of fairness measures, as the numberof users in the system increases. Axiom 4 states that fairnesscomparison is independent of the way the resource allocationvector is reached as the system grows.

4) Axiom of Partition. Consider a partition of a system intotwo sub-systems. Let x =

[x1,x2

]and y =

[y1,y2

]be two resource allocation vectors, each partitioned andsatisfying

∑j xi

j =∑

j yij for i = 1, 2. There exists a

mean function h such that their fairness ratio is the meanof the fairness ratios of the subsystems’ allocations, forall partitions such that sum resources of each subsystemare the same across x and y:

f(x)f(y)

= h

(f(x1)f(y1)

,f(x2)f(y2)

)(3)

where according to the axiomatic theory of mean func-tion [13], a function h is a mean function if and only ifit can be expressed as follows:

h = g−1

(2∑

i=1

si · g(

f(xi)f(yi)

)), (4)

where g is any continuous and strictly monotonic func-tion, referred to as the Kolmogorov-Nagumo function,and si are positive weights such that

∑i si = 1.

5) Axiom of Starvation For n = 2 users, we have f(1, 0) ≤f( 1

2 , 12 ), i.e., starvation is no more fair than equal

allocation.

Axiom 5 is the only axiom that involves a value statementon fairness: starvation is no more fair than equal distributionfor two users. This value statement also holds for all existingfairness measures, e.g., various, spread, deviation, max-minratio, Jain’s index, α-fair utility, and entropy. We will later seein the proofs that Axiom 5 specifies an increasing direction offairness and is used to ensure uniqueness of f(x). We couldhave picked a different axiom to achieve similar effects, but theabove axiom for just 2 users and involving only starvation andequal allocation is the weakest statement, thus the “strongestaxiom”.

With the five axioms presented, we first discuss someimplications of Axiom 4. This axiom can construct fairnessmeasure f from lower dimensional spaces. If we choosey = [w(x1), w(x2)] (where w(xi) =

∑j xi

j is the sumof resource in sub-system i) in Axiom 4 and use the fact

3

that |f(w(xi))| = 1 for scalar inputs as implied by Axiom2, we show that Axiom 4 implies a hierarchical constructionof fairness, which allows us to derive a fairness measuref : Rn

+ → R of n users recursively (with respect to a generatorfunction g(y)) from lower dimensions, f : Rk

+ → R andf : Rn−k

+ → R for integer 0 < k < n.Lemma 1: (Hierarchical construction of fairness.) Consider

an arbitrary partition of vector x into two segments x =[x1,x2

]. For a fairness measure f satisfying Axiom 4, we

have

f(x) = f(w(x1), w(x2)

) · g−1

(2∑

i=1

si · g(∣∣f(xi)

∣∣))

. (5)

As a special case of Lemma 1, if we denote the resourceallocation at level 1 by a vector z = [w(x1), w(x2)] and if theresource allocation at level 2 are the same x1 = x2 = y, it isstraight forward to verify that Axioms 2 and 4 imply

f(y ⊗ z) = f(y · w(z)) · g−1

(∑

i

si · g (|f(z)|))

= f(y) · |f(z)| , (6)

where ⊗ is the direct product of two vectors. Intuitively,Equation (6) means that fairness of a direct product of tworesource allocations equals to the product of their fairnessvalues.

Initial:n∑

i=1

xi

↙ ↘

Level 1:k∑

i=1

xi

n∑

i=k+1

xi

↙ ↘ ↙ ↘Level 2: x1, . . . , xk xk+1, . . . , xn

TABLE IIILLUSTRATION OF THE HIERARCHICAL COMPUTATION OF FAIRNESS.

The recursive computation is illustrated by a two-levelrepresentation in Table II. Let x1 = [x1, . . . , xk] and x2 =[xk+1, . . . , xn]. The computation is performed as follows. Atlevel 1, since the total resource is divided into two “sub-systems”, fairness across the sub-systems obtained in this levelis measured by f

(w(x1), w(x2)

). At level 2, the two sub-

systems of resources are further allocated to k and n−k users,achieving fairness f(x1) and f(x2), respectively. To computeoverall fairness of the resource allocation x = [x1, x2, . . . , xn],we combine the fairness obtained in the two levels using (5).Axiom 4 states that the construction in (5) must be independentof the partition and generates the same fairness measure forany choice of 0 < k < n.

The functions g giving rise to the same fairness measures fmay not be unique, e.g., logarithm and power functions. Thesimplest case is when g is identity and si = 1/n for all i. Anatural parameterization of the weight si in (3) is to choose

the value proportional to the sum resource of sub-systems:

si =wρ(xi)∑j wρ(xj)

, ∀i (7)

where ρ ≥ 0 is an arbitrary exponent. As shown in SectionII-C, the parameter ρ can be chosen such that the hierarchicalcomputation is independent of partition as mandated in Axiom4.

By definition, axioms are true, as long as they are consistent.They should also be non-redundant. However, not all setsof axioms are useful: unifying known notions, discoveringnew measures and properties, and providing useful insights.We first demonstrate the following existence (the axioms areconsistent) results.

Theorem 1: (Existence.) There exists a fairness measuref(x) satisfying Axioms 1–5.

Uniqueness results contain two parts. First, we show that,from any Kolmogorov-Nagumo function g(y), there is aunique f(x) thus generated. Such an f(x) is a well-definedfairness measure if it also satisfies Axioms 1–5. We then referto the corresponding g(y) as the generator of the fairnessmeasure.

Definition 1: Function g(y) is a generator if there exists af(x) satisfying Axioms 1–5 with respect to g(y).Second, we further show that only logarithm and power func-tions are possible generator functions. Therefore, we find, inclosed-form, all possible fairness measures satisfying axioms1-5. The uniqueness result is summarized as the followingTheorem.

Theorem 2: (Uniqueness.) The family of f(x) satisfyingAxioms 1–5 is uniquely determined by logarithm and powergenerator functions,

g(y) = log y and g(y) = yβ (8)

where β ∈ R is a constant.

B. Constructing Fairness Measures

Let sign(·) be the sign function. We derive the fairnessmeasures in closed-form in the following theorem.

Theorem 3: (Constructing all fairness measures satisfyingAxioms 1-5.) If f(x) is a fairness measure satisfying Axioms1-5 with weights in (7), f(x) is of the form

f(x) = sign(r)n∏

i=1

(xi∑n

j=1 xj

)−r

(xi∑n

j=1 xj

)

(9)

f(x) = sign(r(1− βr)) ·

n∑

i=1

(xi∑j xj

)1−βr

1β

(10)

where r ∈ R is a constant exponent. In (10), the twoparameters in the mean function h have to satisfy rβ +ρ = 1,where ρ is in the weights of the weighted sum in (7)and β is inthe definition of the power generator function. It determinesthe growth rate of maximum fairness as population size nincreases, as will be shown in Corollary 3:

f(1n) = nr · f(1). (11)

4

The fairness measure in (9) can be obtained as a specialcase of (10) as β → 0. Obviously, sign(r(1− rβ)) → sign(r)as β → 0. By removing the sign of (10) and taking a logarithmtransformation, we have

limβ→0

log

n∑

i=1

(xi∑j xj

)1−βr

1β

= limβ→0

log[∑n

i=1

(xi∑j xj

)1−βr]

β

= limβ→0

n∑

i=1

−r log

(xi∑n

j=1 xj

)·(

xi∑nj=1 xj

)1−rβ

= logn∏

i=1

(xi∑n

j=1 xj

)−rxi∑n

j=1 xj

(12)

where the second step uses L’Hospital’s Rule. Therefore, wederive (9) by taking an exponential transformation in the laststep of (12). This shows that fairness measure in (9) is aspecial case of (10) as β → 0. Another way to see this isas follows: logarithm generator can be derived from the limitof the following power generators as β → 0:

log y = limβ→0

yβ − 1β

. (13)

As a result of uniqueness in Theorem 2, their correspondingfairness measures in (9) and (10) also become equivalent asβ → 0. From here on, we consider the family of fairnessmeasures in (10), which are parameterized by β in the powergenerator function.

For different values of parameter β, the fairness measuresderived above are equivalent up to a constant exponent r:

fβ,r(x) = [fβr,1]r (x), (14)

if we denote fβ,r to be the fairness measure with parametersβ and r.

According to Theorem 3, r determines the growth rate ofmaximum fairness as population size n increases. In most ofthe following developments, we choose r = 1, so that maxi-mum fairness per user is f(1n)/n = nr−1 = 1, independentof population size.

We now present a unified representation of the constructedfairness measures:

fβ(x) = sign(1− β) ·

n∑

i=1

(xi∑j xj

)1−β

1β

. (15)

In the rest of this section, we will show that this unifiedrepresentation leads to many implications.

We summarize the special cases in Table III, where βsweeps from −∞ to ∞ and H(·) denotes the entropy function.For some values of β, the corresponding mean function h hasa standard name, and for some values of β, known approachesto measure fairness are recovered, while for β ∈ (0,−1)and β ∈ (−1,−∞), new fairness measures are revealed. Forexample, when β = −1 (i.e., harmonic mean is used in Axiom

4), we recover Jain’s index J(x) = fβ(x)/n. For β ≥ 0, werecover the component of the α-fair utility functions that isrelated to fairness, as we will show in Section II-E that theα-fair utility function is equal to the product of our fairnessmeasure and a function of total resource for any β = α ≥ 0.The mapping from generators g(y) to fairness measures fβ(x)is illustrated in Figure 1.

g(y) = yβ ,

g(y) = log(y)

Fairness measure fβ(x)Generator g(y)

Shannon entropy∀β ∈ R ......

Jain’s indexα-fair utility

Fig. 1. The mapping from generators g(y) to fairness measures fβ(x).Among the set of all continuous and increasing functions, fairness measuressatisfying axioms 1-5 can only be generated by logarithm and power functions.A logarithm generator ends up with the same form of f as a special case ofthose generated by a power function.

This axiomatic theory unifies many existing fairness indicesand utilities. In particular, for β < 1, we recover the index-based approach in [1]–[6]. For β ≥ 0, we recover the utility-based approach in [7], [8], [10]–[12], which quantify fairnessfor the maximizers of the α-fair utility functions. For theinteresting range β ∈ [0, 1), the two approaches overlap.Notice that for β > 1 the measures are in fact negative.This is a consequence of the requirement in Axiom 5 that themeasure be monotonically increasing towards fairer solutions.As an illustration, for a fixed resource allocation vectorsx = [1, 2, 3, 5] and y = [1, 1, 2.5, 5], we plot fairnessfβ(x) and fβ(y) for different values of β in Figure 2. Differentvalues of β clearly changes the fairness comparison ration, andmay even result in different fairness orderings: fβ(x) ≥ fβ(y)for β ∈ (−∞, 4.6], and fβ(x) ≤ fβ(y) for β ∈ [4.6,∞).

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

Parameter β

Fai

rnes

s f β(x

)

x=[1,2,3,5]y=[1,1,2.5,5]

β>=0utility

β<1index

Fig. 2. For a given x, the plot of fairness fβ(x) and fβ(y) in (15) againstβ for x = [1, 2, 3, 5] and y = [1, 1, 2.5, 5]: β ≥ 0 recovers theutility-based approach, and β < 1 recovers the index-based approach.

5

Value of β Type of Mean Our Fairness Measure Known Names

β →∞ maximum −maxi

{ ∑i xi

xi

}Max ratio

β ∈ (1,∞) −[(1− β)Uα=β

(x

w(x)

)] 1β

α-fair utility

β ∈ (0, 1)[(1− β)Uα=β

(x

w(x)

)] 1β

α-fair utility

β → 0 geometric eH

(x

w(x)

)Entropy

β ∈ (0,−1)

[∑ni=1

(xi

w(x)

)1−βr] 1

β

No name

β = −1 harmonic (∑

i xi)2

∑i xi

2 = n · J(x) Jain’s index

β ∈ (−1,−∞)

[∑ni=1

(xi

w(x)

)1−βr] 1

β

No name

β → −∞ minimum mini

{ ∑i xi

xi

}Min ratio

TABLE IIIPREVIOUS RESULTS OF FAIRNESS METRICS IN NETWORKING ARE RECOVERED AS SPECIAL CASES OF OUR AXIOMATIC CONSTRUCTION . FOR β ∈ (0,−1)

AND β ∈ (−1,−∞), NEW FAIRNESS MEASURES OF GENERALIZED JAIN’S INDEX ARE REVEALED. FOR β ∈ [0,∞] THE FAIRNESS COMPONENT OFα-FAIR UTILITY FUNCTION IS RECOVERED, SEE SECTION II-E1. IN PARTICULAR, PROPORTIONAL FAIRNESS AT α = β = 1 IS OBTAINED FROM

limβ→1

|fβ(x)|β − n

|1− β| .

C. Properties of Fairness Measures

In this section, we prove a set of properties for all fairnessmeasures satisfying Axioms 1-5. We start an intuitive corollaryfrom the five axioms that will be useful for the rest of thepresentation. The first few can be readily proved directly fromthe axioms, without the explicit form in (15).

Corollary 1: (Symmetry.) A fairness measure satisfying Ax-ioms 1–5 is symmetric over x:

f(x1, x2, . . . , xn) = f(xi1 , xi2 , . . . , xin), (16)

where i1, . . . , in is an arbitrary permutation of indices1, . . . , n.

The symmetry property shows that the fairness measuref(x) satisfying Axioms 1–5 is independent of labeling ofusers. This corollary of the axioms may be viewed as objec-tionable by some. In Section IV, we will modify Axiom 4, sothat the resulting fairness measure will depend on user-specificweights.

We now connect of our axiomatic theory to a line of workon measuring statistical dispersion by vector majorization,including the popular Gini coefficient [17]. Majorization [15]is a partial order over vectors to study whether the elements ofvector x are less spread out than the elements of vector y. x ismajorized by y, and we write x ¹ y, if

∑ni=1 xi =

∑ni=1 yi

(always satisfied due to Axiom 2) andd∑

i=1

x↑i ≤d∑

i=1

y↑i , for d = 1, . . . , n, (17)

where x↑i and y↑i are the ith elements of x↑ and y↑, sortedin ascending order. According to this definition, among thevectors with the same sum of elements, one with the equalelements is the most majorizing vector.

Intuitively, x ¹ y can be interpreted as y being a fairerallocation than x. It is a classical result [15] that x is majorizedby y, if and only if, from x we can produce y by a finitesequence of Robin Hood operations.1

However, majorization alone cannot be used to define afairness measure since it is a partial order and can fail tocompare vectors. Still, if resource allocation x is majorizedby y, it is desirable to have a fairness measure f such thatf(x) ≤ f(y). A function satisfying this property is knownas Schur-concave. In statistics and economics, many measuresof statistical dispersion or diversity are known to be Schur-concave, e.g., Gini Coefficient and Robin Hood Ratio [17],and we show our fairness measure also is Schur-concave.More discussions on the relationship of our axioms and othereconomics theories are provided in Appendix B.

Theorem 4: (Schur-concavity.) A fairness measure satisfy-ing Axioms 1–5 is Schur-concave:

f(x) ≤ f(y), if x ¹ y. (18)

Next we present several properties of fairness measuressatisfying the axioms, whose proofs rely on Schur-concavity.

Corollary 2: (Equal-allocation is most fair.) A fairnessmeasure f(x) satisfying Axioms 1–5 is maximized by equal-resource allocations, i.e.,

f(1n) = maxx∈Rn

f(x). (19)

Corollary 3: (Equal-allocation fairness value is indepen-dent of g.) Fairness achieved by equal-resource allocations 1n

1In a Robin Hood operation, we replace two elements xi and xj < xi

with xi − ε and xj + ε, respectively, for some ε ∈ (0, xi − xj).

6

is independent of the choice of generator function g, i.e.,

f(1n) = nr · f(1), (20)

where r is a constant exponent.

Corollary 4: (Inactive users do not change fairness.) Whena fairness measure f(x) satisfying Axioms 1–5 is generatedby ρ > 0 in (7), adding or removing users with zero resourcesdoes not change fairness:

f(x,0n) = f(x), ∀n ∈ Z+. (21)

The following corollaries’ proofs will directly use theunified representation in (15). First, by specifying a level offairness, we can limit the number of starved users in a system.

Corollary 5: (Fairness value bounds the number of inactiveusers.) Fairness measures satisfying Axioms 1-5 count thenumber of inactive users in the system. When fβ < 0,f(x) → −∞ if any user is assigned zero resource. Whenf > 0,

Number of users with zero resource ≤ n− f(x).(22)

Corollary 6: (Fairness value bounds the maximum resourceto a user.) Fairness measures satisfying Axioms 1-5 bound themaximum resource to a user when f > 0:

Maximum resource to a user ≥∑

i xi

f(x). (23)

Corollary 7: (Perturbation of fairness value from slightchange in a user’s resource.) If we increase resource allocationto user i by ε > 0, while not changing other users’ allocation,fairness measures satisfying Axioms 1-5 increase if and only

if xi < x =( ∑

j xj∑j x1−β

j

) 1β

and for all 0 < ε < x− xi.

This corollary implies that x serves as a threshold for iden-tifying “poor” and “rich” users, since assigning an additionalε amount of resource to user i improves fairness if xi < x,while the same assignment reduces fairness if xi > x.

Corollary 8: (Box constraints of resources allocation boundfairness value.) If a resource allocation x = [x1, x2, . . . , xn]satisfies box-constraints, i.e., xmin ≤ xi ≤ xmax for all i,fairness measures satisfying Axioms 1-5 are lower boundedby a constant that only depends on β, xmin, xmax:

f(x) ≥ sign(1− β) ·(µΓ1−β + 1− µ

) 1β

(µΓ + 1− µ)1β−1

, (24)

where Γ = xmax

xminand µ = Γ−Γ1−β−β(Γ−1)

β(Γ−1)(Γ1−β−1). The bound is tight

when µ fraction of users have xi = xmax and the remaining1− µ fraction of users have xi = xmin.

This corollary provides us with a lower bound on evaluationof the fairness measure, with which one might benchmarkempirical values.

D. Implication 1: Generalizing Jain’s Index

When β = −1 (i.e., harmonic mean is used in Axiom 4),f becomes a scalar multiple of the widely used Jain’s indexJ(x) = 1

nf(x). Upon inspection of (15) and the specific casesnoted in Table III, we note that any β ∈ (−∞, 1) the range offairness measure fβ(x) lies between 1 and n. Equivalently, wecan say that the fairness per user resides in the interval

[1n , 1

].

We refer to this subclass of our family of fairness measuresfor β ∈ (−∞, 1) as the generalization of Jain’s index.

Definition 2: Jβ(x) = 1nfβ(x) is a generalized Jain’s index

parameterized by β ≤ 1.The common properties of our fairness index proven in

Section II-C carry over to this generalized Jain’s index. Forβ = −1, J−1(x) reduces to the original Jain’s index. Theparameter β determines the choice of generator functiong(y) = yβ in Axiom 4. We use fβ(x) to denote the fairnessmeasures in (15), parameterized by β. We first prove that, fora given resource allocation x, fairness fβ(x) is monotonic asβ → 1. Its implication is discussed next.

Theorem 5: (Monotonicity with respect to β.) The fairnessmeasures in (15) is negative and decreasing for β ∈ (1,∞),and positive and increasing for β ∈ (−∞, 1):

∂fβ(x)∂β

≤ 0 for β ∈ (1,∞), (25)

∂fβ(x)∂β

≥ 0 for β ∈ (−∞, 1). (26)

As β → 1, f point-wise converges to constant values:

limβ↑1

fβ(x) = n and limβ↓1

fβ(x) = −n. (27)

The monotonicity of fairness measures fβ(x) on β ∈(−∞, 1) gives an interpretation of β. Figure 3 plots fairnessfβ(θ, 1−θ) for resource allocation x = [θ, 1−θ] and differentchoices of β = {−4.0,−2.5,−1.0, 0.5}. The vertical bars inthe figure represent the level sets of function f , for valuesfβ(θi, 1− θi) = i

10 (fmax − fmin) , i = 1, 2, . . . , 9. For fixedresource allocations, since f increases as β approaches 1, thelevel sets of f are pushed toward the region with small θ (i.e.,the low-fairness region), resulting in a steeper incline in theregion. In the extreme case of β = 1, all level set boundariesalign with the y-axis in the plot. The fairness measure f point-wise converges to step functions fβ(θ, 1− θ) = 2. Therefore,parameter β characterizes the shape of the fairness measures:a smaller value of |1− β| (i.e., β closer to 1) causes the levelsets to be more condensed in the low-fairness region.

Since the fairness measure must still evaluate to a numberbetween 1 and n here, the monotonicity, and the resultingchange in the resolution of fairness measure associated withvarying β, also lead to differences in how unfairness isevaluated. At one extreme, as β → 1 any solution where nouser receives an allocation of zero is fairest. On the other hand,as β → −∞ the relationship between fβ(x) and θ becomeslinear, suggesting a stricter concept of fairness – for the sameallocation, as β → −∞ more fairness is lost. Therefore, theparameter β can tune generalized Jain’s index f for differenttradeoffs between resolution and strictness of fairness measure.

7

0 0.1 0.2 0.3 0.4 0.51

1.52

β=+0.5f

0 0.1 0.2 0.3 0.4 0.51

1.52

β=−1.0

f

0 0.1 0.2 0.3 0.4 0.51

1.52

β=−2.5

f

0 0.1 0.2 0.3 0.4 0.51

1.52

β=−4.0

f

0 0.1 0.2 0.3 0.4 0.5−10

2

−100

β=+1.5

f

0 0.1 0.2 0.3 0.4 0.5−10

2

β=+3.0

f

0 0.1 0.2 0.3 0.4 0.5−10

2

−100

β=+4.5

f

0 0.1 0.2 0.3 0.4 0.5−10

2

−100

β=+6.0

f

Fig. 3. Plot of the fairness measure fβ(θ, 1− θ) against θ, for resource allocation x = [θ, 1− θ] and different choices of β = {−4.0,−2.5,−1.0, 0.5}and β = {1.5, 3.0, 4.5, 6.0}, respectively. It can be observed that fβ(θ, 1− θ) is monotonic as β → 1. Further, smaller values of |1−β| results in a steeperincline over small θ, i.e., the low-fairness region.

If the fairness measure f is used for classifying differentresource allocations, a larger β is desirable, since it givesmore quantization levels in low-fairness region and providesfiner granularity control for unfair resource. On the other hand,if the fairness measure f is used as an objective function, asmaller β is desirable, since it has a steeper incline in thelow-fairness region and give more incentive for the system tooperate in the high-fairness region.

E. Implication 2: Understanding α-Fairness

Due to Axiom 2, our fairness measures are homogeneousfunctions of degree zero and only express desirability overthe (n − 1)-dimension subspace orthogonal to the 1n vector.Hence, they do not capture any notion of efficiency of anallocation. The component of resource vectors along the vector1n describes another quantity used to capture the efficiency ofan allocation, as a function of the sum w(x) of resources.

We focus in this subsection on the widely applied α-fairutility function:

∑

i

Uα(xi), where Uα(x) =

{x1−α

1−α α ≥ 0, α 6= 1log(x) α = 1

.

(28)We first show that the α-fairness network utility function canbe factored into two components: one corresponding to thefamily of fairness measures we constructed and one corre-sponding to efficiency. We then demonstrate that, for a fixed α,the factorization can be viewed as a single point on the optimaltradeoff curve between fairness and efficiency. Furthermore,this particular point is one where maximum emphasis isplaced on fairness while maintaining Pareto optimality of theallocation. This allows us to quantitatively interpret the beliefof “larger α is more fair” across all α ≥ 0, not just forα ∈ {0, 1,∞}. .

