A new approach to procedural freedom in game forms

European Journal of Political Economy 26 (2010) 392–402

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European Journal of Political Economy

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A new approach to procedural freedom in game forms

Marlies Ahlert⁎Department of Economics, Martin-Luther-University Halle-Wittenberg, D-06099 Halle (Saale), Germany

a r t i c l e i n f o

⁎ Tel.: +49 345 55 23440; fax: +49 345 55 27127E-mail address: [email protected].

1 The approach to freedom that I take complementsabsence of freedom (for example, Gwartney and Lawsthrough precision about the meaning of freedom. Freeconceptual approaches to freedom include Demsetz (

0176-2680/$ – see front matter © 2009 Elsevier B.V.doi:10.1016/j.ejpoleco.2009.11.003

a b s t r a c t

Article history:Received 2 November 2009Accepted 3 November 2009Available online 12 November 2009

This paper presents a new framework for ranking procedures in terms of freedom of choice. Theconcept of game forms is used to model procedures as a structure of individuals' interactions.Sets of outcomes for an individual are represented by the individual's own perceptions of thesocial states that are generated by the interaction of all individuals. I condense the informationgiven by a game form and by the perceptions of outcomes to two sets for each individual, firstthe set of perceived outcomes the individual can actively determine and secondly the set ofperceived outcomes the individual can actively exclude from happening. Techniques that areknown from the literature on ranking opportunity sets in terms of freedom of choice areapplied to the pairs of determination and exclusion sets. I propose different rankings of gameforms in terms of procedural freedom, some of which I characterize axiomatically. The modeland the rankings are illustrated by classical examples from Game Theory and Social ChoiceTheory.

© 2009 Elsevier B.V. All rights reserved.

JEL classification:D 63C 72

Keywords:Social choiceFreedom of choiceProcedural freedomGame forms

1. Introduction and overview

Economists may conceive of themselves as standing on the shoulders of Adam Smith. However, in many respects Hobbes isthe giant on whose shoulders modern rational choice theorists stand. With Hobbes the ongoing ethics and economics traditionof the British Moralists, of whom Smith is one, starts. Perhaps surprisingly the present discussion of freedom forms a case inpoint as well. According to the opening line of chapter X Leviathan “(t)he POWER of a Man, (to take it Universally,) is his presentmeans, to obtain some future apparent Good” (Hobbes, 1651/1968). Modern concepts of freedom seem in large measure to be re-incarnations of this power concept. Quite in line with this, the present paper emphasizes control aspects and perceptual aspects offreedom in that it conceives of the freedom to choose in terms of the control that a decision-making individual can exert in pursuitof “a future apparent good”. 1

The degree of freedom an individual experiences when he or she has to decide on an action depends on many circumstances.First, the outcome of her action may not always be a consequence of her own action, only. There is the strategic uncertainty thatresults from the fact that the individual interacts with other individuals and may lack information concerning their actions. Thisuncertainty remains, even if we assume that the rules of the interaction are common knowledge. Different procedures anddifferent information structures will lead to different degrees of freedom. Secondly, the perception of possible outcomes by anindividual has an influence on the degree of freedom the individual experiences. For instance, an individual that is only interestedin her own action or in aspects of outcomes that are only related to her ownwellbeing, ceteris paribus, perceives a higher degree of

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other literature that defines measures of freedom and seeks to evaluate the consequences of the presence oron, 2003; Heckelman and Stroup, 2005). The conceptual issues than I consider are implicit in this literaturedom is also related to the distributive consequences of institutions (for example, Carmignani, 2009). Other1981).

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393M. Ahlert / European Journal of Political Economy 26 (2010) 392–402

freedom than an individual who is also interested in choices of others and aspects of others' lives. Thirdly, the actions chosen andthe interactions that will emerge depend on the preferences of the individuals involved. The same “game form” with differentplayers will make a difference to the strategic analysis of the game because of different perceptions and different preferences.

This paper will concentrate on the first two aspects. I consider social interactions in game forms and propose a concept ofindividual perceptions of outcomes in social interactions. I model a positive and a negative freedom of choice and derive andcharacterize rankings in terms of procedural choice. The general model does not require the existence of individual preferences onoutcomes, “payoffs” are not given. Therefore the concepts cannot and in fact do not rely on rationality concepts specifying how theindividuals will play the game.