1) Factorization of α-fair Utility Function: Re-arrangingthe terms of the equation in Table III, we have

Uα=β(x) =1

1− β|fβ(x)|β

(∑

i

xi

)1−β

= |fβ(x)|β · Uβ

(∑

i

xi

), (29)

where Uβ (∑

i xi) is the uni-variate version of the α-fair utilityfunction with α = β ∈ [0,∞). Our fairness measure fβ(x)captures proportional fairness at α = β = 1 and max-minfairness at α = β →∞.

Equation (29) demonstrates that the α-fair utility functionscan be factorized as the product of two components: a fairnessmeasure, |fβ(x)|β , and an efficiency measure, Uβ (

∑i xi).

The fairness measure |fβ(x)|β only depends on the normalizeddistribution, x/(

∑i xi), of resources (due to Axiom 2), while

the efficiency measure is a function of only the sum resource∑i xi.

Allocation: x

↙ ↘Factorize: x/

∑i xi

∑i xi

↓ ↓Measure: fβ

(x/

∑i xi

)Uβ

(∑i xi

)

↘ ↙Combine: Uα=β(x)

TABLE IVILLUSTRATION OF THE FACTORIZATION OF THE α-FAIR UTILITY

FUNCTIONS INTO A FAIRNESS COMPONENT OF THE NORMALIZEDRESOURCE DISTRIBUTION AND A EFFICIENCY COMPONENT OF THE SUM

RESOURCE.

The factorization of α-fair utility functions is illustrated inTable IV and decouples the two components to tackle issuessuch as fairness-efficiency tradeoff and feasibility of x undera given constraint set. For example, it helps to explain thecounter-intuitive throughput behavior in congestion controlbased on utility maximization [12]: an allocation vector that

8

maximizes the α-fair utility with a larger α may not be lessefficient, because the α-fair utility incorporates both fairnessand efficiency at the same time.

The additional efficiency component in (29) can skew theoptimizer (i.e., the resource allocation resulting from α-fairutility maximization) away from an equal allocation. Forthis to happen there must exist an allocation that is feasible(within the constraint set of realizable allocations) with a largeenough gain in efficiency over all equal allocation allocations.Hence, the magnitude of this skewing depends on the fairnessparameter (α = β), the constraint set of x, and the relativeimportance of fairness and efficiency.

Guided by the product form of (29), we consider a scalar-ization of the maximization of the two objectives: fairness andefficiency:

Φλ(x) = λ` (fβ (x)) + `

(∑

i

xi

), (30)

where β ∈ [0,∞) is fixed, λ ∈ [0,∞) absorbs the exponent βin the fairness component of (29) and is a weight specifyingthe relative emphasis placed on the fairness, and

`(y) = sign(y) log(|y|). (31)

The use of the log function later recovers the product in thefactorization of (29) from the sum in (30).

2) What Does “Larger α is More Fair” Mean?: It is com-monly believed that larger α is more fair, but it is not exactlyclear what this statement means for general α ∈ [0,∞).Guided by the factorization above and the axiomatic construc-tion of fairness measures, we provide two interpretation ofthis statement that justify it from the viewpoints of Paretooptimality and geometry of the constraint set.

An allocation vector x is said to be Pareto dominated by y ifxi ≤ yi for all i and xi < yi for at least some i. An allocationis called Pareto optimal if it is not Pareto dominated by anyother feasible allocation. If the relative emphasis on efficiencyis sufficiently high, Pareto optimality of the solution can bemaintained. To preserve Pareto optimality, we require that ify Pareto dominates x, then Φλ(y) > Φλ(x).

Theorem 6: (Preserving Pareto optimality.) The necessaryand sufficient condition on λ, such that Φλ(y) > Φλ(x) ify Pareto dominates x, is that λ must be no larger than athreshold, which only depends on β:

λ ≤ λ ,∣∣∣∣

β

1− β

∣∣∣∣ . (32)

Consider the set of maximizers of (30) for λ in the rangein Theorem 6:

P ={x : x = arg max

x∈RΦλ(x), ∀λ ≤

∣∣∣∣β

1− β

∣∣∣∣}

. (33)

When weight λ = 0, the corresponding points in P is mostefficient. When weight λ =

∣∣∣ β1−β

∣∣∣, it turns out that thefactorization in (29) becomes the same as (30).

Summarizing the first interpretation of “larger α is morefair”: α-fairness corresponds to the solution of an optimization

that places the maximum emphasis on the fairness measureparameterized by β = α while preserving Pareto optimality.

Allocations in P corresponding to other values of λ achievea tradeoff between fairness and efficiency, while Pareto opti-mality is preserved.

x2

x1

A

B

Fair

1

2

4

(a)

2 2.5 3 3.5 4

−101

∑

i xi

fβ(x

)

A

λ =∣

∣

∣

β

1−β

∣

∣

∣

B

β = 3

(b)

Fig. 4. (a) Feasible region (i.e., the constraint set of the utility maximizationproblem) where overemphasis of fairness violates Pareto dominance, and (b)its fairness-efficiency tradeoff for β = 3. Curve A corresponds to Paretooptimal solutions. Curve B is when the condition of Theorem 6 is violated,and resource allocations are more fair but no longer Pareto optimal.

Figure 4(b) illustrates an optimal fairness-efficiency tradeoff

curve{[

fβ(x), Σixi

], ∀x = arg maxx∈RΦλ(x), ∀λ

}cor-

responding to the constraint set shown in Figure 4(a). The setof optimizers P in (33), which is obtained by maximizing Φin (30), is shown by curve A in Figure 4(b). Curve B is whenthe condition of Theorem 6 is violated, and resource allocationis more fair but no longer Pareto optimal. Relationship withRawl’s theory of fairness will be explored in Appendix B.

3) Another Meaning of “Larger α is More Fair”.: We justdemonstrated the factorization (29) is an extreme point on thetradeoff curve between fairness and efficiency for fixed β = α.What happens when α becomes bigger?

We denote by 5x the gradient operator with respect to thevector x. For a differentiable function, we use the standardinner product (〈x,y〉 =

∑i xiyi) between the gradient of

the function and a normalized vector to denote the directionalderivative of the function.

Theorem 7: (Monotonicity of fairness-efficiency reward ra-tio.) Let allocation x be given. Define η = 1

n1n − x∑xi

asthe vector pointing from the allocation to the nearest fairnessmaximizing solution. Then the fairness-efficiency reward ratio:

⟨5xUα=β(x),

η

‖η‖⟩

⟨5xUα=β(x),

1n

‖1n‖⟩ , (34)

is non-decreasing with α, i.e., higher α gives a greater relativereward for fairer solutions.

The choice of direction η is a result of Axiom 2 andCorollary 2, which together imply that η is the directionthat increases fairness most and is orthogonal to increases inefficiency. Figure 5 illustrates Theorem 7 by showing the two

9

6 6.5 7 7.5 8 8.5 9 9.5 102.5

3

3.5

4

4.5

5

5.5

Gradient

x

η

θ2

θ1

Equal Allocation

1Maximum Throughput

Fig. 5. This figure illustrates the two directions in Theorem 7: η (whichpoints to the nearest fairness maximizing solution) and 1 (which points to thedirection for maximizing total throughput).

directions: η (which points to the nearest fairness-maximizingsolution) and 1 (which points to the direction for maximizingtotal throughput).

An increase in either fairness or efficiency is a “desirable”outcome. The choice of α dictates exactly how desirable oneobjective is relative to the other (for a fixed allocation). Theo-rem 7 states that, with a larger α, there is a larger component ofthe utility function gradient in the direction of fairer solutions,relative to the component in the direction of more efficiency.Economically, this could be seen as decreasing the pricingof fairer allocations relative to a fixed price on allocationsof increased throughput. Notice, however, that comparison isin terms of the ratio between these two gradient componentsrather than the magnitude of the gradient, and both fairnessand efficiency may increase simultaneously.

Theorem 7 provides the second interpretation of “larger αis more fair”, again, not just for α ∈ {0, 1,∞}, but for anyα ∈ [0,∞). Figure 6 depicts how this ratio increases withα = β for some examples allocations.

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

α

Fai

rnes

s−E

ffici

ency

Rew

ard

Rat

io

x = [25,

3

5]

x = [14,

1

2]

x = [ 49

100,

51

100]

Fig. 6. Monotonic behavior of the ratio (34) as a function of α. Three fixedallocations over two users are considered, and solutions that are already morefair have a lower ratio.

This concludes the discussion of the first set of axioms offairness. Obviously many interesting issues are missing: there

is no time evolution of resource allocation, no differentiationamong users, and no mentioning of efficiency in fβ . Some ofthe propositions, such as equal allocation maximizes fairness,may also be viewed as objectionable, thus motivating us torevisit the axioms. The next three sections carry out some ofthese extensions and revisions.

III. CONTINUOUS-DIMENSIONAL ALLOCATIONS

A. Constructing Fairness MeasuresConsider a resource allocation over time [0, T ], represented

by Xt∈[0,T ]. The evolution over time may be due to a controlalgorithm or network dynamics. How do we quantify thefairness of Xt∈[0,T ] using fairness measures?

In the previous section, we studied fairness measures f :Rn

+ → R, ∀n ∈ Z+, where resource allocation is representedby a non-negative vector x. Utilizing these fairness measures,in this section we derive a new class of fairness measures,whose input Xt∈[0,T ] is integrable over time [0, T ]. Themain idea is to divide [0, T ] into n subintervals, regard eachsubinterval as a resource element, and approximate the fairnessof Xt∈[0,T ] by the fairness of an n-dimensional resource vectoras n →∞.

We start with fairness measure given by (15) in Section II.Due to Corollary 2 and Corollary 3, the maximum of fairnessis achieved by equal resource allocations and is dependenton the number of resource elements, i.e., f(1n) = nr. Wenormalize the fairness measure by nr, so that 0 ≤ n−r ·f(x) ≤1 for all n > 0 and non-negative x. We have

n−rf(x1, . . . , xn)

= sign(r(1− βr)) · n−r ·

∑ni=1 x1−βr

i(∑j xj

)1−βr

1β

.(35)

We partition the interval [0, T ] by

0 = t0 < t1 < t2 < . . . < tn = T, (36)

where ti = iT/n = i∆ and ∆ = T/n. We regard each subin-terval by a resource element and approximate the resourceallocation Xt by an n-dimensional resource vector

[x1, . . . , xn] = [Xt1 , . . . ,Xtn ]. (37)

Using (35) and (37), we have

n−rf(x1, . . . , xn)= n−rf(Xt1 , . . . ,Xtn)

= sign(r(1− βr)) · n−r ·

∑ni=1 X1−βr

ti(∑j Xtj

)1−βr

1β

= sign(r(1− βr)) · ∆r

T r·

∑ni=1 X1−βr

ti∆1−βr

(∑j Xtj ∆

)1−βr

1β

=sign(r(1− βr))

T r·

∑ni=1 X1−βr

ti∆

(∑j Xtj ∆

)1−βr

1β

(38)

10

Since Xt∈[0,T ] is integrable, we define a fairness measure forresource allocation over time by taking n →∞:

F(Xt∈[0,T ])

, limn→∞

sign(r(1− βr))T r

·

∑ni=1 X1−βr

ti∆

(∑j Xtj

∆)1−βr

1β

=sign(r(1− βr))

T r·

∫ T

t=0X1−βr

t dt(∫ T

t=0Xtdt

)1−βr

1β

= sign(r(1− βr)) ·[

1T

∫ T

t=0

(Xt/µ)1−βrdt

] 1β

(39)

where µ is the average resource over [0, t] and is given by

µ , 1T

∫ T

t=0

Xtdt. (40)

Equation (39) defines a family of fairness measures forresource allocation Xt∈[0,T ], as β sweeps from −∞ to ∞.It is well-defined due to integrability of Xt∈[0,T ]. It can beviewed as a continuous version of fairness measure (15) inSection II-B. For β = −1, fairness (39) extends Jain’s indexto resource allocation over time [0, T ], while for β ≥ 0,it recovers α-fair utility functions. Similar to its counterpart(15), fairness F(X) satisfy a number of properties, whichgive interesting engineering implications. We first prove twoequivalent representations for fairness in (39), with respect tocumulative distribution and to Lorenz curve [18] of resourceallocation Xt∈[0,T ]:

Theorem 8: (Equivalent representations.) For arbitrary r, β,fairness F(X of resource allocation Xt∈[0,T ] is equivalent tothe following two representations, with respect to cumulativedistribution Px(y) and to Lorenz curve Lx(u), respectively:

F = sign(r(1− βr)) ·[∫ ∞

y=0

(y

µ

)1−βr

dPx(y)

] 1β

(41)

F = sign(r(1− βr)) ·[∫ 1

u=0

(P−1

x (u)µ

)−βr

dLx(u)

] 1β

(42)

where cumulative distribution of Xt∈[0,T ] is given by

Px(y) =1T·∫

{Xt≤y}dt (43)

and Lorenz curve [18] of Xt∈[0,T ] is given by

Lx(u) =1µ·∫

{Px(y)≤u}ydPx(y). (44)

This theorem shows that the fairness over time can becalculated through cumulative distribution or Lorenz curveof Xt∈[0,T ]. Not only does it facilitate the computation offairness over time, it also extends majorization and Schur-concavity, which is proven in Section III-C for n-dimensionalinputs, to resource allocation Xt∈[0,T ]. We define generalized

majorization of resource allocations according to their Lorenzcurves. We say that Xt∈[0,T ] is majorized by Yt∈[0,T ]:

Xt∈[0,T ] ¹ Yt∈[0,T ], if Lx(u) ≤ Ly(u), ∀u ∈ [0, 1] (45)

where Lx and Ly are the Lorenz curve of Xt∈[0,T ], as definedby (44). According to this definition, it is easy to see that,among resource allocation over [0, T ], one with the equalresource over time is the most majorizing allocation.

B. Properties and Implications

We present several properties of fairness measures. Someof the proofs rely on Schur-concavity, which is proven in thefollowing corollary.

Theorem 9: (Schur-concavity.) The fairness measure F in(39) is Schur-concave:

F(Xt∈[0,T ]) ≤ F(Yt∈[0,T ]), if Xt∈[0,T ] ¹ Yt∈[0,T ]. (46)

Corollary 9: (Equal-allocation is most fair.) Fairness mea-sure F is maximized by an equal allocation over time, i.e.,{Xt = c : t ∈ [0, T ], c > 0}.

Corollary 10: (Equal-allocation fairness value is indepen-dent of T .) The fairness achieved by equal allocations overtime is independent of the length of time T , i.e.,

maxXt∈[0,T ]

F(Xt∈[0,T ]) ={

1 if 1− rβ ≥ 0−1 o.w.

(47)

Corollary 11: (Negative shift reduces fairness.) If a fixedamount c > 0 of the resource is subtracted for all t ∈ [0, T ],the resulting fairness measure decreases

F(Xt∈[0,T ] − c) ≤ F(Xt∈[0,T ]), ∀c > 0, (48)

where c > 0 must be small enough such that F(Xt∈[0,T ])− cis positive over t ∈ [0, T ].

Corollary 12: (Starvation time reduces fairness.) For allr, β ∈ R, adding a time interval of length τ > 0 with zeroresource reduces fairness by a factor of (1 + τ/T )−r

F(Xt∈[0,T ])⋃

0t∈[T,T+τ ]) = (1 + τ/T )−r · F(Xt∈[0,T ]). (49)

Corollary 13: (Fairness value bounds the length of starva-tion time intervals.) When 1 − rβ > 0, fairness measure Fgives an upper bound of starvation time intervals (with zeroresource) in the system, i.e.,

Length of starvation time ≤ T · [|F|−1/r − 1]. (50)

Corollary 14: (Fairness value bounds peak resource.)When 1 − rβ > 0, fairness F bounds the peak resourceallocation over time [0, T ]:

maxt∈[0,T ]

Xt ≥ µ

2− |F|−1/r(51)

Corollary 15: (Box constraints of resources allocationbound fairness value.) If a resource allocation Xt∈[0,T ] satis-fies box-constraints, i.e., xmin ≤ Xt ≤ xmax for all t ∈ [0, T ],

11

the fairness measures F is lower bounded by a constant thatonly depends on β, rxmin, xmax:

F ≥ sign(r(1− β))

[θx1−rβ

max + (1− θ)x1−rβmin

(θxmax + (1− θ)xmin)1−rβ

] 1β

, (52)

where θ is the fraction of time that Xt∈[0,T ] equals to xmax

and is given by

θ =1

1− ρ·[(Γ− 1)ρ

Γρ − 1− 1

](53)

where Γ = xmax/xmin and ρ = 1 − rβ. The bound is tightwhen for θ fraction of time Xt∈[0,T ] equals to xmax and orthe remaining 1− θ fraction of time Xt∈[0,T ] equals to xmin.

Corollary 16: (Monotonicity with respect to β.) For a fixedresource allocation Xt∈[0,T ] with T = 1, fairness measure Fis negative and decreasing for β ∈ (1,∞), and positive andincreasing for β ∈ (−∞, 1):

∂F(Xt∈[0,T ])∂β

≤ 0 for β ∈ (1,∞), (54)

∂F(Xt∈[0,T ])∂β

≥ 0 for β ∈ (−∞, 1). (55)

The above results extend properties of f in the last section tofairness measures F . Through Corollary 16 to Corollary 17,by specifying a level of fairness, we can limit the time ofstarvation. Corollary 18 implies that peak resource over timet ∈ [0, T ] is bounded by average resource and fairness -For given average resource, a smaller fairness requires anallocation with larger peak resource. Finally, Corollary 19provides us with a lower bound on evaluation of the fairnessmeasure, with which one may better benchmark empiricalvalues.

C. Two-Dimensional Fairness Over Time and Users

In order to extend the model to be two-dimensional, wecomposite two different fairness measures among users andacross time, respectively. This technique is also recently usedin [44] to introduce multiscale fairness measures.

Given a resource allocation Xt∈[0,T ] =[X1,t∈[0,T ], . . . , Xn,t∈[0,T ]

]over time [0, T ] and for n

users. We can first apply F derived in this section to measureeach user’s fairness over time and then aggregate thesefairness values using previous f with respect to user-specificweights [q1, . . . , qn]. Therefore, we define the followingtwo-dimensional fairness measures:

f ◦ F(Xt∈[0,T ]) (56)

, f([F(X1,t∈[0,T ]), . . . ,F(Xn,t∈[0,T ])

], [q1, . . . , qn]

)

An alternative approach is to first use f to compute fairness foreach time t ∈ [0, T ] and then apply F to measure its dynamicover time. This second approach gives a different definition oftwo-dimensional fairness measures:

F ◦ f(Xt∈[0,T ]) (57)

, F (f

([X1,t∈[0,T ], . . . , Xn,t∈[0,T ]

], [q1, . . . , qn]

), ∀t)

A desirable property is to choose f and F , such that thetwo-dimensional fairness measures in (56) and (57) becomethe same, i.e., f ◦ F(Xt∈[0,T ]) = F ◦ f(Xt∈[0,T ]), for alldistributions Xt∈[0,T ]. Therefore, the same fairness value isguaranteed no matter which dimension of fairness is measuredfirst.

Corollary 17: (Two-dimensional fairness measures.) For alldistributions Xt∈[0,T ], we have f ◦ F(Xt∈[0,T ]) = F ◦f(Xt∈[0,T ]) if and only if both fairness measures f and Fchoose the same set of parameters: 1/λ = 1, r = −1, andβ ∈ R. The resulting two-dimensional fairness measures aregiven by

sign(1− β) ·[

n∑

i=1

qi

T·∫ T

t=0

(Xi,t)β

dt

] 1β

. (58)

IV. ASYMMETRIC USERS

If we want to construct fairness measures with propertiesdifferent from those proven in Section II, we have to revisitthe five axioms in Section II and modify or remove some ofthem. For instance, one limitation of Axioms 1-5 in SectionII is that the resulting fairness measures are symmetric. As aresult, equal resource allocation is always most fair, irrelevantof users’ requirements and contributions to the system. To begeneral enough in many contexts, fairness measures shouldincorporate asymmetric functions. Towards this end, in thissection we propose a new set of five axioms, which result ina different family of asymmetric fairness measures. We startwith two vectors now: x the resource allocation and q theuser-specific weights, which may be arrived at from differentconsiderations including the needs and contributions of eachuser.

A. A Second Set of Five Axioms

Given two n-dimensional vectors, x and q, let fairnessmeasure {fn(x,q),∀n ∈ Z+} be a sequence of mapping from(x,q) to real numbers, i.e., {fn : Rn

+ ×Rn+ → R, ∀n ∈ Z+}.

Again we suppress n for notational simplicity. In the follow-ing, we propose a new set of Axioms by modifying Axiom2 (i.e., the Axiom of Partition) to introduce user-specificweights qj for each user j in the axiom, which provides avalue statement on relative importance of users in quantifyingfairness of the system. Axiom 5 is restated for two users withequal weights, since it may not hold in general when fairnessdepends on user-specific weights.

The Axioms of Continuity, Homogeneity, and Saturation inSection II remains unchanged.

1′) Axiom of Continuity. Fairness measure f(x,q) is con-tinuous on Rn

+ for all n ∈ Z+.

2′) Axiom of Homogeneity. Fairness measure f(x,q) is ahomogeneous function of degree 0:

f(x,q) = f(t · x,q), ∀ t > 0. (59)

Without loss of generality, for a single user, we take|f(x1, q1)| = 1 for all q1, x1 > 0, i.e., fairness is aconstant for n = 1.

12

3′) Axiom of Saturation. Fairness value of equal allocationand equal weights is independent of number of users asthe number of users becomes large, i.e.,

limn→∞

f(1n+1,1n+1)f(1n,1n)

= 1. (60)

4′) Axiom of Partition. Consider an arbitrary partition ofa system into two sub-systems. Let x =

[x1,x2

]and y =

[y1,y2

]be two resource allocation vectors,

each partitioned and satisfying∑

j xij =

∑j yi

j , fori = 1, 2. Suppose q = [q1,q2] and p = [p1,p2] arecorresponding weights for x and y and also satisfy∑

j qij =

∑j pi

j , for i = 1, 2. There exists a meanfunction [13] generated by g, such that their fairnessratio is the mean of the fairness ratios of the subsystems’allocations, for all possible partitions:

f(x,q)f(y,p)

= g−1

(2∑

i=1

si · g(

f(xi,qi)f(yi,pi)

))(61)

where g is any continuous and strictly monotonic func-tion. si are positive weights given by

si =1c

∑

j∈Ii

qj · wρ(xi) for i = 1, 2. (62)

where 1/c is a normalization parameter, such that∑i si = 1, and Ii is the set of user indices in sub-

system i.

The weights si for i = 1, 2 in (62) are constants for agiven partition of sub-systems. Axiom 4′ is similar to Axiom4 in Section II, except that a positive constant weight qj isassigned to each user j, and that weights si in the generalizedmean value in (61) are now proportional to the product of sumresource and total user-specific weights.

Similar to Section II, we regard g as a generator functionfor fairness if the resulting f satisfies all the axioms. . Weshow that Axiom 4′ also implies a hierarchical constructionof fairness,

f(x, q) = f([w(x1), w(x2)], [w(11), w(q2)]

)

·g−1

(2∑

i=1

si · g(∣∣f(xi,qi)

∣∣))

, (63)

which allows us to derive a fairness measure f : Rn+ → R of

n users recursively (with respect to a generator function g(y))from lower dimensions, f : Rk

+ → R and f : Rn−k+ → R for

integer 0 < k < n.

5′) Axiom of Starvation For n = 2 users with q1 = q2 = q,we have f([1, 0], [q, q]) ≤ f(

[12 , 1

2

], [q, q]), i.e., starva-

tion is no more fair than equal allocation under equalweights.

Since fairness measures for n = 2 users with q1 = q2

becomes symmetric, Axiom 5′ is the weakest statement, thusthe strongest axiom, that is consistent with Axiom 5 in SectionII. We will later see that it specifies an increasing direction offairness and is used to ensure uniqueness of f(x,q).

We demonstrate the following existence and uniquenessresults. Similar to Section II, we prove that, from logarithmor power generator function g(y), we can generate a uniquef(x,q), satisfying Axioms 1′–5′. No other generator functiong(y) can satisfy axioms 1′-5′ with any f(x,q).

Theorem 10: (Existence.) There exists a fairness measuref(x,q) satisfying Axioms 1′–5′.

Theorem 11: (Uniqueness.) The family of f(x,q) satisfy-ing Axioms 1′–5′ is uniquely determined by logarithm andpower generator functions,

g(y) = log y and g(y) = yβ (64)

where β ∈ R is an arbitrary constant.

B. Constructing Asymmetric Fairness Measures

Without loss of generality, we assume that user-specificweights are normalized

∑j qj = 1. We derive the fairness

measures in closed-form in the following theorem.Theorem 12: (Constructing all fairness measures satisfying

Axioms 1′-5′.) If f(x,q) is a fairness measure satisfyingAxioms 1′-5′ with weight in (62), f(x,q) is of the form

f(x,q) = sign(−r)n∏

i=1

(xi∑n

j=1 xj

)−rqi

(65)

f(x,q) = sign(−r(1 + rβ))

n∑

i=1

qi

(xi∑n

j=1 xj

)−rβ

1β

(66)

where r ∈ R is a constant exponent. In (66), the twoparameters in the mean function h have to satisfy rβ +ρ = 0,where ρ is in the weights of the weighted sum and β is in thedefinition of the power generator function. It determines thefairness of equal resource allocation, which only depends onthe number of users:

f(1n,1n) = nr · f(1, 1). (67)

The fairness measure in (65) can be obtained as a specialcase of (66) as β → 0. Obviously, sign(−r(1 + rβ)) →sign(−r) as β → 0. By removing the sign of (66) and takinga logarithm transformation, we have

limβ→0

log

n∑

i=1

qi

(xi∑n

j=1 xj

)−rβ

1β

= limβ→0

log[∑n

i=1 qi

(xi∑n

j=1 xj

)−rβ]

β

= limβ→0

n∑

i=1

−rqi log

(xi∑n

j=1 xj

)·(

xi∑nj=1 xj

)−rβ

= logn∏

i=1

(xi∑n

j=1 xj

)−rqi

(68)

where the second step uses L’Hospital’s Rule. Therefore, wederive (65) by taking an exponential transformation in the last

13

step of (68). This shows that fairness measure in (65) is aspecial case of (66) as β → 0.

The mapping from generators g(y) to fairness measuresfβ(x) is illustrated in Figure 7. Among the set of all con-tinuous and increasing functions, fairness measures satisfyingaxioms 1′-5′ can only be generated by logarithm and powerfunctions. A logarithm generator ends up with the same formof f of a power function as the exponent β → 0.

weighted geometric mean

......

Fairness measure f(x,q)

∀β ∈ R

Generator g(y)

g(y) = log(y)

g(y) = yβ ,weighted p-norm

Fig. 7. The mapping from generators g(y) to fairness measures f(x,q).Among the set of all continuous and increasing functions, fairness measuressatisfying axioms 1′-5′ can only be generated by logarithm and powerfunctions. A logarithm generator ends up with the same form of f as a specialcase of those generated by a power function.

−10 −8 −6 −4 −2 0 2 4 6 8 10−12

−10

−8

−6

−4

−2

0

2

4

6

8

Parameter β

Fai

rnes

s f β(x

, q)

x=[1,2,3,5]y=[1,1,2.5,5]

β<=1inverse of p−norm

Fig. 8. For a given x, the plot of fairness fβ(x,q) in (69) for differentvalues of β: β ≤ −1 recovers inverse of p-norms for p = −β. Differentvalues of β may result in same fairness ordering: fβ(x,q) ≥ fβ(y,q) forβ ∈ (−∞,−0.8] ∪ [1, 1.8], and fβ(x,q) ≤ fβ(y,q) for β ∈ [−0.8, 1] ∪[1.8,∞).

We denote fβ,r to be the fairness measure with parameters βand r. According to Theorem 12, r determines the growth rateof maximum fairness as population size n increases. Similar toSection II, we choose r = 1 and derive a unified representationof the constructed fairness measures:

fβ(x,q) = sign(−1− β) ·

n∑

i=1

qi

(xi∑n

j=1 xj

)−β

1β

. (69)

From here on, in this section we consider the family of fairnessmeasures in (69) parameterized by β. Notice that (69) includes(multiplicative) inverse of weighted p-norm as a special casefor β ≤ −1. In particular, for β = −1, we recover inverse oftaxicab norm; for β = −2, we recover inverse of Euclidean

norm; and for β = −∞, we recover inverse of infinity normor maximum norm.

Axioms 1′-5′ differ from Axioms 1-5 in Section II mainlyin that generalized mean value in Axioms 1′-5′ depends onsum resource, as well as total user-specific weights, while inAxioms 1-5 it depends only on sum resource of partition.Table V summarizes the two choices of weights and theresulting fairness measures. The axiomatic theory now pro-vides a unification of both symmetric and asymmetric fairness.As an illustration, for a fixed resource allocation vectorsx = [1, 2, 3, 5] and y = [1, 1, 2.5, 5], and user-specificweights q = [0.4, 0.2, 0.2, 0.2], we plot fairness f(x,q) andf(y,q) in the unified representation (69) against β in Figure8. Again, different values of β may result in different fairnessorderings: fβ(x,q) ≥ fβ(y,q) for β ∈ (−∞,−0.8]∪ [1, 1.8],and fβ(x,q) ≤ fβ(y,q) for β ∈ [−0.8, 1] ∪ [1.8,∞).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Weight q1

Res

ourc

e x 1

Equal−Fairness Curves

f=1.4

f=1.6

f=1.8

f=1.2

f=1.2

f=1.4

f=1.6

f=1.8

f=2.0

f=2.8

f=2.6

f=2.4

f=2.2

f=2.2f=2.4

f=2.6f=2.8

Fig. 9. Plot of equal fairness curves for n = 2 users. Different combinationsof resource allocation x = [x1, 1−x1] and weight q = [q1, 1− q1] achievethe same fairness value for f = 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8.

Different combinations of resource allocation x and weightq can achieve the same fairness value fβ(x,q). To illustratethis, Figure 9 plots equal fairness curves for n = 2 users withx = [x1, 1− x1] and q = [q1, 1− q1], i.e.,

{(x1, q1) : fβ([x1, 1− x1], [q1, 1− q1]) = f} , (70)

where β = −2 and f = 1.2, 1.4, . . . , 2.8. Unlike symmetricfairness measures in Section II, equal fairness curves forasymmetric fairness measures offer an extra degree of designfreedom: we can choose different user-specific weights fromvarious considerations including the needs and contributionsof each user.

C. Properties and ImplicationsWe prove the following properties for fairness measures

derived in this section. Some properties are similar to theircounterparts in Section II-C.

Corollary 18: (Weighted Symmetry.) Let π be a permutationof n elements. For a fairness measure satisfying Axioms 1′–5′,we have

f(x,q) = f(π(x), π(q)),

14

Axioms Weight si Generators Fairness Measures

logarithm sign(r)∏n

i=1

(xi∑n

j=1 xj

)−r

(xi∑n

j=1 xj

)

1–5 wρ(xi)

power sign(r(1− βr)) ·[∑n

i=1

(xi∑j xj

)1−βr] 1

β

logarithm sign(−r)∏n

i=1

(xi∑n

i=j xj

)−rqi

1′–5′ (∑

j∈Iiqj) · wρ(xi)

power sign(−r(1 + rβ)) ·[∑n

i=1 qi

(xi∑n

j=1 xj

)−rβ] 1

β

TABLE VCOMPARING FAIRNESS MEASURES SATISFYING AXIOMS 1-5 AND AXIOMS 1′-5′ , WITH DIFFERENT CHOICES OF WEIGHTS IN GENERALIZED MEAN

VALUE. NOTE THAT WEIGHTS IN THIS TABLE REMAINS TO BE NORMALIZED, SO THAT∑

i si = 1.

for all permutation π.

Corollary 19: (Robin Hood Operations.) For small enoughε, replacing two elements xi and xj with xi − ε and xj + ε

improves fairness if and only if xi/q1

rβ+1i < xj/qj

1rβ+1 for

fairness measures satisfying Axioms 1′–5′.Corollary 20: (Most Fair Allocation.) Fairness measure sat-

isfying Axioms 1′–5′ is optimized by a resource allocationvector [q

1rβ+11 , q

1rβ+12 , . . . , q

1rβ+1n ].

Corollary 21: (Perturbation of fairness value from slightchange in a user’s resource.) For fairness measures satisfyingAxioms 1′–5′, if we increase resource allocation to user i bya small amount ε, while not changing other users’ allocation,the fairness measure increases if and only if xi < q

1rβ+1i · x =(

qi

∑j xj/

∑j qjx

−rβj

) 1rβ+1

and 0 < ε < q1

rβ+1i · x− xi.

Corollary 22: (Monotonicity with respect to β.) The fair-ness measures in (69) is negative and decreasing for β ∈(−1,∞), and positive and increasing for β ∈ (−∞,−1):

∂fβ(x,q)∂β

≤ 0 for β ∈ (−1,∞), (71)

∂fβ(x,q)∂β

≥ 0 for β ∈ (−∞,−1). (72)

Most properties have similar forms to their counterparts inSection II-C, while their implications become user-dependent.The properties of fairness measure (65) can be derived as aspecial case from that of fairness measure (66) as β → 0. Theweighted-symmetric property illustrates that user differentia-tion in Axioms 1′-5′ is introduced in terms of user-specificweights. Users with equal weights are treated equally forfairness. Corollary 22 is an extension of the Robin HoodOperation and Schur-concavity property proven in Section II.

D. Towards A Fairness Evaluation Tool

Our axiomatic study might lead to a fairness evaluation tool,which turns the axiomatic theory into an easy-to-understanduser-interface. We outline some possible functionalities of sucha tool here.

The tool can (a) perform a consistency check of fairnessbased on discovering a person’s underlying fairness function;(b) models user-specific weights q for different applicationsand settings; and (c) tune the tradeoff, via λ, between fairnessand efficiency in system design objectives.

In this section, we present a human-subject experi-ment as a step towards using fairness theory as a con-sistency checking tool. The tool can be accessed fromhttp://www.princeton.edu/∼chiangm/fairnesstool.html. In atypical variation of the experiment, each subject is told thatthere are two workers, A and B. A works twice as fast asB, and their monthly incomes are represented as a 1 × 2vector. Through a sequence of simple Y/N questions, we willdiscover the underlying β and q used by the subject. We runthe following three steps in our experiment.

1) Ask the subject a sequence of binary questions: is vectorx (a particular income vector for workers A and B) morefair than y (another income vector)? For a given answerto the first question, plot the region of possible (q1, β)region compatible with that answer, i.e., if x is morefair than y, we choose

{(q1, β) : fβ(x,q) > fβ(x,q),q = [q1, 1− q1]} (73)

Then adaptively pick another vector that can help shrinkthat region further. Iterate till we have a very thin sliceof feasible region in (q1, β) plane compatible with allthe answers.

2) Ask the person what range of weights would she give touser 1 (and she may say, eg, [0.6, 0.7]). Then we knowthe range of β that is compatible with the answers sofar.

3) Pick extreme values of beta in the above range, andtell the person that, according to her implicit fairnessmeasure, vector x must be more fair than vector y. Askher if this consequence sounds alright to her? Based onher answer, further shrink the (q1, β) range. This can beviewed as a mathematical expression of the principle of“reflective equilibrium” in political philosophy’s studyof distributive fairness [33], [55]. Finally, we can nail

15

down to a narrow range of (q1, β) as her underlyingfairness function.

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Parameter β

Para

met

er q

1

Fig. 10. Plot of all possible fairness parameter β and user-specific weightq = [q1, 1 − q1], such that the resulting fairness measures in (69)) satisfyconstraints (74)-(78). Steps 2 and 3 could be repeated to further shrink the(q1, β) range.

Consider the the family of fairness measures fβ(x,q) in(69). We take 8 different samplings of feasible allocationvectors, which are assigned the following fairness orders by ahuman subject:

[40, 60] is less fair than [60, 40], (74)[50, 50] is less fair than [58, 42], (75)[67, 33] is less fair than [50, 50], (76)q1 ∈ [0.5, 0.9] (77)[54, 46] is less fair than [62, 38]. (78)

We use an exhaustive search to find out which fairness mea-sures (i.e., which fairness parameter β and user-specific weightq = [q1, 1− q1] in (69)) satisfy fairness constraints (74)-(78).The resulting set of {(β, q1)} is plotted in Figure 10. It showsthat the human subject implicitly assumes a fairness parameterβ ∈ [0.1, 5.5] and a weight vector q = [q1, 1 − q1] satisfyingq1 ∈ [0.59, 0.9] in the test. No other fairness measures in (69)is consistent with the answers. Steps 2 and 3 could be repeatedto further shrink the (q1, β) range. In many experiments, weobserve that the set of compatible fairness measures can beefficiently narrowed down using information provided by only5 questions.

Through the above experiment, we do not need to askexplicitly for a subject’s underlying β, which would have beena practically difficult question to answer. Instead, through asequence of binary questions comparing two given alloca-tion vectors, we can reverse-engineer the fairness functionimplicitly used by the subject. For example, a small scaleexperiment was recently run with the staff and students inChiang’s research group, and showed that the β value of thesepeople concentrate between 0 and 1, which turns out to be the“busiest” part of the β spectrum.

Even when fairness measures are put on an axiomaticground, different people will still prefer different fairness

measures. What the unique, axiomatic construction, togetherwith the discovery of β, can do is to enforce consistency ofusing the same β by the same person as she evaluates differentresource allocations.

V. FAIRNESS-EFFICIENCY UNIFICATION

By removing some of the axioms — for example, Axiom (2and 2′) of Homogeneity that decouples efficiency from fairness— what kind of new fairness measures would result? We nowdevelop an alternative set of fewer axioms that lead to theconstruction of fairness measures and do not automaticallydecouple fairness from efficiency and feasibility of resourceallocation.

A. A Third Set of Four Axioms

We propose a set of alternative axioms, which includes Ax-ioms 1–5 and Axioms 1′–5′ as special cases2. Let F : Rn

+ → Rfor all n ∈ Z+ be a sequence of fairness measures satisfyingfour axioms as follows.

1′′) Axiom of Continuity. Fairness measure F (x) is contin-uous on Rn

+ for all integer n ∈ Z+.2′′) Axiom of Saturation. Fairness measure F (x) of equal

resource allocations eventually becomes independent ofthe number of users:

limn→∞

F (1n+1)F (1n)

= 1. (79)

3′′) Consider an arbitrary partition of a system into twosub-systems. Let x =

[x1,x2

]and y =

[y1,y2

]be

two resource allocation vectors, each partitioned. Thereexists a mean function [13] generated by g, such thattheir fairness ratio is the mean of the fairness ratios ofthe subsystems’ allocations, for all possible partitionsand positive t > 0:

F (t · x)F (t · y)

= g−1

(2∑

i=1

si · g(

F (xi)F (yi)

))(80)

where g is any continuous and strictly monotonic func-tion. Weights si are chosen according to (7) and (62).

4′′) Axiom of Starvation For n = 2 users, we have F (1, 0) ≤F ( 1

2 , 12 ), i.e., starvation is no more fair than equal

allocation.Axioms 1′′, 2′′, and 4′′ remain the same Axioms 1, 3, 5 and

Axioms 1′, 3′, and 5′ in previous sections. Since the Axiom 2and (2′) of Homogeneity is removed, fairness measure F (x)may depend on the absolute magnitude of resource vector x.Axiom 3′′ requires (80) to be satisfied by all partitions andpositive scaling factor t. It defines a hierarchical structure ofresource allocation in a stronger sense than Axiom 4 in SectionII and Axiom 4′ in Section IV, which are special cases ofAxiom 3′′ for t = 1.

2Weights q Axioms 1′–5′ are suppressed for a unified representation offairness F .

16

B. Deriving a Fairness-Efficiency Unification

Using Axiom 3′′, we can prove that F (x) is a homogeneousfunction of real degree. Furthermore, if the order of homogene-ity is zero, Axioms 1′′-4′′ is equivalent to the two axiomaticsystems, Axioms 1-5 in Section II and Axioms 1′-5′ in SectionIV for different choice of si in (7) and (62), respectively. Thismeans that the new axiomatic system is more general thanprevious ones.

Theorem 13: (Existence and Uniqueness.) If fairness F (x)satisfies Axioms 1′′-4′′ with weights in (7) or (62), it is of theform

F (x) = f(x) ·(∑

i

xi

) 1λ

, (81)

where 1λ ∈ R is the degree of homogeneity, and f(x) is a

symmetric fairness measure satisfying Axioms 1-5 in SectionII or an symmetric fairness measure satisfying Axioms 1′-5′

in Section IV.It is easy to verify that some properties, like that of

symmetry in Section II-C also hold for fairness measureF (x) when weight si is chosen by (7), yet some propertiesof fairness measures satisfying Axioms 1–5 are changed inthe generalization. In particular, the new fairness measuremay not be Schur-concave, and equal allocation may not befairness-maximizing. For instance, it can be readily verifiedthat F (1, 1) < F (0.2, 10) for β = 2 and 1

λ = 10.From Theorem 13 and uniqueness results in Sections 2

and 4, power generators g(y) = yβ and logarithm generatorsg(y) = log y (viewed as a special case of power generators asβ → 0) are unique. We can thus construct a family of fairnessmeasures Fβ,λ(x), which is parameterized by both λ and β,

Fβ,λ(x) = fβ(x) ·(∑

i

xi

) 1λ

. (82)

where fβ(x) is the unified representation (15) in Section IIand (69) in Section IV.

This recovers our results in previous sections: GeneralizedJain’s index is a special case of Fβ,λ(x) for si in (7), 1/λ = 0,and β < 1; fairness measure fβ(x) is a subclass of Fβ,λ(x)for si in (7) and λ = 0; inverse of p-norm is another subclassof Fβ,λ(x) for si in (62) and β ≤ −1; and α-utility isobtained for si in (7), 1/λ = β/(1 − β), and β > 0. Thedegree of homogeneity 1/λ determines how Fβ,λ(x) scalesas throughput increases. The decomposition of fairness andefficiency in Section II-E is now an immediate consequencefrom Axioms 1′′-4′′.

There is a useful connection with the characterization ofα-fair utility function in Section III. The absolute value |λ|is equivalent to the parameter used for defining the utilityfunction (30) in Section II-E. From Theorem 6, we concludethat fairness measure Fβ,λ(x) is Pareto optimal if and only if

1|λ| ≥

∣∣∣∣1− β

β

∣∣∣∣ . (83)

For a given β, there is a minimum degree of homogeneity suchthat Pareto optimality can be achieved. When inequality (83)

is not satisfied, Fβ,λ(x) loses Pareto optimality and producesless throughput-efficient solutions if it is used as an objectivefunction in utility maximization.

The degree of homogeneity of a fairness measure satisfy-ing Axioms 1′′-4′′ parameterizes a tradeoff between fairnessand efficiency. Moreover, when power functions are used asgenerating functions, the degree of homogeneity is equivalentto 1

λ in (30). Therefore, the intuition behind our result ona maximum |λ| (minimum degree of homogeneity) to ensurePareto optimality can be extended to the general optimization-theoretic approach to fairness, i.e., for a fairness measureF generated from any g, there is a minimum degree ofhomogeneity 1

λ to produce a Pareto optimal solution. Whileour first set of axioms generalizes Jain’s index and revealsnew fairness measures, the third set of axioms offers a richfamily of objective functions for resource allocation based onoptimization formulations.

This family of fairness measures unifies a wide range ofexisting fairness indices and utilities in diverse fields brieflydiscussed in Appendix B. The unification is illustrated inFigure 11, including all known fairness measures that areglobal (i.e., mapping a given allocation vector to a singlescalar and decomposable (i.e., subsystems’ fairness values canbe somehow collectively mapped into the overall system’sfairness value).

−∞ -1 0 1 +∞β

Min-Max Max-Min

Theil Index, Atkinson Index (ε)

Jain’s Index Proportional FairnessJasso IndexEntropy

Renyi Entropy (α)

α-fairness (α)

Foster-Sen Welfare Function

{f(x,q), β = 1− ε, qi = 1n, r = −1− 1

β}

{F (x,q), λ = −1, β = 1− α, qi = 1n, r = 1− 1

β}

{F (x,q), λ = 1, β = 1− ε, qi = 1n, r = 1− 1

β}

{F (x,q), λ = 1−ββ

, β = α, qi = 1n, r = 1− 1

β}

p-norm (p){f(x,q), β = p, r = 1}

Fig. 11. F unifies different fairers measures from diverse disciplines.

VI. CONCLUSION

For a given resource allocation vector, different people willhave different evaluations of how fair it is. These evaluationsmay correspond to image of the same fairness measure F butwith different parameters β and λ. Realizing which parametersare implicitly used helps to ensure consistency where the sameFβ,λ is used to evaluate fairness of other allocation vectors.

An axiomatic quantification to the fundamental concepts offairness also illuminates many issues in resource allocation.Extending a basic system of five axioms in Section II, wehave shown that one way to re-examine axioms of fairness isto refute their more controversial corollaries. For example, thefirst set of axioms imply that fairness is always maximized by

17

equal resource allocation and has no user-specific value or thenotion of efficiency, a simplification that clearly is inadequatein many scenarios. In Section IV and Section V, we providetwo alternative sets of axioms, which incorporates asymmetricfairness measures (by introducing user-specific weights) andunifies the notion of efficiency (i.e., making more resourceavailable by scaling x in all coordinates could lead to largerfairness value), respectively. Another approach is to expandthe data structure.

We started with an extremely simple one: given a vector,look for a scalar-valued function. In Section III, we extendedfairness measures to continuous dimensional inputs, whereresource allocations become non-negative functions over time.In Section V, we start with both a vector of resources anda vector of user weights, which may be arrived at fromvarious considerations. Time-varying and user-specific valuesof resource can also be modeled via Sections IV and V.

There are in general two main approaches to fairnessevaluation: binary (is it fair according to a particular criterion,e.g., proportional fair, no-envy?) or continuous (quantifyinghow fair is it, and how fair is it relative to another alloation?).In the latter case, there are three sub-approaches:• (A) A system-wide, global measure: (A1) f(x) where f

is our fairness function, or (A2) f(U1, U2, . . . , Un) whereUi is a utility function for each user that may depend onthe entire x.

• (B) Individual, global measures: the set of {fi(x)} ={fβi(x)} (user i cares about the entire allocation).

• (C) Individual, local measures: the set of {fi(xi)} (useri only cares about her resource via some function f ).

Approaches B and C may also involve some notion of expec-tation, based on need or contribution, and user-specific utilityevaluation of resource.

We have so far been focusing on approach (A1) , whereasin fairness studies in computer science, economics, and sociol-ogy, discussed in Appendix B, approaches B and C have alsobeen studied, in which one can think of two ways to aggregatethe individual evaluations:• Scalarization: W (f1, f2, . . . , fn) for some increasing

function W , with a special case being∑

i fi, and anotherspecial case being (A2), if Ui is fi for all i.

• Binary summary: e.g., envy-freeness, which reduces to abinary evaluation criterion that is true if no user wants tochange her resource with any other user.

In what ways will system-wide fairness evaluation and anaggregate of individual fairness evaluations be compatible?What will be (W,q) choice’s impact on the difference betweenfβ(x,q) and W (f1, f2, . . . , fn)? Addressing this question willalso be a step towards bringing individual strategic actiontogether with an allocator’s decision over time.

Another distinction can be drawn between two types offairness measures:• Global fairness measures that assume “decomposability”:

the fairness value of a vector can be somehow recon-structed from the fairness values of the subvectors. Thisis the case in our (f, F ), as well as in other cases inAppendix B such as Renyi entropy and Atkin’s index.

• Local fairness measures that take an axiom of “local-closedness”: an evaluation about x1 and x2 should beindependent of x3. These fairness measures include Giniindex, Nash Bargaining Solution, and Shapley Value astypical examples.

A summary of the unification of these global, decomposablefairness measures into Fβ,λ(x,q) is shown in Figure 11.

This paper presents an axiomatic theory, and is far fromthe end of axiomatic theories of fairness. For instance, wehave assumed that there is only one type of resource, thisresource x is infinitely divisible, and fairness measure is nota function of the feasible region of allocations. The followingtwo axioms have remained unchanged throughout the paper:the Axiom of Continuity and the Axiom of Saturation, thusbecoming interesting targets to investigate. Furthermore, wehave ignored the process of resource allocation, such as theissues of who makes the allocation and whether it is made withautonomity of users. Time and time again, centralized systemswith unchecked power claims to achieve fairness-maximizingallocation and actually produces extreme unfairness. We havealso ignored the long-term evolution of the system whereusers, intelligent and sometimes irrational, anticipate and reactto the allocation at any given time, which further influencesthe long term efficiency and fairness of the system. Sharpenedconnections with other axiomatic theories in information,economics, and politics are thus called for.

VII. ACKNOWLEDGEMENT

We are grateful to David Kao and Ashu Sabharwal forcollaborating on an earlier version of the axioms and fairness-efficiency unification [40], to Rob Calderbank, Sanjeev Kulka-rni, Xiaojun Lin, Linda Ness, Jennifer Rexford, and KevinTang for helpful comments, and to Paschalis Tsiaflakis fordata on the spectrum management examples.

APPENDIX

A. Numerical Illustrations in Networking

This paper was originally motivated by problems of resourceallocation in communication networks. We use classical prob-lems in congestion control, routing, power control, and spec-trum management to numerically illustrate some of the keypoints in the paper, including the use of generalization of α-fairness, fairness-efficiency tradeoff, fairness during transientbehavior of iterative algorithms, and the impact of user-specificweights.

Fig. 12. A linear network with L links and n = L+1 flows. All links havethe same capacity of 1 unit.

1) Congestion Control: TCP congestion control in theInternet has been studied via a utility maximization modelsince [7], [20], with an extensive research literature including

18

papers focusing on α-fairness. Overviews can be found ine.g., [21], [22]. The typical optimization problem modelingcongestion control is as follows:

maxx

n∑

i=1

Ui(xi) (84)

s.t. x ∈ Rwhere U is a utility function and R is a set of all feasibleresource allocation vectors. We consider the classical exampleof a linear network with L links, indexed by l = 1, . . . , L,and n = L + 1 flows, indexed by i = 0, 1, . . . , L, shownin Figure 12. Flow i = 0 goes through all the links andsources i ≥ 1 go through links l = i. All links have the samecapacity of 1 unit. We denote xi to be the rate of flow i. Wewill illustrate two points: how a given x can be evaluatedby different fairness measures, and how F acting as theobjective function broadens the trade-off between efficiencyand fairness.

The first example compares different fairness measures for agiven resource allocation x in the linear network with L = 5:

x =[13,23,23,23,23,23

]. (85)

We illustrate the unified representation Fβ,λ(x) in (82), whichincludes symmetric and asymmetric fairness measures devel-oped in Sections II and IV as subclasses. Figure 13 plotsfairness Fβ,λ(x) against β for 1/λ = 0, 1, 2. Larger λ putsmore emphasis on fairness relative to efficiency. For asym-metric fairness, user specific weights are given by q0 = 0.5and qi = 0.1 for i ≥ 1.

−5 −4 −3 −2 −1 0 1 2 3 4 5−150

−100

−50

0

50

100

Parameter β

Fai

rnes

s F β,

λ( x)

Symmetric fairness, 1/λ=0,1,2Asymmetric fairness, 1/λ=0,1,2

1/λ=2

1/λ=1

1/λ=0

1/λ=2

1/λ=1

1/λ=0

Fig. 13. Fairness Fβ,λ(x) derived in (82) in Section V, for β ∈ [−4, 4]and 1/λ = 0, 1, 2. Both Symmetric and asymmetric fairness measures aremonotonically increasing on β < −1 and decreasing on β > 1.

We observe that symmetric fairness measures change signat β = 1, while asymmetric fairness measures change sign atβ = −1. Both symmetric and asymmetric fairness measuresare monotonically increasing on β < −1 and decreasing onβ > 1. For −1 < β < 1, they have different signs and mono-tonicities. As in Theorem 13 in Section V, fairness Fβ,λ(x)can be factorized as the product of a fairness component (i.e.,fβ(x) or fβ(x,q)) and an efficiency component (

∑i xi)

1λ . For

fixed resource allocation x, parameter β determines the valueof the fairness component, while 1/λ decides how Fβ,λ(x)scales as efficiency

∑i xi increases.

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

Degree of homogeneity 1/λ

Rat

e or

Fai

rnes

s

Fairness component fβ (x*)

Efficiency component Σi x*

i

Optimal rate for long flow x0*

Optimal rate for short flows xi*

x0*

xi*

Fig. 14. Optimal rate allocations, their fairness components, and efficiencycomponents, for congestion control problem (86) with different degreesof homogeneity 1/λ of the fairness measure. As 1/λ grows, efficiencycomponent

∑i x∗i increases and skews the optimal rate allocation x∗ away

from an equal allocation, while fairness component fβ(x∗) is monotonicallydecreasing.

Next, we consider a congestion control problem in linearnetwork, which is formulated as a maximization of symmetricfairness Fβ,λ(x), a generalization of α-fair utility function,under link capacity constraints:

maxx

Fβ,λ(x) = fβ(x) ·(∑

i

xi

) 1λ

(86)

s.t. x0 + xi ≤ 1, for i = 1, . . . , L.

For β ≥ 0 and 1/λ = β/(1 − β), the optimal rate allocationmaximizing (86) achieves α-fairness for α = β. For 1/λ ≥β/(1−β), problem (86) is concave after a logarithm change ofobjective function, i.e., log Fβ,λ(x). We fix β = 1/2 and solve(86) for different values of 1/λ. Figure 14 plots optimal rateallocations x∗ (in terms of x∗0 and x∗i for i ≥ 1), their fairnesscomponents fβ(x∗), and their efficiency components

∑i x∗i ,

all against 1/λ. As 1/λ grows, efficiency component∑

i x∗iincreases and skews the optimal rate allocation away fromthe equal allocation: the long flow x0 in the linear networkgets penalized, while short flows xi are favored. At the sametime, fairness component of the objective function decreases.These observations illustrate the results in Section II-E andin Section V: degree of homogeneity 1/λ serves as a tradeofffactor between fairness and efficiency in congestion control.Different values of 1/λ result in various source algorithms indual-based TCP congestion protocols [21], [22].

2) Multi-Commodity Flow Routing: We consider one clas-sical type of routing problems: maximum multi-commodityflow on an undirected graph G = (V,E), with given capacityc(u, v) on each edge (u, v) ∈ E. A multi-commodity flowinstance on G is a set of ordered pairs of vertices (si, di)for i = 1, 2, . . . , n. Each pair (si, di) represents a commodity

19

Fig. 15. An undirected graph G = (V, E) with n = 2 commodities, (s1, d1)and (s2, d2). The number next to each edge represents its capacity. Due tothe existence of edge (s2, 1), source s2 can contribute more to maximummulti-commodity throughput than source s1.

with source si and destination di. The objective of a maxi-mum multi-commodity flow problem is to maximize the totalthroughput of flows traveling from the sources to the corre-sponding destinations, subject to edge capacity constraints. Letxi be the throughput of commodity (si, di) and φi(u, v) denoteits flow along edge (u, v) ∈ E. Maximum multi-commodityflow problem is the following one extensively studied sincethe 1950s:

maxx,φ

n∑

i=1

xi (87)

s.t.n∑

i=1

φi(u, v) ≤ c(u, v), ∀(u, v) ∈ E

φi(u, v) = −φi(v, u), ∀(u, v) ∈ E∑

u∈V

φi(u, v) = 0, if v 6= si, di

n∑

u∈V

φi(si, v) =n∑

u∈V

φi(u, di) = xi. ∀i

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.35

0.4

0.45

0.5

0.55

0.6

Edge capacity c(s2,1)=y

Fair

ness

fβ(

x, q)

q=[0.1 0.9] q=[0.2 0.8] q=[0.3 0.7] q=[0.4 0.6] q=[0.5 0.5] q=[0.6 0.4]

Fig. 16. Fairness of the maximum multi-commodity flow solutions againstedge capacity c(s2, 1) = y ∈ [0, 1]. Asymmetric fairness measures fβ(x,q)in (65) are employed, with parameters r = −1, β = −2, and commodityspecific weights q1 = 1−q2 and q2 = 0.9, 0.8, 0.7, 0.6, 0.5, 0.4. Increasingy improves fairness only if the weight q2 assigned to commodity 2 is largeenough. For each choice of weights, the point that achieves maximum fairness(over choices of y) are marked by triangles.

For the graph with n = 2 commodities shown in Figure 15,we vary capacity y from 0 to 1 on edge (s2, 1), to createan asymmetry between the two commodities. We solve the

maximum multi-commodity flow problem (87) and evaluatefairness of the resulting solutions x = [x1, x2], using asym-metric fairness measures fβ(x,q) derived in (65) in SectionIV with r = −1, β = −2, and commodity specific weightsq1 = 1 − q2 and q2 = 0.9, 0.8, 0.7, 0.6, 0.5, 0.4. The resultsare plotted in Figure 16.

Due to the existence of edge (s2, 1), source s2 contributesmore to maximum multi-commodity throughput than sources1. Figure 16 shows that this contribution improves fairnessonly if the weight q2 assigned to commodity 2 is large enough.When q1 ≥ q2, increasing capacity y always reduces fairness,an objectionable outcome. For y ∈ [0, 1], we derive thesolution to the maximum commodity flow problem (87) asx = [1, 1 + y]. Using Corollary 24 in Section IV, in orderto have a monotonic relationship between fairness fβ(x,q)and capacity y ∈ [0, t], we need to choose weights [q1, q2]satisfying

q2

q1≥ (1 + t)1+rβ . (88)

This closed-form bound quantifies the argument that sourcescontributing more to maximum multi-commodity throughputshould be given higher weights, if we want their contribution tohave a positive impact on fairness. Larger the range of capacityy for which this needs to hold, larger the ratio q2/q1 needs tobe.

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

7

time t

SIN

R

User 1User 2User 3

Fig. 17. Evolution of users’ SINR over time t, using the distributed powercontrol algorithm [19]. Transmit power is updated according to (90) every0.05 second. Channel matrix G is updated every 1 second.

3) Power Control: Transmit power control is a key designchallenge in wireless cellular networks since the 1990s, witha vast literature on optimization problem formulations andalgorithms, e.g., as surveyed in [23]. We use different fairnessmeasures to evaluate a wireless power control system. Con-sider a wireless cellular network with n users, each servedby a different logical link from the mobile device to the basestation. Let pj be the transmit power of user j and Gij denotethe channel gain from the transmitter of logical channel j tothe receiver of logical channel i. Therefore, interference causedfrom user j to user i has power Gij ·pj . Signal-To-Interference-

20

Noise-Ratio (SINR) for user i’s link is defined as:

γi =Gii · pi∑

j 6=i Gij · pj + σ2, for i = 1, . . . , n, (89)

where σ2 denotes noise power on this link. Much of the studyon cellular network power control started in 1992 with a seriesof results that minimize total transmit power

∑i pi for a fixed,

feasible SINR target γ∗.We consider the widely adopted distributed power control

algorithm [19], which iteratively updates user’s transmit powerto achieve SINR target γ∗. In this iterative algorithm with eachiteration indexed by integer k, each user only needs to monitorits individual received SINR and then update its transmit powerby

pi[k] =γ∗i

γ[k − 1]pi[k − 1], for i = 1, . . . , n, (90)

independently and possibly asynchronously. Intuitively, eachuser i increases its power when its SINR is below target γ∗i anddecreases it otherwise. We run this algorithm, for n = 3 and achannel matrix G draw from a i.i.d. Gaussian distribution with0 mean and variance 1. The channels are changed every seconddue to variations of the wireless propagation environment. Wechoose SINR target γ∗ = [1, 2, 3] and update transmit poweraccording to (90) every 0.05 second. The resulting evolutionof users’ SINR over time t is illustrated in Figure 17.

0 0.5 1 1.5 2 2.5 3 3.5 4−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

time t

Fai

rnes

s f

Fairness (9)Fairness (10), β=−2,−5Fairness (63)Fairness (64), β=−2,−5

Fig. 18. Evaluation of a wireless power control system, which runs a iterativepower update algorithm for a fixed and feasible SINR assignment [19], usingdifferent fairness measures. Fairness for system snapshots are plotted againsttime t. As |β| increases, fairness value becomes more sensitive to resourcechanges during system transit states.

First, we take a snapshot of the network at each time t andevaluate fairness of the instantaneous SINR assignment, i.e.,f(γ) and f(γ,q). We choose r = −1 and β = −2,−5 forfairness measures (10) and (66), generated by power functions.For asymmetric fairness measures (65) and (66), we chooseweight q = [1/6, 1/3, 1/2] and q = [0.03, 0.22, 0.75] in (65)and (66) respectively, so that fairness f(γ,q) are maximizedat target SINR using Corollary 23.

As shown in Figure 18, at time t = 1, 2, 3, fairness ofthe system fluctuates due to the fluctuation of channel matrixG. Fairness measures (10) and (66) with β = −5 are more

sensitive during transit states than that of β = −2, whilefairness measures (9) and (65) with β = 0 are least sensi-tive. This illustrates the analysis in Section II-D that smallerparameter β suggests a stricter notion of fairness: fairnessvalue becomes more sensitive to resource change. If we chooseuser-specific weights in asymmetric fairness measures so thatfairness is maximized at target SINR, the distributed powercontrol algorithm can be viewed as maximizing (asymmetric)fairness of the network.

1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Users

Fai

rnes

s f

β=−2β=−5

Fig. 19. Use F(γi,t∈[0,T ]) in (39) to quantify fairness of users’ SINRdistributions over time. We choose r = 1 and compute fairness F for usersi = 1, 2, 3 for β = −2 and β = −5. User 1’s SINR distribution over timeis most fair.

Another approach to analyze fairness values aggregated overtime is to start with the time series of users’ SINR and quantifyfairness of SINR distributions over time. Let γi,t∈[0,T ] be useri’s SINR distribution over time [0, T ]. We use fairness measure(39) in Section III to evaluate γi,t∈[0,T ]:

F(γi,t∈[0,T ]) = sign(r(1− βr)) ·[

1T

∫ T

t=0

(γi,t/µ)1−βrdt

] 1β

.

We choose r = 1 and compute fairness F for users i = 1, 2, 3for two different choices of parameters: β = −2 and β = −5.Fairness measure F for this range is generalized Jain’s indexwith continuous-dimensional resource distributions. Fairnessvalue for different users’ SINR distributions are shown inFigure 19. In this case, what matters is how stable (smallmagnitude of fluctuation) is γ during the transient stage afterchannel change. User 1’s SINR is most stable over the transientstage in Figure 17. Therefore, it chieves the maximum fairnessof F = 0.89 for β = −2 among the three users in thenetwork. User 3’s SINR is most fluctuating over time. It ismost unfair with F = 0.18. Although β = −2 and β = −5gives different fairness values, the fairness ordering of usersremains unchanged.

4) Rate-Power Tradeoff in Spectrum Management: Whenpower control takes place over multiple frequency bands, itis often called spectrum management. It arises in both Or-thogonal Frequency-Division Multiplexing (OFDM) wirelessnetworks and Digital-Subscriber-Line (DSL) broadband accessnetworks. In the past decade, it has become one of the mosteffective solutions in engineering next generation DSL [24].

21

Fig. 20. A network consisting of n = 6 DSL users (i.e., lines). Near-farcrosstalk occurs when a user (e.g., line 6) enjoying a good channel close tothe receiver interferes with the received signal of a user (e.g., line 4) furtheraway having a worse channel. Crosstalk may be mitigated by power controlover the frequency bands, i.e., spectrum management.

We consider a network consisting of n interfering DSLusers (i.e., lines), depicted in Figure 20, with a standardsynchronous discrete multi-tone (DMT) modulation of Ktones (i.e., frequency bands). Let pk

i be the transmit power ofuser i on tone k and pi =

∑k pk

i be his total transmit power.We denote Gk to be an n × n channel gain matrix, whose(i, j)’th element is the channel gain from transmitter modemi to receiver modem j on tome k. The diagonal elements ofGk are the direct channels and the off-diagonal elements arethe crosstalk channels. The data rate of user i on tone k isdefined as:

bki = ψk

i (Gk,pk) , log

(1 +

1Γ

Gkii · pk

i∑j 6=i Gk

ij · pkj + σk

i

), (91)

where ψki (·) is a data rate function for user i on tone k, σk

i

denotes the corresponding noise power, and Γ is the SNR-gap to capacity, which is a function of bit error rate, codinggain, and noise margin. Let η be the DMT symbol rate. Totalachievable data rate of user i is given by

Ri = η ·K∑

k=1

bki = η ·

K∑

k=1

ψki (Gk,pk) (92)

The goal of DSL spectrum management is to allocate eachuser’s transmit powers over K tones to optimize certain designobjectives and satisfy certain constraints. One of the recentspectrum management challenges is to strike a proper tradeoffbetween power-consumption and total rate of the network. Inorder to reduce power consumed, rates achieved by the usersneed to be decreased, and decreased by different amountsfor different users. Since rates are interference-limited as in(91) and location of users put them into different interferenceenvironment (i.e., different Gk

ij), an interesting question isto quantify fairness of rate-reduction. In [25], the followingoptimization problem is formulated and solved to enforce a

certain notion of fairness:

max{pk

i ,Ri}

n∑

i=1

Ri (93)

s.t. pi =K∑

k=1

pki ≤ pi,max, for i = 1, . . . , n

Ri = η ·K∑

k=1

ψki (Gk,pk), for i = 1, . . . , n

Ri

Roi

/pi

pi,max= θ, for i = 1, . . . , n

where [Ro1, R

o2, . . . , R

on] is a given rate allocation vector some-

where on the boundary of feasible rate region, pi,max is thegiven transmit power budget of user i, and θ > 0 is a positiveconstant.

The notion of fairness, enforced by the third constraintin optimization problem (93), is based on the following adhoc argument: the ratio of each user’s percentage data ratedecrement to her percentage transmit power decrement shouldbe the same across the users.

We use the algorithm and data in [25] to illustrate furtheruses of the axiomatically constructed fairness measures in thepaper. The topology is as shown in Figure 20. The channelmodel in [25] has the following parameters: Twisted pairlines have a diameter of 0.5 mm. Transmit power budget ispi,max =20.4 dBm for each user. SNR gap Γ is 12.9 dB,corresponding to a coding gain of 3 dB, a noise margin of6 dB, and a target symbol error probability of 10−7. Tonespacing is 4.3125 kHz. DMT symbol rate is η =4 kHz.

There are three points we will illustrate. First, the “propor-tional” style fairness in (93) can be quantified using fairnessmeasure f . Second, using f we can quantify the tradeoffbetween reducing power and the fairness of doing so. Third,asymmetric fairness measures are usefuls in weighing differentusers based on her location.

1 2 3 4 5 60

0.5

1

Users

Nor

mal

ized

dat

a ra

tes

or p

ower

usa

ges

Fairness of normalized data rates = Fairness of normalized power usages = 5.9062

Ri/R

io

pi/p

i,max

1 2 3 4 5 60

5

10

15

Users

Dat

a ra

tes

(Mbp

s)

or p

ower

usa

ges

(dB

m)

Fairness of data rates = 4.5313, Fairness of power usages = 5.9062

Ri

pi

Fig. 21. Plot optimal (normalized and unnormalized) data rates and powerusages, solving (93). Normalized data rates and power usages achieve a largefairness value f = 5.9062, while fairness of data rates is smaller, due to annear-far effect in the network.

22

First, we solve problem (93) with ρ = 1.75 and plot thedistributions of the optimal data rate and transmit power acrossdifferent users in Figure 21. The ratio of normalized data ratesto normalized power usage is proportional for all users, dueto the third constraint in (93). The optimal solution to (93)achieves the same fairness for the rate reduction vector andthe power reduction vector , i.e.,

f

(R1

Ro1

, . . . ,R1

Ro1

)= f

(p1

p1,max, . . . ,

pn

pn,max

)= 5.9062 (94)

where we choose f to be the symmetric fairness measure (10)in Section II with β = −2. This fairness value is only 0.1 (i.e.,2%) away from maximum fairness f = 6. It is (almost) β =−2 fair. We also observe that fairness of data rate distributionis much smaller:

f (R1, . . . , Rn) = 4.5313. (95)

This is due to the near far near-far effect in the DSL network.Next, we solve (93) with different constant ρ > 0 and let

the resulting normalized total power∑

i pi/∑

i pi,max in theoptimal solution vary between 0.2 and 1. Figure 22 plots thecorresponding fairness of (normalized and unnormalized) datarate distributions in each optimal solution, using the samesymmetric fairness measure with β = −2. It illustrates atradeoff between fairness and the amount of power reduction:fairness improves as normalized total power

∑i pi/

∑i pi,max

increases from 0.2 (i.e., an 80% power reduction) to 1 (i.e.,no power reduction). Again, data rates are less fair thannormalized data rates due to the near-far effect.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

4

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

5.8

6

Degree of energy reduction (Σi p

i / Σ p

i,max)

Fair

ness

f

f(R1/R

1o,…,R

n/R

no): Fairness of normalized rates

f(R1,…,R

n): Fairness of rates

Fig. 22. Tradeoff between reducing power and fairness in doing so,i.e., fairness of (normalized and unnormalized) data rate distributions vs.normalized total power

∑i pi/

∑i pi,max. Fairness improves as normalized

total power increases from 0.2 (80% power reduction) to 1 (no powerreduction).

Finally, consider R∗i , user i’s optimal data rate in prob-lem (93). It is larger for the more advantageously-locatedusers. If we consider the family of asymmetric fairness mea-sures (66) in Section IV, according to Corollary 23, {R∗i }also maximizes the following asymmetric fairness measuref([R1, . . . , Rn], [q1, . . . , qn]), where a higher weight is as-signed to a user with larger optimal data rate, with the

following choice of user specific weights:

qi = (R∗i )rβ+1, for i = 1, . . . , n. (96)

B. Related Axiomatic Theories and Fairness Studies

Forcing metrics of important notions to be the inevitableconsequence of simple statements is a practice commonlyfound in the mathematical way of thinking. Here we compareand contrast with several axiomatic theories in information,economics, and political philosophy, some of which are quan-titative. We show that fairness measures in this paper is ageneralization of Renyi entropy, and a special case of Lorenzcurve ordering. Axioms here are contrasted with those in Nashbargaining solution and in Shapley value. We also surveyseveral studies of fairness in other disciplines.

1) Renyi entropy: Renyi entropy is a family of functionalsquantifying the uncertainty or randomness of generalized prob-ability distributions, developed in 1960 [26]. Renyi entropy isderived from a set of five axioms as follows:

1) Symmetry.2) Continuity.3) Normalization.4) Additivity.5) Mean-value property.

Comparing Renyi’s axioms to ours, we notice that theAxiom of Continuity and the Axiom of Normalization areequivalent to our Axiom of Continuity and Axiom of Ho-mogeneity, respectively. The Axiom of Symmetry becomesCorollary of Symmetry in Section II, due to our Axiom ofPartition. Next, the Axiom of Additivity and Axiom of Mean-value are replaced by our Axiom of Partition. More precisely,the Axiom of Additivity can be directly derived from ourAxiom of Partition as in (6) for direct product. The Axiomof Mean-value, which states that the entropy of the unionof two incomplete distributions is the weighted mean valueof the entropies of the two distributions, plays a role similarto our recursive construction of f in the Axiom of Partition.The key difference is that in our Axiom of Partition and itsimplications, fairness of a vector is constructed by the productof a “global” fairness across partitions, and a weighted mean of“local” fairness within each partition. The Axiom of Saturationand the Axiom of Starvation are unique to our system.

For β ≤ 1, it is straightforward to verify that our fairnessmeasure satisfying the Axioms 1′′-5′′ in Section V includesRenyi entropy as a special case. If we choose β = 1− α andλ = −1 in our fairness measure Fβ,λ, then Fβ,λ equals to theexponential of Renyi entropy with parameter α, i.e.,

Fβ=1−α,λ=−1(x) = ehα(x). (97)

Our proof for the existence and uniqueness of fairnessmeasures in Theorems 1-3 in Appendix C extends those ofRenyi entropy [26] and Shannon entropy [27]: A key step inthe proofs is to show that any function satisfying the (fairnessor entropy) axioms is an additive number-theoretical function[26]. For fairness we show that log f(1n) is an additive

23

number-theoretical function over integer n ≥ 1, i.e.,

logf(1mn)f(1)

= logf(1n)f(1)

+ logf(1m)f(1)

, (98)

limn→∞

logf(1n+1)

f(1)− log f(1n)

f(1)= 0. (99)

where 1n is an all-one resource allocation vector of length n.Conditions (98) and (99) follow from the Axiom of Partitionand the Axiom of Saturation, respectively. Using the resultin [26] we have log f(1n) = r log n, which, together withthe Axiom of Homogeneity, leads to an explicit expression offairness measure f(x) for the case of two users and resourcevectors of rational numbers. The result is then generalizedto resource vectors of real numbers using the Axiom ofContinuity, and to arbitrary number of users based on Axiomof Partition. The Axiom of Saturation and the Axiom ofStarvation are then used to ensure uniqueness of fairnessmeasures.

Renyi entropy, a generalization of Shannon entropy, hasbeen further extended by others since the 1960s, e.g., smoothRenyi entropy for information sources [28] and quantum Renyientropy [29]. This work is a different generalization of Renyientropy: first along asymmetric users with user-specific valuesand then along parameter plane of general λ and β > 1. Renyientropy and α-fairness, which are now special cases of ouraxiomatic system (for β ≤ 1 and β ≥ 0 respectively), arelargely “complementary” along β axis, but also with a rangeof overlap for β ≤ 1.

2) Lorenz Curve: Schur-concavity of fairness measuresproven in Theorem 3 for discrete-dimensional inputs and inTheorem 12 for continuous-dimensional inputs is a criticalproperty for justifying fairness measures by establishing anordering on the set of Lorenz curves [18]. Let Px(y) be thecumulative distribution of a resource allocation x. Its Lorenzcurve Lx, defined by

Lx(u) =1µ·∫

{Px(y)≤u}ydPx(y), (100)

is a graphical representation of the distribution of x, andhas used to characterize the social welfare distributions andrelative income differences in economics.

In 2001, an axiomatic characterization of Lorenz curveorderings was proposed based on a set of four axioms [18]:

1) Order. (The ordering is transitive and complete.)2) Dominance. (The ordering is Shur-concavity.)3) Continuity.4) Independence.

It is shown that a Lorenz curve orderings Lx º Ly satisfies thefour axioms above if and only if there exists a continuous andnon-increasing real function p(u) defined on the unit interval,such that

Lx º Ly ⇔∫ 1

0

p(u)dLx(u) ≥∫ 1

0

p(u)dLy(u) (101)

Our fairness measures have an equivalent representation inSection III.1, i.e.,

F = sign(r(1− βr)) ·[∫ 1

u=0

(P−1

x (u)µ

)−βr

dLx(u)

] 1β

.(102)

Comparing (102) to (101), it is easy to see that our fairnessmeasure defines a Lorenz curve ordering for all β ≥ 0 andr ≥ 0, by setting

p(u) =(

P−1x (u)µ

)−βr

, (103)

which is continuous and non-increasing over u ∈ [0, 1].Therefore, the Lorenz curve ordering defined by our fairnessmeasure satisfies the four axioms in [18], including Shur-concavity property proven in Section II and Section III.

3) Cooperative Economic Theories: A number of theorieshave been developed to study the collective decisions ofgroups. Many of these theories have also been uniquelyassociated with sets of axioms [30]. While applications ofthese theories range from political theory to voting methods,here we focus on two well-known axiomatic constructions:the the Nash bargaining solution in 1950 [31] and the Shapleyvalue in 1953 [32].

The Nash bargaining solution was developed from a set offour axioms:

1) Invariance to affine transformation.2) Pareto optimality.3) Independence of irrelevant alternatives (IIA).4) Symmetry.

Symmetry is shown as a corollary in our theory, and Paretooptimality not imposed as an axiom given our focus onfairness, but rather used as a criterion in balancing fairnessand efficiency in Section V. Nash’s axiom of IIA contributesmost to his uniqueness result, and it is also often consideredas a value statement. It has been shown by many others,that replacing the “local-closedness” axiom of IIA with othervalue statements may result in solution classes different fromthe bargaining solution. Given a feasible region of individualutilities, the Nash bargaining solution is also equivalent to amaximization of the proportional fairness utility function.

Another well-known concept in economic study of groups isthe Shapley value [32]. In a coalition game, individual decidewhether or not to form coalitions in order to increase themaximum utility of the group, while ensuring that their shareof the group utility is maximized. The Shapley value concernsan operator that maps the structure of the game, to a set ofallocations of utility in the overall group. Given a coalitiongame, four axioms uniquely define the Shapley value:

1) Pareto Optimality.2) Symmetry.3) Dummy.4) Additivity.

Again, Pareto optimality is included as an axiom. Although thestructure of a coalition game is different than a simple divisionof resources, some parallels are apparent. For instance, theDummy axiom refers to a scenario where an individual doesnot increase any groups utility by joining it. In this case, theShapley value assigns a utility of zero to that individual. Thisis not imposed in our theory as an axiom, but is similar toCorollary 4, where an inactive user has no impact on fairnessvalue. Shapley’s axiom of additivity provides a method ofbuilding up a single coalition game with potentially many

24

individuals, from smaller games, which may have only twoplayers. This is similar to our Axiom if Partition, wherein thefairness measure is recursively constructed from the fairnessattained by subsets of the overall allocation.

In both Nash bargaining solution and Shapley value, ef-ficiency is an axiom in defining the solution. In contrast,the first family of fairness measures, f in Section II, isconfined to homogeneous functions of degree zero. One mightsuspect that it is possible to extend our axiomatic structure tothe optimization theoretic fairness approach by relaxing theAxiom of Homogeneity to homogeneous functions of arbitrarydegree. This is indeed the case as developed in Section V.

The distinction between fairness measures with local-closedness axiom and those with decomposability axiom hasbeen discussed in the Conclusion. It remains to be seen ifthe “fair” operating point as defined by NBS/Shapley canbe incorporated into a user reaction model together with ourfairness measure.

4) User Reaction Models and Fair Taxation: In [83], [84],a dynamic system model was developed to study the long-termimpact and equilibrium behaviors of users planning economicactivities in anticipation of tax policies, defined as the medianvoter action. Fairness is an important part of the model, andis defined as the effort-based (rather than corruption-basedcomponent in the sum of total income) and measured byvariance. Again, there is some ideal point of operation thatis fair based on the source of income.

A more general notion of fairness may be inserted inthe control dynamic system. What would be the equilibriumproperties (existence, number, income distribution, efficiency)if our fairness measure is used, with q modeling the user’srelative position in effort-based vs. corruption-based income?This is also related to optimal taxation from a fairness pointof view.

The standard theory of optimal taxation maximizes a socialwelfare function subject to a range of constraints [85]. Theobjective function is almost always the sum utilities, and oftenlinear utilities. We briefly discuss some implications of ourfairness theory on taxation. There are many more questions tobe addressed beyond the initial study below.

We may quantify the impact of tax on fairness by analyzingfairness of pre-tax, post-tax resource allocations, and taxesthemselves, using fairness measures derived from Axioms 1-5.Let x = [x1, . . . , xn] be a pre-tax resource allocation vectorand c : R+ → R+ be a tax policy. The corresponding post-taxresource allocation [x1−c(x1), . . . , xn−c(xn)] will be denotedby x− c(x) with a slight abuse of notation. We consider a taxpolicy c satisfying the following two assumptions:

(a) Zero tax are collected from inactive users, i.e., c(0) = 0.(b) Users’ resource ranking remains unchanged, i.e., xi −

c(xi) > xj − c(xj) for all xi > xj .It is easy to see that these two assumptions guarantee positivityof post-tax allocation, i.e., xi − c(xi) > 0 for all xi ∈ R+.

For a given fairness measure f satisfying Axioms 1-5, wesay that a tax policy is fair if f(x − c(x)) ≥ f(x) for allx ∈ Rn

+ and unfair if f(x−c(x)) ≤ f(x) for all x ∈ Rn+. We

derive the following corollaries for characterizing fairness ofpre-tax and post-tax resource allocations that hold for all fβ

Corollary 23: (Fair tax policy.) A tax policy c is fair if andonly if c(z)/z is non-decreasing for z ∈ R+.

Corollary 24: (Unfair tax policy.) A tax policy c is unfairif and only if c(z)/z is non-increasing for z ∈ R+.

Corollary 23 shows that when fairness of a tax policy c isevaluated using fbeta, it coincides with the intuition that fairtax must be progressive. It is straightforward to verify thatany convex and non-decreasing function c satisfies Corollary23 and is therefore fair. On the other hand, Corollary 24 showsthat any tax policy c less progressive than a linear taxation isunfair. A special case is collecting a constant tax from all users.More precisely, if a constant amount t > 0 of the resource issubtracted from each user (i.e., xi − t for all i), the resultingfairness decreases, i.e., f(x− t · 1n) ≤ f(x), where t > 0 issmall enough such that xi − t is non-negative.

Corollary 25: (Fairness of c.) A tax policy c is fair only iff(c(x)) ≤ f(x). Similarly, c is unfair only if f(c(x)) ≥ f(x).

It is necessary to have the taxation vector as less fair than thepre-tax resource vector. But this condition is only necessary.f(c(x)) ≤ f(x) does not always imply that tax policy c isfair. For instance, we have f(c(x)) < f(x) and f(x−c(x)) ≤f(x), when x = [6, 3, 2], c(x) = 3, 1, 1, and β = 2.5.

Corollary 26: (Ordering of tax policies.) Let c1 and c2

be two tax policies. We say that c1 is more fair than c2 iff(x − c1(x)) ≥ f(x − c2(x)) for all x ∈ Rn

+. This holds ifc1(x)/c2(x) is non-decreasing, plus any of the following threesufficient conditions:

(a) c1 is fair and c2 is unfair.(b) Both c1 and c2 are fair and c1(z) ≥ c2(z) for all z ∈ R+.(c) Both c1 and c2 are unfair and c1(z) ≤ c2(z) for all

z ∈ R+.

This corollary establishes a complete ordering of tax poli-cies with respect to fairness of post-tax allocations. Given thattax policy c1 is more progressive than tax policy c2, threesufficient conditions for c1 to be more fair than c2 are derived.When two tax policies are both fair or both unfair, their impacton fairness is affected by their magnitudes.

5) Ultimatum Game, Cake-Cutting, and Fair Division:In Ultimatum Game [81], player A divides a resource intotwo parts, one part for herself and the other for player B.Player B can then choose to accept the division, or reject it,in which case neither player receives any resource. Withouta prior knowledge about player B’s reaction, player A maydivide anywhere between [0.5, 0.5] and [1, 0]. Running thisgame as a social experiment in different cultures have lead todebates about the exact implications of the results on people’sperception on fairness: how fair does it take for player B’sperception, and player A’s guess of that, to accept player A’sdivision? This has also been contrasted with the perception offairness in the related Dictator Game [75], [76], where playerB has no option but to accept the division by player A.

A classic generalization of Ultimatum Game that has re-ceived increasing attention in the past decade is the cake-cutting problem. As reviewed in books [74], [80], the cakeis a measure space, and each player uses a countably-additive,non-atomic measure to evaluate different parts of the cake.

25

Among the work studying the cake-cutting problem, e.g., [79],the primary focus has been on two criteria: efficiency (Paretooptimality) and fairness (envy-freeness). Achievability resultsfor 4 users or more are still challenging.

As reviewed in [82], cake-cutting can be generalized to “fairdivision” problem, a typical example being the joint problemof room-assignment and rent-division. Each user contributesto the resource pool, and as explained in [77], there are manysurprises in the solution structures if envy-freeness is againtaken as the definition of fairness.

Fairness in cake-cutting and fair-division is traditionallydefined as envy-freeness. It is a binary summary based on eachindividual’s local evaluation: the allocation of cake is fair ifno user wants to trade her piece with another piece. In [78],this restrictive viewpoint on fairness is expanded to includeproportional allocation of the left-over piece after each usergets 1/nth of the cake (in her own evaluation). It is shownthat Pareto optimality and proportional sense of fairness maynot be compatible for 3 players or more.

Equipped with the axiomatically constructed fairness mea-sures fβ , we see that some questions are in order. Is there a 2-person, 1-cut procedure (with the help of a referee and possiblynot strategy proof) that is envy-free, Pareto optimal, andfairness-value-y achieving ? For 3-person, is there a procedurethat is Pareto optimal and fairness-value-y achieving? For whatβ in fβ(x) would envy-free be compatible with fairness? Moregenerally, if envy-freeness is replaced by fairness in the senseof achieving a target value of fβ(U1(x1), . . . , Un(xn)), whereUi is user i’s own preference/utility function, will existenceresults and constructive procedures in efficient and fair cuttingof cake be more readily achieved?

6) Rawls’ Theory of Justice and Distributive Fairness:In 20th century political philosophy, the work of Rawls onfairness has been arguably the most influential and provocativesince its original publication in 1971 [33]. Starting withthe “original position” behind the “veil of ignorance”, thearguments posed by Rawls are based on two fundamentalprinciples (i.e., axioms stated in English), as stated in his 2001restatement [55]:

1) “Each person is to have an equal right to the mostextensive scheme of equal basic liberties compatiblewith a similar scheme of liberties for others.”

2) “Social and economic inequalities should be arrangedso that they are both (a) to the greatest benefit of theleast advantaged persons, and (b) attached to offices andpositions open to all under conditions of equality ofopportunity.”

The first principle governs the distribution of liberties andhas priority over the second principle. One may interpret it as aprinciple of distributive fairness in allocating limited resourcesamong users. It can now be captured as a theorem (rather thanan axiom) that says any fairness measures satisfying Axioms1-5 will satisfy the following property: adding an equal amountof resource to each user will only increase fairness value, i.e.,

fβ (x + c · 1n,q) ≥ fβ (x,q) , for qi =1n

, ∀c ≥ 0, ∀β,

where equal weights qi = 1n can be viewed as a quantification

of “equal right” in the first principle of Rawls’ theory.First part of Rawls’ second principle concerns with the

distribution of opportunity while the second part is the cel-ebrated “difference principle”: an approach different fromstrict egalitarianism (since it is on the absolute value ofthe least advantaged user rather than the relative value) andutilitarianism (when narrowly interpreted where the utilityfunction does not capture fairness). Now it is axiomaticallyconstructed as a special case of a continuum of generalizednotions trading-off fairness with efficiency. Our results canbe viewed as an axiomatic quantification of Rawls’ theory,especially as a mathematical extension and representation ofthe equal rights principle and the difference principle. This isbest illustrated by annotating Rawls’ own graph in Figure 23.The point representing the difference principle is now seen asthe consequence of concatenating two steps of pushing to theextremum on both β and λ: β →∞, and λ as large as possiblewhile retaining Pareto-efficiency, i.e., λ → λ.

Fig. 23. Annotated version of the only quantitative graph in Rawls [55]on p.62. The colored circles and arrows are added by the PI. Since theefficiency frontier between the More Advantaged Group (MAG) user and theLess Advantaged Group (LAG) user does not pass through 45 degree line,strict egalitarian point is the origin. Difference principle generates a pointD on the efficient frontier, whereas utility maximization generates point B.Nash Bargaining Solution point N may lie between D and B. α-fair utilityfunction, being the point on the efficient frontier closest to the 45 degree line,will produce D as well. Fairness function Fβ,λ(x,q) can generate any pointon the efficient frontier when λ is restricted to be less than the threshold forPareto efficiency to retain.

Among the other main branches of fairness in philosophy,the utilitarian or welfare-based principle [56], [58] advocatesutility maximization, but the form of utility function can bebroad, indeed, as broad as Fβ,λ that captures both fairnessand efficiency. Still, specification and implementation can bechallenging, there as well as in our fairness theory: e.g., user-specific utility modeling, inter-user utility “conversion rate”,computational challenge of solving constrained optimization,and potential non-uniqueness of optimal solution.

Desert-based principles [60], [62] provide helpful anglesfor us to model the user-weights q, which can be based oncontribution, effort, or compensation. Libertarian principlesoffer other viewpoints on fairness. One of these is the 3-part entitlement theory by Nozick [63], complemented by the“Lockean proviso” [64], where an exclusive acquisition of aresource is just if there is “enough and as good left in commonfor others” afterwards. The theory becomes less clear in case

26

of scarce resources where the constraint set does not allowsuch a scenario [65]

7) Normative Economics and Welfare Theory: Rawls’ the-ory also has intricate interactions with normative economics,where many results are analytic in nature [68]. In additionto stochastic dominance, Arrow’s Impossibility Theorem, andthe cooperation game theories of Nash and of Shapley, thereare several major branches [57], [69], [71]. Another set is theethical axioms of transfers. For example, the Pigou-Daltonprinciple states that inequality decreases via Robin Hoodoperation that does not reverse relative ranking. The principleof proportional transfers states that what the donor gives andthe beneficiary receives should be proportional to their initialpositions.

Bergson-Samuelson social welfare function [66], [72]W (U1(x), . . . , Un(x)) aims at enabling complete and consis-tent social welfare judgement on top of individual preference-based utility functions Ui. Its existence raises the question ofcomparing

∑i wixi with fβ(x) · (∑i xi)λ in our theory.

Kolm’s theory of fair allocation [70] uses the criterion ofequity as “no-envy”, and it is well- known that competitiveequilibrium with equal budget is the only Pareto-efficient andenvy-free allocation if preferences are sufficiently diverse andform a continuum [59]. It is unclear if it we can replace envy-freeness with a system-wide fairness criterion like fβ(x).

8) Connections with Sociology and Psychology: InequalityIndices. Quantifying inequality/injustice/unfairness using in-dividual, local measures, has been pursued in sociology. Forexample, Jasso in 1980 [46] advocated justice evaluation indexas log of the ratio between actual allocation and “just” allo-cation. Allocation can be done either in quantity or in quality(in which case ranking quantifies the quality allocation). Manyproperties were derived in theory and experimented with indata about income distribution in different countries [48]. Inparticular, probability distribution of the index is induced bythe probability distribution of the allocation. This index isderived based on two principles and three laws in the paper,including “equal allocation maximizes justice” and “aggregatejustice is arithmetic mean of individual ones”. These arespecial cases of the corollaries or axioms in our development,and the index can indeed be viewed as a special case of fβ :a logarithm function with a constant offset.

In [47], two other injustice indexes, JI1 and JI2, were devel-oped based on the above. One interesting feature is that JI1differentiates between under-reward and over-reward as twotypes of injustice. Another useful feature is the decompositionof the total amount of perceived injustice into injustice dueto scarcity and injustice due to inequality. This is similar tothe two components on efficiency and fairness in Fβ,λ(x).In [49], inequality between two individuals is decomposedinto inequality within the groups they belong to and inequalitywithin each group. This decomposition is generalized in ourrecursive construction of fairness measures implied by Axiomof Partition.

They are further unified with Atkinson’s measure of in-equality [57]: 1 minus the ratio of geometric mean andarithmetic mean. At the heart of these indices is the approachof taking combinations of arithmetic and geometric means of

an allocation to quantify the its spread. In this view, it is notsurprising that Atkinson’s index is also a special case of ourfairness measures, as shown in Figure 11. The list of axiomsbehind Atkinson’s index, however, is both longer and morerestrictive.

One may be able to view actual allocation as the optimizerof Fβ,λ(x) and the just allocation as say NBS. Then thejustice evaluation by each individual thus calculated may besubstituted into dynamic system describing the allocation andreaction process.

Procedural Fairness. Fairness may be as much about theoutcome as about the procedure. Procedural fairness has beenextensively studied in legal system and has generated muchinterest in psychology and organization studies since the1960s. It is sometimes coupled with user-weight assignment indistributive fairness, e.g., in [50] equity means proportionalitybetween resource allocation and individual contributions.

Substantial amount of work in procedural fairness initiatedwith Thibaut and Walker’s book [54] in 1975, with specificfocus on dispute resolution via a third party, as well as psycho-logical and cultural underpinnings of preferences for variousabjudication procedures. Methodological issues in laboratorystudies since then have been surveyed in Lind and Tyler’s book[51].

In two publications [52], [53] on general theory of procedu-ral fairness, Leventhal advocated six criteria: consistency, biassuppression, accuracy of information, correctability, represen-tativeness, and ethicality. Some of these elements are clearlycontext and culture-dependent. Several interesting questionsare in order. Is it possible to combine distributive and proce-dural justice to reach general assessment of fairness? Whenwill procedural justice concerns override distributive justiceassessment? Can procedural fairness measures be axiomati-cally constructed out of the key criteria and maximized usingpractical protocols?

C. Collection of Proofs

1) Proof of Theorems 1, 2, and 3: We prove that fairnessmeasures satisfying Axioms 1-5 can only be generated by log-arithm and power generator functions. Then, for each choiceof logarithm and power generator function, a fairness measureexist and is uniquely given by either (9) or (10). A fairnessmeasure is a sequence of mapping {fn : Rn

+ → R, ∀n ≥ 1}.To simplify notations, we suppress n in fairness measures anddenote them by f . It is easy to verify that if f > 0 satisfiesAxioms 1-4, so does −f < 0. We first assume that f > 0during this proof and then determine sign to fairness measuresfrom Axioms 5.

Our proof starts with a lemma, which states that fairnessachieved by equal-resource allocations 1n is independent ofthe choice of g. This Lemma will be used repeatedly in theproof.

Lemma 2: From Axioms 1–5, fairness achieved by equal-resource allocations 1n is given by

f(1n) = nr · f(1), ∀ n ≥ 1, (104)

where r ∈ R is a constant exponent.

27

Proof:We first prove that f(1mn) · f(1) = f(1n) · f(1m) for all

m,n ∈ Z+. For m = 2 and arbitrary n ∈ Z+, we have

f(12n)f(12)

=f(12n)

f(n · 12)

= g−1

(2∑

i=1

si · g(

f(1n)f(n)

))

= g−1

(g

(f(1n)f(1)

))

=f(1n)f(1)

, (105)

where the first step uses homogeneity in Axiom 2, the secondstep uses Axiom 4 with x = [1n,1n] and y = [n, n], and thethird step uses

∑i si = 1.

We use induction and assume that for m ≤ k and n ∈ Z+,we have

f(1mn) · f(1) = f(1n) · f(1m) (106)

For m = k + 1 and n ∈ Z+, we derive that

f(1(k+1)·n)

f(1k+1)=

f(1(k+1)·n)f(n · 1k+1)

= g−1

(s1 · g

(f(1kn)

f(n · 1k)

)+ s2 · g

(f(1n)f(n)

))

= g−1

((s1 + s2) · g

(f(1n)f(1)

))

=f(1n)f(1)

, (107)

where the second step uses Axiom 4 with x = [1kn,1n] andy = [n ·1k, n], and the third step uses (106) with m = k. Thismeans that f(1mn)·f(1) = f(1n)·f(1m) holds for m ≤ k+1and n ∈ Z+. Therefore, it holds for all m,n ∈ Z+.

From (106), we can show that log f(1mn)/f(1) is anadditive number-theoretical function [34], i.e.,

logf(1mn)f(1)

= logf(1n)f(1)

+ logf(1m)f(1)

(108)

Further, from Axiom 3, we derive

limn→∞

[log

f(1n+1)f(1)

− logf(1n)f(1)

]= 0 (109)

Using the result in [34], (108) and (109) implies thatlog f(1n)/f(1) must be a logarithmic function. We have

logf(1n)f(1)

= r log n, (110)

where r is a real constant. This is (104) after taking anexponential on both sides.

(Uniqueness of logarithm and power generator functions.)We first show that logarithm and power generator functionsare necessary. In other words, no other fairness can possiblysatisfy Axioms 1-5 with weight si in (15).

We consider arbitrary positive X1 + X2 = 1, whichcan be written as the ρ-th power of rational numbers, i.e.,X1 = (a1/b1)ρ and X2 = (a2/b2)ρ, for some positive

integers a1, b1, a2, b2. Let u, v be two positive integers andK = (u/v)1/ρ be a rational number to be used as a short-hand notation later. We choose the sum resource of the twosub-systems to be W1 = a1b2v and W2 = a2b1v, so that

Xi =Wρ

i

Wρ1 +Wρ

2

, i = 1, 2. (111)

Consider the following two resource allocation vectors: Forresource allocation vector x, sub-system 1 has t = b1b2u users,each with resource W1/t; sub-system 2 has t = b1b2u users,each with resource W2/t. In other words, we have

x1 = [W1

t, . . . ,

W1

t︸ ︷︷ ︸t times

] and x2 = [W2

t, . . . ,

W2

t︸ ︷︷ ︸t times

] (112)

For resource allocation vector y, sub-system 1 has a1b2v users,each with resource 1; sub-system 2 has a2b1v users, each withresource 1. This means

y1 = [ 1, . . . , 1︸ ︷︷ ︸a1b2v times

] and y2 = [ 1, . . . , 1︸ ︷︷ ︸a2b1v times

] (113)

The sum resource of the two sub-systems Wi = vtai/bi

is in-variant. Let ξi = Wρi /(Wρ

1 + Wρ2 ), for i = 1, 2, be

an auxiliary variable. It follows from Axiom 4 and (104) inLemma 2 that

f(y1,y2)f(x1,x2)

= g−1

(2∑

i=1

si · g(

f(yi)f(xi)

))

= g−1

(ξ1 · g

(ar1b

r2v

r

tr

)+ ξ2 · g

(ar2b

r1v

r

tr

))

= g−1

(X1 · g

(ar1v

r

br1u

r

)+ X2 · g

(ar2v

r

br2u

r

))

= g−1(X1 · g

((X1/K)

rρ

)+ X2 · g

((X2/K)

rρ

))(114)

where the second step uses Axiom 2 and (104) in Lemma 2,i.e.,

f(xi) = f

(Wi

t· 1t

)= f (1t) = tr (115)

and

f(y1) = f

(W1

t· 1a1b2v

)= (a1b2v)r (116)

f(y2) = f

(W2

t· 1a2b1v

)= (a2b1v)r (117)

Next, we apply (5) in Lemma 1 to f(x1,x2) and obtain

f(x1,x2)

= f

(W1 +W1

t· 1t

)· g−1

(g

(f

(W1

t,W2

t

)))

= f (1t) · f(W1,W2)= tr · f(W1,W2)

= tr · f(X1ρ

1 , X1ρ

2 ) (118)

28

where the entire system is partitioned into t sub-systems3, eachwith a resource allocation vector [W1/t,W1/t]. The last stepuses (111). Again, using Axiom 2 and (5) in Lemma 1, wederive

f(y1,y2) = f (1a1b2v+a2b1v) = (a1b2v + a2b1v)r (119)

Combining (114), (118), and (119), we derive

(X1ρ

1 + X1ρ

2 )r

f(X1ρ

1 , X1ρ

2 )K− r

ρ

=(a1b2v + a2b1v)r

f(X1ρ

1 , X1ρ

2 ) · tr

= g−1

(2∑

i=1

Xi · g((

Xi

K

) rρ

))(120)

For different K > 0, (120) further implies that

g−1

(2∑

i=1

Xi · g((

Xi

K

) rρ

))

= Krρ · g−1

(2∑

i=1

Xi · g((

Xi

K

) rρ

))(121)

for rational X1, X2,K > 0 and arbitrary ρ, r. Due to Axiom 1,it is easy to show that (121) also holds for real X1, X2,K > 0.Without loss of generality, we assume that g(1) = 0, becausewe may replace g(y) by g(y)−g(1) in (121), and the equationstill holds.

It then follows from Theorem 4.3 in [35] that

g(X ·K) = p(K)g(X) + q(K) (122)

where p(K), q(K) are functions of K for real X, K > 0.Using g(1) = 0 and choosing X = 1, we derive

g(K) = q(K) (123)

Substituting it into (122), we find that

g(X ·K) = p(K)g(X) + g(K) (124)

for all positive X,K > 0. Similarly, we have

g(X ·K) = p(X)g(K) + g(X). (125)

Combining (124) and (125) gives

p(X)− 1g(X)

=p(K)− 1

g(K)(126)

for all X, K > 04. Each of these functions must reduces to aconstant c, so that p(X)− 1 = c · g(X). It then follows from(124) that

g(X ·K) = cg(X) · g(K) + g(X) + g(K). (127)

3Although Axiom 4 is defined for partitioning into 2 subsystems, it can beextended to partitioning into t identical sub-systems, using the same inductionin proof of Lemma 2.

4It holds only for X 6= 1 and K 6= 1. However, since (127) is true ifX = 1 or K = 1, the exception is irrelevant for proving (126).

To determine possible generator functions g, we need toconsider two cases. (a) If c = 0, (127) reduces to g(X ·K) =g(X) + g(K), whose solution is

g(y) = z · log(y) (128)

where z 6= 0 is an arbitrary constant. (b) If c 6= 0, we putc · g(y) + 1 = h(y) and (127) reduces to h(X ·K) = h(X) ·h(K), whose general solution is h(y) = yβ . Hence, we have

g(y) =yβ − 1

c(129)

where β ∈ R is a constant exponent.(Existence of fairness measures for logarithm generator.)

We derive fairness measure for the logarithm generator func-tion in (128) and show that it satisfies Axioms 1-5. By choos-ing K = u/v = 1 and a change of variable xi = X

1ρ

i = ai/bi

in (120), we plug in the logarithm generator (128) and find

f(x1, x2) =(x1 + x2)r

g−1(∑2

i=1xρ

i

xρ1+xρ

2· g(xr

i ))

=2∏

i=1

(xi

x1 + x2

)−rx

ρi

xρ1+x

ρ2 (131)

To generate fairness measure for n > 2 users, we use inductionand assume that for n = k users, we have

f(x1, . . . , xk) =k∏

i=1

(xi∑k

j=1 xj

)−rx

ρi∑

j xρj

(132)

which holds for n = k = 2. Let uk =∑k

i=1 xi. For n = k+1,we apply (5) in Lemma 1 and derive (130), where the secondstep uses Axiom 2 with f(xk+1) = f(1) = 1. Comparing thelast step of (130) to (132) with n = k+1, induction of fairnessmeasure from n = k to n = k + 1 holds if and only if

uρk =

(k∑

i=1

xi

)ρ

=k∑

i=1

xρi . (133)

Therefore, we must have ρ = 1. Fairness measure generatedby logarithm generator in (128) can only have the followingform

f(x1, . . . , xk) =k∏

i=1

(xi∑k

j=1 xj

)−rxi∑j xj

(134)

which is (9) in Theorem 3.The proof so far assumes positive fairness measure f > 0.

To determine the sign of f , we add sign(r) in (134), applyAxiom 5, and find that for r > 0

f(0, 1) = 1 < 2r = f(12,12), (135)

and for r < 0:

f(0, 1) = −1 < −2r = f(12,12). (136)

Therefore, Axiom 5 is satisfied. We derive the fairness measurein (9).

29

f(x1, . . . , xk, xk+1)

= f

(k∑

i=1

xi, xk+1

)· [f(x1, . . . , xk)]

(∑k

i=1 xi)ρ

(∑k

i=1 xi)ρ+x

ρk+1 · [f(xk+1)]

xρk+1

(∑k

i=1 xi)ρ+x

ρk+1

= f

(k∑

i=1

xi, xk+1

)· [f(x1, . . . , xk)]

uρk

uρk+x

ρk+1 · 1

=

( ∑ki=1 xi∑k+1i=1 xi

) −ruρk

uρk+x

ρk+1

·(

xk+1∑k+1i=1 xi

) −rxρk+1

uρk+x

ρk+1

·k∏

i=1

(xi∑k

j=1 xj

)−rx

ρi∑

j xρj· u

ρk

uρk+x

ρk+1

(130)

It remains to prove that fairness measure (134) satisfiesAxioms 1-4 with arbitrary r ∈ R. It is easy to see that Axioms1–3 are satisfied. To verify Axiom 4, we consider two arbitraryvectors x = [x1,x2] and y = [y1,y2] with w(xi) = w(yi).Since generator g is logarithm function, we show that

f(x1,x2)f(y1,y2) =

g−1(∑2

i=1 si · g(f(xi)

))

g−1(∑2

i=1 si · g (f(yi)))

= exp

{2∑

i=1

si ·[log

(f(xi)

)− log(f(yi)

)]}

= exp

{2∑

i=1

si ·[log

(f(xi)f(yi)

)]}

= g−1

(2∑

i=1

si · g(

f(xi)f(yi)

))(137)

which establishes Axiom 4 for any partition. This completesthe proof for logarithm generators.

(Existence of fairness measures for power generator.) Wederive fairness measure for power generator function in (129)and show that it satisfies Axioms 1-5. By choosing K =u/v = 1 and a change of variable xi = X

1ρ

i = ai/bi in(120), we plug in the power generator (129) and find

f(x1, x2) =(x1 + x2)r

g−1(∑2

i=1xρ

i

xρ1+xρ

2· g(xr

i ))

=(x1 + x2)r (xρ

1 + xρ2)

1β

(xρ+βr

1 + xρ+βr2

) 1β

. (138)

To derive the fairness measure for three users, we con-sider two different partitions of the resource allocation vector[x1, x2, x3] as [x1, x2], [x3] and [x1], [x2, x3]. Using (5) inLemma 1, we obtain two equivalent form of the fairnessmeasure for n = 3 users in (139) and (140).

As in Axiom 4, the fairness measure is independent ofpartition. Hence, (139) and (140) should be equivalent for allx1, x2, x3 ≥ 0. Comparing the terms in (139) and (140), wemust have r = 0 or ρ + βr = 1. Since r = 0 is a special caseρ + βr = 1 by choosing ρ = 1 and β 6= 0, we only need toconsider a unified representation for ρ+βr = 1. We concludethat the fairness measure for n = 3 must have the following

form

f(x1, x2, x3) =

(∑3i=1 x1−βr

i

) 1β

(∑3i=1 xi

) 1β−r

, (141)

where r = 1−ρβ is a proper exponent. Let uk =

∑ki=1 xi be

the sum of the first k elements in vector [x1, . . . , xk, xk+1].Then, using (5) in Lemma 1 inductively, we obtain

f(x1, . . . , xk, xk+1)

= f(k∑

i=1

xi, xk+1) · g−1 (s1g (f(x1, . . . , xk)) + s2g (1))

=

(u1−βr

k + x1−βrk+1

) 1β

(uk + xk+1)1β−r

·

uρk

∑ki=1 x1−βr

i

u1−βrk

+ xρk+1

uρk + xρ

k+1

1β

=

(u1−βr

k + x1−βrk+1

) 1β

(uk + xk+1)1β−r

·[∑k

i=1 x1−βri + x1−βr

k+1

u1−βrk + x1−βr

k+1

] 1β

=

(∑k+1i=1 x1−βr

i

) 1β

(∑k+1i=1 xi

) 1β−r

, (142)

where induction holds from n = k to n = k + 1. The laststep of (142) is almost (10) in Theorem 3, except for a termsign(r(1−rβ)). The term is added in order to satisfy Axiom 5.To show this, we consider four possible cases: for 1− rβ > 0and r > 0

f(0, 1) = 1 < 2r = f(12,12), (143)

and for 1− rβ > 0 and r < 0

f(0, 1) = −1 < −2r = f(12,12). (144)

When 1 − rβ < 0, we have r > 0 ⇔ β > 0 and r < 0 ⇔β < 0. We find that for 1− rβ < 0 and r > 0

f(0, 1) = −∞ < −2r = f(12,12), (145)

and for 1− rβ > 0 and r < 0

f(0, 1) = 0 < 2r = f(12,12). (146)

30

f(x1, x2, x3) = f(x1 + x2, x3) · g−1 (s1g (f(x1, x2)) + s2g (1))

=(x1 + x2 + x3)r ((x1 + x2)ρ + xρ

3)1β

((x1 + x2)ρ+βr + xρ+βr

3

) 1β

·

(x1+x2)ρ+βr(xρ

1+xρ2)

xρ+βr1 +xρ+βr

2+ xρ

3

(x1 + x2)ρ + xρ3

1β

=(x1 + x2 + x3)r

((x1 + x2)ρ+βr + xρ+βr

3

) 1β

·[

(x1 + x2)ρ+βr(xρ1 + xρ

2)

xρ+βr1 + xρ+βr

2

+ xρ3

] 1β

(139)

f(x1, x2, x3) = f(x1, x2 + x3) · g−1 (s1g (f(1)) + s2g (x2, x3))

=(x1 + x2 + x3)r (xρ

1 + (x2 + x3)ρ)1β

(xρ+βr

1 + (x2 + x3)ρ+βr) 1

β

·

(x2+x3)ρ+βr(xρ

2+xρ3)

xρ+βr2 +xρ+βr

3+ xρ

1

(x2 + x3)ρ + xρ1

1β

=(x1 + x2 + x3)r

(xρ+βr

1 + (x2 + x3)ρ+βr) 1

β

·[

(x2 + x3)ρ+βr(xρ2 + xρ

3)

xρ+βr2 + xρ+βr

3

+ xρ1

] 1β

(140)

It is straightforward to check that fairness measure in (142)satisfies Axioms 1-4. This completes the proof for powergenerators.

2) Proof of Corollary 1: We prove symmetry from Axioms1-5 directly, without resorting to the fairness expression in (9)and (10). For n = 2 users, symmetry follows directly from(120), which implies

f(x1, x2) = f(x2, x1), ∀x1, x2 ≥ 0. (147)

To start the induction, assume symmetry holds for n users.Let x = [x1, . . . , xn, xn+1] be a resource allocation vectorand i1, . . . , in, in+1 be an arbitrary permutation of the indices1, . . . , n, n + 1. When in+1 6= 1, applying (5) in Lemma 1,we derive (148).

Next, we can use (148) to show that

f(xi1 , . . . , xin , xin+1) = f(x1, . . . , xn, xn+1). (149)

When in+1 = 1, using the same technique, we have

f(xi1 , . . . , xin , xin+1) = f(xi1 , . . . , xin+1 , xin)= f(x1, . . . , xn, xn+1). (150)

Then symmetry also holds for n + 1 users.3) Proof of Theorem 4: Because vector x is majorized

by vector y, if and only if, from x we can produce y bya finite sequence of Robin Hood operations [15], where wereplace two elements xi and xj < xi with xi − ε and xj + ε,respectively, for some ε ∈ (0, xi − xj), it is necessary andsufficient to show that such an Robin Hood operation alwaysimproves a fairness measure defined by Axioms 1–5.

Toward this end, we consider partitioning a resource allo-cation vector x of n users into two segments: x1 = [xi, xj ]and x2 = [x1, . . . , xi−1, xi+1, . . . , xj−1, xj+1, . . . , xn]. Lety = [y1,x2] where y1 = [xi−ε, xj +ε] be the vector obtainedfrom x1 by the Robin Hood operation. Using Axiom 4, we

have

f(x)f(y) =

f(x1,x2)f(y1,x2)

= g−1

(s1 · g

(f(x1)f(y1)

)+ s2 · g

(f(x2)f(x2)

))

= g−1

(s1 · g

(f(xi, xj)

f(xi − ε, xj + ε)

)+ s2 · 1

)

≤ 1,

where the last step follows form the monotonicity of g andthe monotonicity of fairness measure with two-users in (120),which gives

f(xi, xj) ≤ f(xi − ε, xj + ε). (151)

Therefore, if x is majorized by y, then we have f(x) ≤ f(y).The fairness measure is Schur-concave.

4) Proof of Corollary 2: The proof for Corollary 2 isimmediate from Theorem 4, because among the vectors withthe same sum of elements, one with the equal elements is themost majorizing vector, i.e., 1

n · 1n majorized all x.5) Proof of Corollary 3: This corollary is proven by Lemma

2, in the proof of Theorems 1, 2, and 3.6) Proof of Corollary 4: Let x be an arbitrary resource

allocation vector, t > 0 be a positive number, and z =∑

i xi

be an axillary variable. From Axiom 1, we have

f(x,0n)

= limt→∞

f

(x,

1t1n

)

= limt→∞

f(z,

n

t

)· g−1

(zρg (f(x))zρ +

(nt

)ρ +

(nt

)ρg (f(1n))

zρ +(

nt

)ρ

)

= limt→∞

f

(∑

i

xi,n

t

)· g−1 (g (f(xn)))

= f(x),

31

f(xi1 , . . . , xin, xin+1)

= f

n∑

j=1

xij, xin+1

· g−1

(s1 · g (f(xi1 , . . . , xin

)) + s2g(f(xin+1)

))

= f

n∑

j=1

xij , xin+1

· g−1

(s1 · g

(f(x1, . . . , xin+1−1, xin+1+1, . . . , xn+1)

)+ s2g

(f(xin+1)

))

= f(x1, . . . , xin+1−1, xin+1+1, . . . , xn+1, xin+1)

= f

in+1−1∑

j=1

xj ,

n+1∑

j=in+1

xj

· g−1

(s1 · g

(f(x1, . . . , xin+1−1)

)+ s2g

(f(xin+1+1, . . . , xin+1)

))

= f

in+1−1∑

j=1

xj ,

n+1∑

j=in+1

xj

· g−1

(s1 · g

(f(x1, . . . , xin+1−1)

)+ s2g

(f(xin+1 , , . . . , xn+1)

))

= f(x1, . . . , xin+1−1, xin+1 , . . . , xn+1) (148)

where the third step follows from (5) in Lemma 1.7) Proof of Corollaries 5 and 6: When f < 0 is negative,

it is easy to show that f(x) → −∞ if xi → 0. When f > 0,suppose that k users are inactive. From (104) and Corollaries1 and 3, we have

f(x) ≤ f(1n−k) = n− k. (152)

which gives k ≤ n−f(x). Further, since the number of activeusers n−k is upper bounded by f(x), the maximum resourceis lower bounded by

∑i xi/f(x).

8) Proof of Corollary 7: Let k(x) =∑n

i=1

(xi∑j xj

)1−β

bean auxiliary function, such that

f(x) = sign(1− β) · k 1β (x). (153)

It is easy to check that fairness measures f(x) in (9) and (10)are differentiable. We have

∂f(x)∂xi

=1β

k1β−1(x) · |1− β|

(∑

j xj)1−β

[x−β

i −∑

j x1−βj∑

j xj

]

Because k(x) > 0 is positive, ∂f(x)∂xi

has a single root at

xi = x =

( ∑j xj∑

j x1−βj

) 1β

. (154)

It is straightforward to show that for any β 6= 1, we have

∂f(x)∂xi

> 0, if xi > x and∂f(x)∂xi

< 0, if xi < x

Therefore, when xj ∀j 6= i are fixed, f(x) is maximized byxi = x.

9) Proof of Corollary 8: To derive an lower bound on f(x)under the box constraints xmin ≤ xi ≤ xmax ∀i, we first arguethat f(x) is minimized only if users are assigned resourcexmin or xmax. Using the box constraints and Corollary 8, we

have

x =

( ∑j xj∑

j x1−βj

) 1β

=

(∑

i

xi∑j xj

· x−βi

)− 1β

≥(∑

i

xi∑j xj

· x−βmin

)− 1β

= xmin. (155)

Similarly, we can show

x ≤ xmax. (156)

According to Corollary 8, f(x) is increasing on xi ∈ [xmin, x]and decreasing on xi ∈ [x, xmax]. Hence, f(x) is minimizedonly if all xi take the boundary values in the box constraints,i.e.,

xi = xmin or xi = xmax. (157)

Let Γ = xmax

xminand µ be fraction of users who receive xmax.

By relaxing the constraint µ ∈ {in ,∀i} to µ ∈ [0, 1], we derive

an lower bound on f(x) as follows

minxi∈[xmin,xmax],∀i

f(x)

= minµ∈{ i

n ,∀i}sign(1− β) · n

[µΓ1−β + (1− µ)

(µΓ + 1− µ)1−β

] 1β

≥ minµ∈[0,1]

sign(1− β) · n[

µΓ1−β + (1− µ)

(µΓ + 1− µ)1−β

] 1β

.(158)

To find the minimizer in the last optimization problem above,we first recognize that at the two boundary points µ = 0and µ = 1 (i.e., all users receive the same amount ofresource), f(x) = n achieves its maximum value. Therefore,the minimum value is achieved by some µ ∈ (0, 1). If µ∗

32

is the minimizer of (158), it is necessary that the first orderderivative of the right hand side of (158) is zero, i.e.,

∂[

µΓ1−β+(1−µ)

(µΓ+1−µ)1−β

]

∂µ= 0. (159)

Solving the above equation, we obtain

(Γ− 1)(1− β)[(

Γ1−β − 1)µ + 1

]

=(Γ1−β − 1

)[(Γ− 1)µ + 1] .

Because this equation is a linear in µ, its root µ∗ is the uniqueminimizer of (158):

µ∗ =Γ− Γ1−β − β(Γ− 1)β(Γ− 1)(Γ1−β − 1)

. (160)

The lower bound in Corollary 8 follows by plugging µ∗ into(158).

10) Proof of Corollary 9: We first prove the followinglemma, which will be used repeatedly in proofs of Corollaries9,10,11,12.

Lemma 3: If u,v ∈ Rn+ are two vectors satisfying u↑i

v↑i≤ u↑j

v↑j

for all i ≥ j, then u is majorized by v and f(u) ≤ f(v). u↑iand v↑i are the ith smallest elements of u and v, respectively.

Proof: Due to Schur-concavity of fairness measure f inTheorem 3, it is sufficient to prove that u is majorized by vunder the given conditions. For any integer 1 ≤ d ≤ n, wehave

∑di=1 u↑i∑ni=1 u↑i

=∑d

i=1 u↑i∑di=1 u↑i +

∑ni=d+1 u↑i

=∑d

i=1 u↑i∑di=1 u↑i +

∑ni=d+1 v↑i · u↑i

v↑i

≤∑d

i=1 u↑i∑d

i=1 u↑i +∑d

j=1 u↑j∑dj=1 v↑j

·∑ni=d+1 v↑i

=∑d

i=1 v↑i∑di=1 v↑i +

∑ni=d+1 v↑i

=∑d

i=1 v↑i∑ni=1 v↑i

, (161)

where the third step uses∑d

j=1 u↑j∑dj=1 v↑j

≤ u↑iv↑i

for all i ≥ d, since

u↑i /v↑i is non-decreasing over its index i. (161) implies that uis majorized by v according to subsection II-C.

To prove Corollary 9, sufficiency is immediate from Lemma3, by setting u = x and v = x− c(x), which are positive dueto the assumptions of tax policies in subsection B4. To verifythe condition in Lemma 3, we have

u↑iv↑i

=x↑i

x↑i − c(x↑i )=

[1− c(x↑i )

x↑i

]−1

, (162)

which is non-decreasing over i due to the monotonicity ofc(z)/z over z ∈ R. The first step in (162) uses the assumptionthat a tax policy does not change users’ resource ranking.Therefore, f(x) ≤ f(x− c(x)) according to Lemma 1.

To prove necessity, we consider x = [x1, x2] with x1 ≤ x2.When c is a fair tax policy, it implies that f(x1− c(x1), x2−c(x2)) ≥ f(x1, x2) and x1 − c(x1) ≤ x2 − c(x2) dueto the assumption that a tax policy does not change users’resource ranking. From Schur-concavity of fairness measuref in Theorem 3, we have

x1

x2≤ x1 − c(x1)

x2 − c(x2). (163)

A simple manipulation of (163) shows that c(x1)/x1 ≤c(x2) ≤ x2 for all x1 ≤ x2. This completes the proof ofCorollary 9.

11) Proof of Corollary 10: We use Lemma 3 with u =x− c(x) and v = x to prove sufficiency. We have

u↑iv↑i

=x↑i

x↑i − c(x↑i )=

[1− c(x↑i )

x↑i

]−1

, (164)

which is non-decreasing over i due to the monotonicity ofc(z)/z over z ∈ R. Therefore, from Lemma 3, we concludethat f(x) ≤ f(x− c(x)).

For necessity, we again consider x = [x1, x2] with x1 ≤ x2.When c is a fair tax policy and f(x1, x2) ≥ f(x1−c(x1), x2−c(x2)). From Schur-concavity of fairness measure f in Theo-rem 3, we have

x1 − c(x1)x2 − c(x2)

≤ x1

x2. (165)

A simple manipulation of (165) shows that c(x1)/x1 ≥c(x2) ≤ x2 for all x1 ≤ x2. This completes the proof ofCorollary 10.

12) Proof of Corollary 11: Corollary 11 is a direct conse-quence of Lemma 3.

13) Proof of Corollary 12: We prove the three sufficientconditions one-by-one. When c1 is fair and c2 is unfair, wehave f(x− c1(x)) ≥ f(x) ≥ f(x− c2(x)). When c1 and c2

are both fair, choosing u = x− c2(x) and u = x− c1(x), wehave

u↑iv↑i

=x↑i − c2(x

↑i )

x↑i − c1(x↑i )

= 1 +c1(x

↑i )/x↑i − c2(x

↑i )/x↑i

1− c1(x↑i )/x↑i

=

[1− c2(x

↑i )

c1(x↑i )

]·[x↑i /c1(x

↑i )− 1

]−1

. (166)

It is easy to see that 1 − c2(x↑i )

c1(x↑i )

in (166) is positiveand non-decreasing over index i, since c1(z) ≥ c2(z)and c1(z)/c2(z) is non-decreasing over z ∈ R. We also

have[x↑i /c1(x

↑i )− 1

]−1

in (166) is also positive and non-decreasing over index i, since tax policy c1 is fair. Therefore,the quantity in (166) is non-decreasing over index i. FromLemma 3, we conclude that f(x − c1(x)) ≥ f(x − c2(x)).When c1 and c2 are both unfair, by setting u = x− c1(x) andu = x− c2(x), the proof is almost the same and will not berepeated here.

33

14) Proof of Theorem 5: We first prove the monotonicityof fβ(x) for β ∈ (−∞, 0). Consider two different values 0 >

β1 ≥ β2. We define the a function φ(y) = yβ2β1 for y ∈ R+.

Since β2/β1 ≥ 1, the function φ(y) is convex in y. Therefore,we have

fβ2(x) =

n∑

i=1

(xi∑j xj

)1−β2

1β2

=

n∑

i=1

xi∑j xj

· φ

(xi∑j xj

)−β1

1β2

≤φ

∑

i=1n

xi∑j xj

(xi∑j xj

)−β1

1β2

=

φ

n∑

i=1

(xi∑j xj

)1−β1

1β2

=

n∑

i=1

(xi∑j xj

)1−β1

1β1

= fβ1(x), (167)

where the third step follows from Jensen’s inequality and β2 <0. This shows that fβ(x) is increasing on (−∞, 0).

The rest of proof is similar. For β ∈ (0, 1), we consider1 > β1 ≥ β2 > 0. Function φ(y) = y

β2β1 is concave. We have

fβ2(x) =

n∑

i=1

(xi∑j xj

)1−β2

1β2

=

n∑

i=1

xi∑j xj

· φ

(xi∑j xj

)−β1

1β2

≤φ

∑

i=1n

xi∑j xj

(xi∑j xj

)−β1

1β2

= fβ1(x). (168)

where the third step follows from Jensen’s inequality and β2 >0. Therefore, fβ(x) is increasing on (0, 1).

For β ∈ (1,∞), we consider β1 ≥ β2 > 1. Function φ(y) =y

β2β1 is concave. We have

fβ2(x) −

n∑

i=1

(xi∑j xj

)1−β2

1β2

= −

n∑

i=1

xi∑j xj

· φ

(xi∑j xj

)−β1

1β2

≥ −φ

∑

i=1n

xi∑j xj

(xi∑j xj

)−β1

1β2

= fβ1(x). (169)

where the third step follows from Jensen’s inequality and β2 >0. Therefore, fβ(x) is decreasing on (1,∞). This completesthe proof of Theorem 5.

15) Proof of Theorem 6: We first assume β > 1 (whichimplies fβ(·) < 0) and show that the condition λ ≤

∣∣∣ β1−β

∣∣∣ isnecessary and sufficient for preserving Pareto optimality. Thecase where β < 1 can be shown using an analogous proof.

To show that the condition λ ≤∣∣∣ β1−β

∣∣∣ is sufficient, weconsider an allocation x and a vector γ such that γi ≥ 0 for alli and

∑i γi =

∑i xi. Clearly, x′ = x + δγ Pareto dominates

x for δ > 0. We now consider the difference between thefunction (30) evaluated for these two allocations. First assumeβ > 1, which implies fβ(·) < 0, and

Φλ(x′)− Φλ(x)

= λ (` (fβ (x′))− ` (fβ (x))) + `

(∑

i

x′i

)− `

(∑

i

xi

)

= − λ (log |fβ (x′)| − log |fβ (x)|)

+ log

((1 + δ)

∑

i

xi

)− log

(∑

i

xi

)

= − λ (log |fβ (x′)| − log |fβ (x)|) + log (1 + δ) .(170)

If x′ is also more fair than x, then showing

−λ (log |fβ (x′)| − log |fβ (x′)|) > 0 (171)

is trivial, and the difference between the objective evaluated atthe two allocations is strictly positive. Therefore, we considerthe case where x′ is less fair.

Continuing from (170) and applying the definition in (15)yields

Φλ(x′)− Φλ(x)

= − λ log

n∑

i=1

(x′i∑j x′j

)1−β

1β

+ log (1 + δ)

+ λ log

n∑

i=1

(xi∑j xj

)1−β

1β

= − λ

βlog

(∑ni=1 (x′i)

1−β

∑ni=1 (xi)

1−β

)+

(1− λ

β − 1β

)log (1 + δ) .

(172)

Because x′i ≥ xi for all i, we know that for β > 1,(x′i)

1−β ≤ (xi)1−β , which implies

−λ

βlog

(∑ni=1 (x′i)

1−β

∑ni=1 (xi)

1−β

)> 0. (173)

Consequently, for the entire difference to be positive, it issufficient that

1− λβ − 1

β≥ 0, (174)

or, equivalently,

λ ≤ β

β − 1. (175)

34

Next, we prove that the condition λ ≤∣∣∣ β1−β

∣∣∣ is necessary.

Suppose β > 1 as before and λ >∣∣∣ β1−β

∣∣∣ for constructing acontradiction. We show that there exists two vectors x and x′,such that Φλ(x′)−Φλ(x) < 0, while x′ Pareto dominates x.

Consider a resource allocation vector x of length n + 1,such that xi = 1 for i = 1, . . . , n and xn+1 = n. Clearly, xis Pareto dominated by another vector x′, where x′i = xi fori = 1, . . . , n and x′n+1 = xi + δ (

∑i xi), for some positive

δ > 0. Let ξ = log(1 + δ) > 0 be an axillary variable. Fromthe last step of (172), we have

Φλ(x′)− Φλ(x)

= − λ

βlog

(∑n+1i=1 (x′i)

1−β

∑n+1i=1 (xi)

1−β

)+

(1− λ

β − 1β

)ξ

= − λ

βlog

(n + (n + 2nδ)1−β

n + n1−β

)+

(1− λ

β − 1β

)ξ

≤ − λ

βlog

(n

n + n1−β

)+

(1− λ

β − 1β

)ξ

= − λ

βlog

(1 + n−β

)+

(1− λ

β − 1β

)ξ

It is straight forward to verify that Φλ(x′)−Φλ(x) < 0, if weset

δ =12

[(1 + n−β

) λλ(β−1)−β

]> 0. (176)

As a result, the condition (32) of the theorem is sufficientand necessary for ensuring Pareto optimality of the solution.

16) Proof of Theorem 7: From the definition of α-fairutility, we compute the numerator and denominator:

⟨5xUα=β(x),

η

‖η‖⟩

=∑

i

x−βi

1N − xi∑

i xi√‖x‖2

(∑

i xi)2− 1

N

=1√

‖x‖2N(∑

i xi)2− 1

· 1√N

·∑

i

x−βi

(1− xi∑

j xjN

),

(177)

and ⟨5xUα=β(x),

1‖1‖

⟩=

∑

i

x−βi

1√N

. (178)

Notice that both values are positive. The ratio between thesethen is⟨5xUα=β(x), η

‖η‖⟩

⟨5xUα=β(x),

1‖1‖

⟩ =1√

‖x‖2N(∑

i xi)2− 1

1−

∑i

xi∑j xj

x−βi

∑i

1N x−β

i

.

(179)

It is easily shown that the factor in front of the bracket isstrictly positive. The only component that varies with β is theratio between two weighted averages of the same vector withdifferent weights: ∑

ixi∑j xj

x−βi

∑i

1N x−β

i

. (180)

The sum in the numerator places more weight ( xi∑j xj

> 1N ) on

elements that decrease more rapidly (or increase more slowlyfor the case xi < 1) with β, implies that the overall numeratordecreases more rapidly (or increases more slowly) than thedenominator as β increases. Therefore, (180) is monotonicallynon-increasing, and Theorem 7 is true.

17) Proof of Theorem 8: To prove the equivalent fairnessrepresentation in (41), we start with the integral in (39):

1T

∫ T

t=0

(Xt/µ)1−βrdt

=n∑

i=1

1T

∫

{yi≤Xt<yi+1}(Xt/µ)1−βr

dt

=n∑

i=1

(yi/µ)1−βr 1T

∫

{yi≤Xt<yi+1}dt

=n∑

i=1

(yi/µ)1−βr [Px(yi+1)− Px(yi)] (181)

where 0 = y1 < y2 < . . . < yn+1 < ∞ is a partition of thedomain of Px, and yi ≤ yi ≤ yi+1 is a proper value in step2. The last step uses the definition of Px in (43). Let n →∞.Due to continuity, we derive

1T

∫ T

t=0

(Xt/µ)1−βrdt =

∫ ∞

y=0

(y/µ)1−βrdPx(y) (182)

which proves (41).Similarly, to prove the equivalent fairness expression in (42),

we start with (182):∫ ∞

y=0

(y/µ)1−βrdPx(y)

=n∑

i=1

∫

{ui≤Px(y)<ui+1}(y/µ)1−βr

dPx(y)

=n∑

i=1

(P−1

x (u)/µ)−βr

∫

{yi≤Xt<yi+1}

y

µdPx(y)

=n∑

i=1

(P−1

x (u)/µ)−βr

[Lx(ui+1)− Lx(ui)] (183)

where where 0 = y1 < y2 < . . . < yn+1 = 1 is a partition ofthe domain of Lx, and ui ≤ ui ≤ ui+1 is a proper value in step2. P−1

x exists, since cumulative distribution Px is monotone.The last step uses the definition of Lx in (44). Let n → ∞.Due to continuity, we derive∫ ∞

y=0

(y

µ

)1−βr

dPx(y) =∫ 1

u=0

(P−1

x (u)µ

)−βr

dLx(u)(184)

which, combined with (182), proves (42).18) Proof of Theorem 9: From (42), we find that

F(Xt∈[0,T ]) ≤ F(Yt∈[0,T ]) holds if

∫ 1

u=0

(P−1

x (u)µ

)−βr

dLx(u) ≤∫ 1

v=0

(P−1

y (v)µ

)−βr

dLy(v)

Since above inequality is an integral of a non-negative functionover Lorenz curve dLx and dLx, the axiomatic theory ofLorenz curve ordering in [18] implies that this inequality

35

defines a complete order over all Lorenz curves. From theTheorem 9 in [18], we have

F(Xt∈[0,T ]) ≤ F(Yt∈[0,T ]), if Lx(u) ≤ Ly(u) ∀u ∈ [0, 1].

This equation, together with (45), completes the proof.19) Proof of Corollaries 13 and 14: From the construction

of F in Section III,

F(Xt∈[0,T ]) = limn→∞

n−rf(Xt1 , . . . ,Xtn), (185)

where 0 = t0 < t1 < t2 < . . . < tn = T is a partition of[0, T ]. Due to Corollary 2 and Corollary 3 in Section II, fis maximized by equal resource allocation and has maximumvalue |f | = nr. We have

n−rf(Xt1 , . . . ,Xtn) ≤ 1, (186)

for all dimension n. Taking n → ∞ in (186), we find thatF(Xt∈[0,T ]) ≤ 1. When F is positive, it is easy to verifyusing (39) that optimal value F(Xt∈[0,T ]) = 1 can be achievedby equal resource allocation over time, i.e., {Xt = c : t ∈[0, T ], c > 0}. The proof for negative F is similar.

20) Proof of Corollary 15: We define Yt = Xt − c forc > 0 and all t ∈ [0, T ] as the shifted resource allocation.From the definition of cumulative distribution in (43), we find

Py(y) =1T·∫

{Yt≤y}dt =

1T·∫

{Xt≤y+c}dt = Px(y + c)(187)

Using the definition of Lorenz curve in (44), we derive

Ly(u) =1

µ− c·∫

{Py(y)≤u}ydPy(y) (188)

=1

µ− c·∫

{Px(y+c)≤u}ydPx(y + c)

=1

µ− c·∫

{Px(z)≤u,z>c}(z − c)dPx(z)

=1µ·∫

{Px(z)≤u,z>c}

µ(z − c)µ− c

dPx(z)

=1µ·∫

{Px(z)≤u}

µ(z − c)µ− c

dPx(z)

=1µ·∫

{Px(z)≤u}

(z +

c

µ− c(z − µ)

)dPx(z)

The second step uses (187). The third step changes variablefrom y to z − c. The fifth step follows from the fact thatPx(z) = 0 for z ≤ c, since c ≤ Xt for all t ∈ [0, T ] (otherwisewe have Yt < 0, which is not possible). Since z = P−1

x (u)is non-decreasing over u, we have

µ =∫

{Px(z)≤1}zdPx(z) ≥ 1

u

∫

{Px(z)≤u}zdPx(z) (189)

which follows from the fact that average of a non-decreasingfunction P−1

x over [0, u] is non-decreasing. Using this inequal-ity, for the last term in (189), we have∫

{Px(z)≤u}(z − µ)dPx(z) =

∫

{Px(z)≤u}zdPx(z)− µu ≤ 0.

Combine above equation and (189). We find

Ly(u) ≤∫

{Px(z)≤u}zdPx(z) = Lx(u) ∀u. (190)

From Schur-concavity property in Theorem 13, we concludethat F(Xt∈[0,T ] − c) ≤ F(Xt∈[0,T ]).

21) Proof of Corollary 16: Let ω be the average ofXt∈[0,T ]

⋃0t∈[T,T+τ ]. It is easy to see that

ω =1

T + τ

∫ T

t=0

Xtdt =T

T + τ· µ. (191)

Let s = sign(r(1 − rβ)) be the sign of fairness measure F .When 1 − βr > 0, from the definition of fairness F in (39),we derive

F(Xt∈[0,T ]

⋃0t∈[T,T+τ ])

= s

[1

T + τ

∫ T

t=0

(Xt/ω)1−βrdt + 0

] 1β

= s

[T rβ

(T + τ)rβ· 1T

∫ T

t=0

(Xt/µ)1−βrdt + 0

] 1β

= (1 + τ/T )−r · s[

1T

∫ T

t=0

(Xt/µ)1−βrdt + 0

] 1β

= (1 + τ/T )−r · F(Xt∈[0,T ]). (192)

Adding a time interval of length τ > 0 with zero resourcereduces fairness by a factor of (1 + τ/T )−r.

22) Proof of Corollary 17: Suppose that Xt > 0 for allt ∈ [0, T ]. We consider a resource allocation

Yt∈[0,T+τ ] , Xt∈[0,T ]

⋃0t∈[T,T+τ ], (193)

whose length of inactive time intervals is exactly τ . First, weconsider r > 0, so that fairness F is positive. From Corollary16 and 17, we have

F(Yt∈[0,T+τ ]) = (1 + τ/T )−r · F(Xt∈[0,T ]) ≤ (1 + τ/T )−r

Rearranging the terms, we conclude that

τ ≤ T · [(F(Yt∈[0,T+τ ]))−1/r − 1], (194)

When r < 0 and fairness F is negative, we have

F(Yt∈[0,T+τ ]) = (1 + τ/T )−r · F(Xt∈[0,T ])≤ −(1 + τ/T )−r (195)

Rearranging the terms and considering that −r > 0, weconclude that

τ ≤ T · [(−F(Yt∈[0,T+τ ]))−1/r − 1], (196)

These inequalities give an upper bound of inactive timeintervals (with zero resource) for 1− rβ > 0 and all r ∈ R.

23) Proof of Corollary 18: Suppose resource allocationXt∈[0,T ] has fairness F . Since µ is the average resource, wefind

µ =1T

∫

{Xt>0}Xtdt

≤ ( maxt∈[0,T ]

Xt) · 1T

∫

{Xt>0}dt

≤ ( maxt∈[0,T ]

Xt) · (2− |F|−1/r) (197)

36

where the last step uses Corollary 17 - the length of in-active time intervals is upper bounded by [|F|−1/r − 1]. If2−|F|−1/r > 0, we conclude that the peak resource allocationover time [0, T ] is lower bounded by

maxt∈[0,T ]

Xt ≥ µ

2− |F|−1/r(198)

24) Proof of Corollary 19: We prove a lower bound onfairness F for box constraints. Define

h(Xt∈[0,T ]) =∫ T

t=0

(Xt/µ)1−βrdt (199)

as an auxiliary function, such that

F(Xt∈[0,T ]) = sign(r(1− βr)) ·[

1T

h(Xt∈[0,T ])] 1

β

. (200)

In the following proof, we will only consider r(1 − rβ) > 0and β > 0, while the proofs for other cases are similar andomitted for simplicity. When r(1 − rβ) > 0 and β > 0,to minimize fairness F over resource allocation Xt∈[0,T ], weformulate the following minimization problem of h(Xt∈[0,T ])for average resource constraint µ and the box constraintxmin ≤ Xt ≤ xmax:

minXt∈[0,T ]

h(Xt∈[0,T ]) (201)

s.t. xmin ≤ Xt ≤ xmax, ∀tµ = (1/T ) ·

∫ T

t=0

Xtdt

We introduce a Lagrangian multiplier λ for the averageresource constraint, and two Lagrangian multiplier functions,φt and ψt, for the two boundaries of the box constraint. Weformulate an Lagrangian for the minimization problem:

L =∫ T

t=0

(Xt/µ)1−βr + λ

(µ− 1

T

∫ T

0

Xtdt

)

+∫ T

0

(Xt − xmax)φtdt +∫ T

0

(xmin −Xt)ψtdt.

Since KKT conditions are necessary for optimality, we derivethe first order condition, i.e., the derivative of L with respectto Xt is zero:

∂L

∂Xt=

X−rβt

µ1−rβ(1− rβ)− λ

T+ φt − ψt = 0, (202)

and complementary slackness conditions:

(Xt − xmax)φt = 0, ∀t, (203)(xmin −Xt)ψt = 0, ∀t. (204)

Let X∗t be an optimizer. We first prove that X∗

t is eitherxmin or xmax for all t ∈ [0, T ]. Suppose xmin < X∗

t < xmax

holds for some t ∈ T , from (203) and (204), we conclude thatφt = 0 and ψt = 0. Substituting this result into (202), we findthat when t ∈ T , X∗

t is a constant, which is denoted by x0

and satisfies

x−rβ0

µ1−rβ(1− rβ)− λ

T= 0. (205)

Because X−rβt is strictly increasing over Xt, we find that

when X∗t = xmax

ψt − φt =x−rβ

max

µ1−rβ(1− rβ)− λ

T

>x−rβ

0

µ1−rβ(1− rβ)− λ

T= 0 (206)

where the first step uses the first order condition for X∗t =

xmax. Since φt, ψt ≥ 0 are non-negative, inequality (206)implies that ψt > φt ≥ 0. Inequality ψt > 0 and X∗

t =xmax > xmin contradict with complementary slackness con-dition (204). Therefore, we conclude that xmin < X∗

t < xmax

is impossible. X∗t must take boundary values of the box

constraint, i.e., it is either xmin or xmax for all t ∈ [0, T ].Define Γ = xmax/xmin and θ to be the fraction of time the

X∗t = xmax. We rewrite the auxiliary function h by

h(X∗t ) =

∫ T

t=0

(X∗t /µ)1−βr

dt

=θTx1−rβ

max + (1− θ)Tx1−rβmin

(θxmax + (1− θ)xmin)1−rβ(207)

We recognize that at the two boundary points θ = 0 and θ = 1(i.e. all users receive the same amount of resource), F = 1achieves its maximum value. Therefore, the minimum fairnessis achieved by some θ ∈ (0, 1). We set ∂h/∂θ = 0,

∂h∂θ h(X∗

t ) =∂

∂θ

[θTx1−rβ

max + (1− θ)Tx1−rβmin

(θxmax + (1− θ)xmin)1−rβ

]

= 0. (208)

Solving the last equality in (208), we obtain

(x1−rβmax − x1−rβ

min )z1−rβ

= (θx1−rβmax + (1− θ)x1−rβ

min )(1− rβ)zrβ(xmax − xmin).

where z = θxmax+(1−θ)xmin. Simplify the equation above,we find a unique optimizer θ as follows:

θ =1

1− ρ·[(Γ− 1)ρ

Γρ − 1− 1

](209)

where Γ = xmax/xmin and ρ = 1 − rβ. This complete theproof.

25) Proof of Corollary 20: We first prove the monotonicityof Fβ(Xt∈[0,T ]) for β ∈ (−∞, 0). Consider two different

values 0 > β1 ≥ β2. We define the a function φ(y) = yβ2β1 for

y ∈ R+. Since β2/β1 ≥ 1, the function φ(y) is convex in y.Since T = 1 and Fβ(Xt∈[0,T ]) is positive for β ∈ (−∞, 0),

37

we have

Fβ2(Xt∈[0,T ]) =

[∫ T

t=0

(Xt/µ)1−β2 dt

] 1β2

=

[∫ T

t=0

(Xt/µ) · φ((Xt/µ)−β1

)dt

] 1β2

≤[φ

(∫ T

t=0

(Xt/µ) · (Xt/µ)−β1 dt

)] 1β2

=

[φ

(∫ T

t=0

(Xt/µ)1−β1 dt

)] 1β2

=

[∫ T

t=0

(Xt/µ)1−β1 dt

] 1β1

= Fβ1(Xt∈[0,T ]), (210)

where the third step follows from Jensen’s inequality and β2 <0. This proves that Fβ(Xt∈[0,T ]) is increasing on (−∞, 0).

The rest of proof is similar. For β ∈ (0, 1), we consider1 > β1 ≥ β2 > 0, such that function φ(y) = y

β2β1 is concave.

For β ∈ (1,∞), we consider β1 ≥ β2 > 1, such that functionφ(y) = y

β2β1 is concave. Apply the chains on inequality (210)

to β ∈ (0, 1) and β ∈ (1,∞) respectively, we can prove themonotonicity of Fβ(Xt∈[0,T ]) for β ∈ R.

26) Proof of Theorems 10, 11, and 12: This proof is anextension of the proof in C.1. We prove that fairness measuressatisfying Axioms 1′-5′ can only be generated by logarithmand power generator functions. Then, for each choice oflogarithm and power generator function, a fairness measureexist and is uniquely given by either (65) or (66). It is easy toverify that if f > 0 satisfies Axioms 1′-5′, so does −f . Wefirst assume that f > 0 during this proof and then determinesign to fairness measures from Axioms 5′. Since the proof ofLemma 2 does not depend on the definition of si, it holds forall q ∈ Zn

+. Therefore, Lemma 2 becomes

f(1n,q) = nr · f(1, 1), ∀q ∈ Z+, n ≥ 1, (211)

(Uniqueness of logarithm and power generator functions.)We first show that logarithm and power generator functionsare necessary. In other words, no other fairness can possiblysatisfy Axioms 1′-4′ with weight si in (62). Consider arbitrarypositive X1 +X2 = 1, which can be written as the ρ-th powerof rational numbers, i.e., X1 = (a1/b1)ρ and X2 = (a2/b2)ρ,for some positive integers a1, b1, a2, b2. Similarly, we choosearbitrary positive Q1 + Q2 = 1, which can be written as theρ-th power of rational numbers, i.e., Q1 = (c1/d1)ρ and X2 =(c2/d2)ρ, for some positive integers c1, d1, c2, d2. Let u, v betwo positive integers and K = (u/v)1/ρ be rational number.We choose the sum resource of the two sub-systems by W1 =

a1b2c2d1v and W2 = a2b1c1d2v, so that

Q1Wρ1

Q1Wρ1 +Q2Wρ

2=

(c1d2a1b2c2d1v)ρ

(c1d2a1b2c2d1v)ρ + (c2d1a2b1c1d2v)ρ

=(a1/b1)ρ

(a1/b1)ρ + (a2/b2)ρ

=X1

X1 + X2

= X1 (212)

where the last step uses X1 + X2 = 1. Similarly, we have

Q2Wρ2

Q1Wρ1 + Q2Wρ

2

= X2 (213)

Consider a partition and two resource allocation vectors asfollows: for x, sub-system 1 has t = b1b2c1c2u users, eachwith weight Q1/t and resource W1/t; sub-system 2 also hast = b1b2c1c2u users, each with weight Q2/t and resourceW2/t. In other words, we have

x1 = [W1

t, . . . ,

W1

t︸ ︷︷ ︸t times

] and x2 = [W2

t, . . . ,

W2

t︸ ︷︷ ︸t times

], (214)

q1 = [Q1

t, . . . ,

Q1

t︸ ︷︷ ︸t times

] and q2 = [Q2

t, . . . ,

Q2

t︸ ︷︷ ︸t times

]. (215)

For y, sub-system 1 has u1 = a1b2c2d1v users, each withweight Q1/u1 and resource 1; sub-system 2 also has u2 =a2b1c1d2v users, each with weight Q2/u2 and resource 1.This means

y1 = [1, . . . , 1]︸ ︷︷ ︸u1 times

and y2 = [1, . . . , 1]︸ ︷︷ ︸u2 times

, (216)

p1 = [Q1

u1, . . . ,

Q1

u1︸ ︷︷ ︸u1 times

] and p2 = [Q2

u2, . . . ,

Q2

u2︸ ︷︷ ︸u2 times

]. (217)

Since w(xi) = w(yi) and w(qi) = w(pi), (218) holds dueto Axiom 4′ and (211). The second step of (218) uses Axiom2 and (211), i.e.,

f(xi,qi) = f

(Wi

t· 1t,

Qi

t· 1t

)

= f

(1t,

Qi

t· 1t

)

= tr (219)

and similarly,

f(y1) = (a1b2c2d1v)r, (220)f(y2) = (a2b1c1d2v)r. (221)

Next, we apply (63) to f(x,q) and obtain

f(x,q)

= f

(W1 +W2

t· 1t,

Q1 + Q2

t· 1t

)

·g−1

(g

(f

([W1

t,W2

t

],

[Q1

t,Q2

t

])))

= tr · f([W1,W2], [Q1/t,Q2/t])

= tr · f([(X1/Q1)1ρ , (X2/Q2)

1ρ ], [Q1/t, Q2/t])(222)

38

f(y,p)f(x,q) = g−1

(2∑

i=1

si · g(

f(yi,pi)f(xi,qi)

))

= g−1

(Q1Wρ

1

Q1Wρ1 + Q2Wρ

2

· g(

ar1b

r2c

r2d

21v

r

tr

)+

Q2Wρ2

Q1Wρ1 + Q2Wρ

2

· g(

ar2b

r1c

r1d

r2v

r

tr

))

= g−1

(X1 · g

(ar1d

r1v

r

br1c

r1u

r

)+ X2 · g

(ar2d

r2v

r

br2c

r2u

r

))

= g−1

(X1 · g

((X1

Q1K

) rρ

)+ X2 · g

((X2

Q2K

) rρ

))(218)

where the entire system is partitioned into t sub-systems5, eachwith a resource allocation vector [W1/t,W1/t]. The last stepuses (212) and (213). Again, using Axiom 2′ and (211), wederive

f([y1,y2],p) = f (1a1b2c2d1v+a2b1c1d2v)= (a1b2c2d1v + a2b1c1d2v)r (223)

Combining (218), (222), and (223), we derive((X1/Q1)

1ρ + (X2/Q2)

1ρ

)r

f([(X1/Q1)1ρ , (X2/Q2)

1ρ ], [Q1/t,Q2/t])

K− rρ

=(a1b2c2d1v + a2b1c1d2v)r

f([(X1/Q1)1ρ , (X2/Q2)

1ρ ], [Q1/t,Q2/t]) · tr

= g−1

(2∑

i=1

Xi · g((

Xi

QiK

) rρ

))(224)

For different K > 0, (224) further implies that

g−1

(2∑

i=1

Xi · g((

Xi

QiK

) rρ

))

= Krρ · g−1

(2∑

i=1

Xi · g((

Xi

QiK

) rρ

))(225)

for rational X1, X2, Q1, Q2,K > 0 and arbitrary ρ, r. Dueto Axiom 1′, it is easy to show that (225) also holds for realX1, X2, Q1, Q2,K > 0. It follows from [35] and the argumentin the proof C.1 that there exist only two possible generatorfunctions:

g(y) = z · log(y) (226)

where z 6= 0 is an arbitrary constant, and

g(y) =yβ − 1

c(227)

where β ∈ R is a constant exponent.(Existence of fairness measures for logarithm generator.)

We derive fairness measure for the logarithm generator func-tion in (226) and show that it satisfies Axioms 1′-4′. By choos-ing K = u/v = 1 and a change of variable xi = X

1ρ

i = ai/bi

5Although Axiom 4′ is defined for partitioning into 2 subsystems, it can beextended to partitioning into t identical sub-systems, using the same inductionin proof of Lemma 2.

in (224), we plug in the logarithm generator (226) and find

f([x1, x2], [q1, q2]) =(x1 + x2)r

g−1(∑2

i=1qix

ρi

q1xρ1+q2xρ

2· g(xr

i ))

=2∏

i=1

(xi

x1 + x2

)−rqix

ρi

q1xρ1+q2x

ρ2 (228)

To generate fairness measure for n > 2 users, we useinduction assume that for n = k users, we have

f(x,q) =k∏

i=1

(xi∑k

j=1 xj

)−rqix

ρi∑

j qjxρj

(230)

which holds for n = k = 2. Let uk =∑k

i=1 xi andQk =

∑ki=1 qi For n = k + 1, we apply (63) and show

that (229) holds. The first step of (229) uses Axiom 2′ withf(xk+1, qk+1) = f(1, 1) = 1. Comparing the last step of(229) to (230) with n = k + 1, induction of fairness measurefrom n = k to n = k + 1 holds if and only if

Qkuρk = (

k∑

i=1

qi)(k∑

i=1

xi)ρ =k∑

i=1

qixρi . (231)

Therefore, we must have ρ = 0. Fairness measure generatedby logarithm generator in (226) can only have the followingform

f(x,q) =k∏

i=1

(xi∑k

j=1 xj

)−rqi

(232)

which is (65) in Theorem 10.The proof so far assumes positive fairness measure f > 0.

To determine the sign of f , we add sign(−r) in (232), applyAxiom 5, and find that for r > 0

f([0, 1], [1, 1]) = 0 < 2r = f

([12,12

], [1, 1]

), (233)

and for r < 0:

f([0, 1], [1, 1]) = −∞ < −2r = f

([12,12

], [1, 1]

). (234)

Therefore, Axiom 5′ is satisfied. We derive the fairness mea-sure in (65).

It remains to prove that fairness measure (232) satisfiesAxioms 1′-4′ with arbitrary r ∈ R. It is easy to see thatAxioms 1′-3′ are satisfied. To verify Axiom 4′, we use the

39

f([x1, . . . , xk, xk+1], [q1, . . . , qk, qk+1])

= f

([k∑

i=1

xi, xk+1

], [

k∑

i=1

qi, qk+1]

)· [f([x1, . . . , xk], [q1, . . . , qk])]

Qkuρk

Qkuρk+qk+1x

ρk+1 · 1

=

( ∑ki=1 xi∑k+1i=1 xi

) −rQkuρk

Qkuρk+qk+1x

ρk+1

·(

xk+1∑k+1i=1 xi

) −rqk+1xρk+1

Qkuρk+qk+1x

ρk+1

·k∏

i=1

(xi∑k

j=1 xj

)−rqix

ρi∑

j qjxρj· Qku

ρk

Qkuρk+qk+1x

ρk+1

(229)

same argument (137) in the proof of Theorems 1, 2, and 3for logarithm generator function. Since (137) is independentof the choice of si, we have

f(x,q)f(y,p)

= g−1

(2∑

i=1

si · g(

f(xi,qi)f(yi,pi)

))(235)

which establishes Axiom 4′ for any partition. This completesthe proof for logarithm generators.

(Existence of fairness measures for power generator.) Wederive fairness measures for power generator function in (227)and show that it satisfies Axioms 1′-5′. By choosing K =u/v = 1 and a change of variable xi = X

1ρ

i = ai/bi in (224),we plug in the power generator (227) and find

f([x1, x2], [q1, q2]) =(x1 + x2)r

g−1(∑2

i=1qix

ρi

q1xρ1+q2xρ

2· g(xr

i ))

=(x1 + x2)r(q1x

ρ1 + q2x

ρ2)

1β

(q1xρ+rβ1 + q2x

ρ+rβ2 )

1β

(236)

To derive the fairness measure for three users, we con-sider two different partitions of the resource allocation vector[x1, x2, x3] as [x1, x2], [x3] and [x1], [x2, x3]. Let q1,2 = q1 +q2, q2,3 = q2+q3, and u = x1+x2+x3 be auxiliary variables.Using the construction in (63), we obtain two equivalent formof the fairness measure for n = 3 users in (237) and (238).

As in Axiom 4′, the fairness measure is independent topartition. Similar to the proof steps in C.1, we will considertwo different partitions of three uses to derive a sufficient andnecessary condition on parameters ρ, r, and β. We find that(237) and (238) should be equivalent for all x1, x2, x3 ≥ 0.Comparing the terms in (237) and (238), we find that (237)and (238) are equivalent if and only if

ρ = −r · β. (239)

Similar to the proof in C.1, (239) is a sufficient and necessaryfor fairness measures to be independent of partition for n = 3.Use (239) to remove ρ in (237) and (238) and simply the terms.We derive the expression of fairness measure for three usersas follows

f([x1, x2, x3], [q1, q2, q3]) =

3∑

i=1

qi

(xi∑j xj

)−rβ

1β

(240)

where β 6= 0 and r can be arbitrary.To generate fairness measure for n > 3 users, we assume

that (231) holds for n = k users. We define axillary variablesuk =

∑ki=1 xi, Qk =

∑ki=1 qi, and ξk =

∑ki=1 qix

−rβi . Let

x = [x1, . . . , xk] and q = [q1, . . . , qk]. For n = k + 1, weapply (3) in Lemma 1 and use (240) to derive

f([x1, . . . , xk, xk+1], [q1, . . . , qk, qk+1])

= f ([uk, xk+1], [Qk, qk+1]) ·(

Qkuρkf(x,q)β + qk+1x

ρk+1

Qkuρk + qk+1x

ρk+1

) 1β

=(Qkuρ

k + qk+1xρk+1)

1β

u−rk+1

·(

uρ−rβk ξk + qk+1x

ρk+1

Qkuρk + qk+1x

ρk+1

) 1β

=(∑k+1

i=1 qix−rβi )

1β

u−rk+1

=

k+1∑

i=1

qi

(xi∑j xj

)−rβ

1β

(241)

where the third step uses ρ − rβ = 0 and the last step usesuk =

∑ki=1 xi. Therefore, (241) holds for n = k + 1 due to

induction.Notice that (241) is almost (66), except for a term

sign(−r(1+rβ)). The term is added in order to satisfy Axiom5′. To show this, we first consider the case of rβ > 0, whichimplies (1 + rβ) > 0. For r > 0 and β > 0, we find

f([0, 1], [1, 1]) = −∞ < −2r = f([12,12

], [1, 1]), (242)

and for r < 0 and β < 0,

f([0, 1], [1, 1]) = 0 < 2r = f([12,12

], [1, 1]). (243)

Next, when rβ < 0, we have

f([0, 1], [1, 1]) = −sign(1 + rβ) · q1β

2

= −sign(1 + rβ) · 2− 1β , (244)

where the last step uses the statement of equal weights q1 =q2 = 1/2 in Axiom 5′. We also obtain

f([12,12

], [1, 1]) = −sign(1 + rβ) · 2r (245)

Comparing (244) and (245) for r > −1/β and r < −1/β,we conclude that f(

[12 , 1

2

], [1, 1]) ≥ f([0, 1], [1, 1]) holds for

rβ < 0. Therefore, we derive fairness measure (66). It isstraightforward to check that fairness measure in (142) satisfiesAxioms 1-4. This completes the proof for power generators.

40

f([x1, x2, x3], [q1, q2, q3]) (237)

= f([x1 + x2, x3], [q1 + q2, q3]) · g−1

((q1,2)(x1 + x2)ρ · g (f([x1, x2], [q1, q2])) + q3x

ρ3 · g (1)

(q1,2)(x1 + x2)ρ + q3xρ3

)

=ur((q1,2)(x1 + x2)ρ + q3x

ρ3)

1β

((q1,2)(x1 + x2)ρ+rβ + q3xρ+rβ3 )

1β

·

(q1,2)(x1 + x2)ρ · (x1+x2)rβ(q1xρ

1+q2xρ2)

(q1xρ+rβ1 +q2xρ+rβ

2 )+ q3x

ρ3·

(q1,2)(x1 + x2)ρ + q3xρ3

f([x1, x2, x3], [q1, q2, q3]) (238)

= f([x1 + x2, x3], [q1 + q2, q3]) · g−1

((q2,3)(x2 + x3)ρ · g (f([x2, x3], [q2, q3])) + q1x

ρ1 · g (1)

(q2,3)(x2 + x3)ρ + q1xρ1

)

=ur((q2,3)(x2 + x3)ρ + q1x

ρ1)

1β

((q2,3)(x2 + x3)ρ+rβ + q1xρ+rβ1 )

1β

·

(q2,3)(x2 + x3)ρ · (x2+x3)rβ(q2xρ

2+q3xρ3)

(q2xρ+rβ2 +q3xρ+rβ

3 )+ q1x

ρ1·

(q2,3)(x2 + x3)ρ + q1xρ1

27) Proof of Corollary 21: For n = 2 users, the weightedsymmetry is immediate from (222), which implies

f([x1, x2], [q1, q2]) = f([x2, x1], [q2, q1]). (246)

We use induction to prove weighted symmetry for n > 2 users.The remaining proof is same as that of Corollary 1.

28) Proof of Corollaries 22, 23, and 24: We prove Corol-laries 22, 23, and 24 for (66) when f > 0 is positive. The

proof for f < 0 is the same. Let κ =∑n

i=1 qi

(xi∑j xj

)−rβ

be an auxiliary function. We have

∂f(x,q)∂xi

=−rβ · κ 1

β−1

(∑

j xj)−r

[qix

−rβ−1i −

∑j qjx

−rβj∑

j xj

](247)

Because κ > 0 is positive, ∂f(x,q)∂xi

has a single root at

xi = q1

rβ+1i · x = q

1rβ+1i ·

( ∑j xj∑

j qjx−rβj

) 1β

. (248)

If a small amount of resource ∆x is moved from user i touser j, we have

∆f =−rβκ

1β−1

(∑

m xm)−r

[qix

−rβ−1i − qjx

−rβ−1j

]·∆x + o(∆x)

which is positive if and only if xi/q1

rβ+1i < xj/qj

1rβ+1 . This

proves Corollary 22. From (247) and (248), it is straightfor-ward to show that

∂f(x,q)∂xi

> 0, if xi > q1

rβ+1i · x, (249)

∂f(x,q)∂xi

< 0, if xi < q1

rβ+1i · x, (250)

which proves Corollary 24. Further, fairness measure is max-imized by resource allocation [q

1rβ

1 , q1

rβ+12 , . . . , q

1rβ+1n ]. Corol-

lary 23 is proven.

29) Proof of Corollary 25: For β ∈ (−∞,−1), fβ(x) isequivalent to inverse of p-norm with p = −β. Monotonicity offβ(x,q) is straightforward from monotonicity of p-norm. Forβ ∈ (−1, 0), we consider two axillary functions h1(y) = y−β

and h2(y) = y1/β , which are both monotonically increasingover β for y ∈ (0, 1). Therefore, fβ(x,q) = −h2(

∑i qi ·

h1(xi)) is monotonically decreasing.To prove monotonicity of fβ(x,q) for β ∈ (0,∞). Consider

two different values 0 < β1 ≤ β2. We define the a functionφ(y) = y

β2β1 for y ∈ R+. Since β2/β1 ≥ 1, the function φ(y)

is convex in y. Therefore, we have

fβ2(x) = −

n∑

i=1

(qi · xi∑

j xj

)−β2

1β2

= −

n∑

i=1

qi · φ

(xi∑j xj

)−β1

1β2

=

φ

n∑

i=1

qi ·(

xi∑j xj

)−β1

1β2

=

n∑

i=1

(xi∑j xj

)−β1

1β1

= fβ1(x,q), (251)

where the third step follows from Jensen’s inequality and β2 >0. This shows that fβ(x,q) is monotonically decreasing onβ ∈ (0,∞). This completes the proof.

30) Proof of Theorem 13: We need to prove that (81) issufficient and necessary. We first show that if F (x) satisfiesAxioms 1′′-4′′, its normalization, denoted by

f(x) = F (x) ·(∑

i

xi

)− 1λ

, (252)

must satisfy Axioms 1-5 for si in (7) and satisfying Axioms 1′-5′ for si in (62). Then, we verify that fairness F (x) given by(252) indeed satisfies Axioms 1′′-4′′.

41

(⇒) The continuity of f(x) follows directly from that ofF (x) in Axioms 1′′. Let z > 0 be a positive real number andv be a vector of arbitrary length. To prove homogeneity, wemake use of Axiom 3′′ for x = [zv, zv], y = [1, 1], and t = 1,

F (zv,zv)f(1,1) = g−1

(s1 · g

(F (zv)F (1)

)+ s2 · g

(F (zv)F (1)

))

=F (zv)F (1)

(253)

where the second step uses the fact that∑

i si = 1. Next, weapply Axiom 3′′ for x = [v,v], y = [1, 1], and z = [z, z],

F (zv,zv)F (t,t) = g−1

(s1 · g

(F (v)F (1)

)+ s2 · g

(F (v)F (1)

))

=F (v)F (1)

(254)

and for x = [t, t], y = [1, 1], and t = 1,

F (t,t)f(1,1) = g−1

(s1 · g

(F (t)F (1)

)+ s2 · g

(F (t)F (1)

))

=F (t)F (1)

(255)

Comparing the above three equations, (253), (254), and (255),we have

F (zv) = F (z) · F (v). (256)

When v is a scalar, using the result in [34], (256) implies thatlog F (z) = 1

λ log(z) must be a logarithmic function with anexponent 1

λ . We have

F (zv) = z1λ · F (v), (257)

which is a homogenous function of order 1λ . Therefore, nor-

malization f(x) in (252) is a homogenous function of degreezero and satisfies Axiom 2 and Axiom 2′.

Using the homogeneity property (257) and Axioms 2′′, weobtain

limn→∞f(1n+1)f(1n) = lim

n→∞F (1n+1)F (1n)

·(

1 +1n

)− 1λ

= limn→∞

F (1n+1)F (1n)

= 1. (258)

This proves Axiom 3 and Axiom 3′. It is easy to see thatAxiom 4 and Axiom 4′) are special cases of Axiom 3′′ fort = 1. From the homogeneity property (257) and Axiom 3′′,we find that

f( 12 , 1

2 ) = F (12,12) · 1− 1

λ ≥ F (1, 0) = f(1, 0),(259)

which proves Axiom 5 and Axiom 5′, i.e., starvation is nomore fair than equal allocation.

(⇐) Suppose that F (x) is homogenous function of order 1λ

and is given by

F (x) = f(x) ·(∑

i

xi

) 1λ

. (260)

where f(x) satisfies Axioms 1-5 for si in (7) and Axioms 1′-5′ for si in (62). We need to prove that such F (x) satisfiesAxioms 1′′-4′′.

Axiom 1′′ is a direct result from continuity of f(x) inAxiom 1 and Axiom 1′. Axiom 2′′ can be obtained fromAxiom 3 and Axiom 3′ due to

limn→∞F (1n+1)F (1n) = lim

n→∞f(1n+1)f(1n)

·(

1 +1n

) 1λ

= limn→∞

f(1n+1)f(1n)

= 1. (261)

To prove Axiom 3′′, from Axiom 4 and Axiom 4′, we have

f(x)f(y) = g−1

(∑2i=1 si · g

(f(xi)f(yi)

))(262)

where x = [x1,x2] and y = [y1,y2] are two resourceallocation vectors satisfying w(xi) = w(yi). We conclude that

f(xi)f(yi)

=w

1λ (xi) · f(yi)

w1λ (xi) · f(yi)

=F (xi)F (yi)

(263)

where the last step uses (260). Similarly, we derive

f(x)f(y)

=t

1λ · w 1

λ (x) · f(yi)t

1λ · w 1

λ (xi) · f(y)=

F (txi)F (tyi)

. (264)

Plugging (263) and (264) into (262), we prove (80) andAxiom 3′′. Finally, Axiom 4′′ follows from (260), Axioms 5,and Axioms 5′:

F (12,12) = f(

12,12) · 1 1

λ ≥ f(1, 0) = F (1, 0). (265)

Therefore, F (x) given by (260) satisfies Axioms 1′′-4′′. Thiscompletes the proof.

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