Bervoets (2007) also proposes a model to describe freedom of choice in game forms. In his framework he uses the traditionalevaluation of outcomes in games by preference relations. He proposes and axiomatically characterizes two rankings that comparethe freedom of choice of an individual, Max andMaxMin. Max ranks game forms in terms of freedom of choice comparing the bestoutcomes the individual could reach. MaxMin ranks game forms with respect to the best outcome an individual can guaranteeherself. MaxMin is similar to an intrapersonal ranking of game forms in terms of guarantees that Ahlert (2008) has developed.However, in Ahlert's approach interpersonal rankings in terms of guarantees are derived, too. The Max ranking compares bestoutcomes in game forms, however, the individual is not free to choose an outcome, she can only choose strategies. The realizedoutcome depends on the strategies chosen by the other individuals, too. So the individual is not free to choose the best outcome.Max compares elements that cannot be controlled by the individual. In contrast to Bervoets's approach, I try to model positive andnegative freedom of choice in opportunity sets such that the individual perceives the options as different and really experiencesthem as choices.

Section 2 introduces the conceptual framework that enables a description of the perception of freedom of an individual inprocedures that are represented by game forms. Two aspects of freedom are modelled, the freedom to determine perceivedoutcomes and the freedom to prevent perceived outcomes. For my concept it is essential that the notion of freedom is applied tosets of perceived outcomes and not to sets of objective outcomes.

Section 3 applies the framework to different perception structures and to examples of procedures that are represented byfamiliar simple games, such as the dictator game, the ultimatum game, and a coordination game.

Section 4 contains a comparison of rankings in terms of procedural freedom, that aggregate the two dimensions, freedom todetermine and freedom to exclude perceived results of a procedure. I give examples of rankings and discuss some of their axiomaticproperties. The main problem I try to solve in this section is how to deal with the trade off between the two criteria. What is moreimportant, to be able to determine a certain outcome or to be able to exclude it? Intuitively, the answer might depend on thepreferences of an individual over outcomes and on the relation to other outcomes that can be determined or excluded. I suggest anapplication of the notion of essential alternatives to formulate some necessary conditions for rankings in terms of proceduralfreedom. For the special case of trichotomous perceptions these conditions uniquely characterize a ranking in terms of proceduralfreedom. Section 5 summarizes and indicates some ideas for further research.

2. The conceptual framework

2.1. The structure of the interaction

I consider strategic interactions of a finite number n of individuals or players. The set of individuals is called N={1,…,n}. Eachstrategic interaction is represented by a game form Γ with n players. The strategy set of individual i in a game form Γ is called Si, astrategy of individual i is denoted by si.ω(s1,…,sn) denotes the outcome that is generated by a strategy combination (s1,…,sn) of allindividuals. I allow strategy sets to be empty (among the n players there may be some that do not act in the game).

In contrast to notational conventions of standard game theory, outcomes of the game form Γ are not given by payoff vectors butby alternatives that are interpreted as social states or social alternatives. This means that I do not consider the individual payoffs orutilities that may be attached to these social states. The set of outcomes of Γ is assumed to be finite and is denoted by Ω(Γ). Iassume that there is a given general set of feasible social states so that Ω(Γ)p for all Γ. Π( ) denotes the set of all finitesubsets of . I do not assume that the set has any special structure, in particular it is not necessary to represent as an m-dimensional space. This assumption would form a special case of my model.

I define a procedure to be a game form Γ such that outcomes are defined by social states.G is the set of all procedures with n players.

2.2. The perception of outcomes

Each individual i has her own perception of social states in which is represented by a function vi: →Xi. This means thatindividual imay only be interested in certain aspects of a social state x in or may condense the complexity of a social state in herindividual way. In themost extreme theoretical case, an individual may aggregate the complexity of a social state to a real number.(Since I do not assume to be m-dimensional I do not model here that an individual is only interested in certain dimensions of asocial state. Again, this would be a special case.) Xi displays individual i's perception of the “world” of all feasible social states(Π(Xi) denotes the set of all finite subsets of Xi.). This implies that there may be two different social states x and y in , such that acertain person i identifies them or perceives them as indifferent (i.e. vi(x)=vi(y)) and another person k perceives differences (i.e.vk(x)≠vk(y)).

394 M. Ahlert / European Journal of Political Economy 26 (2010) 392–402

I assume that each individual evaluates her freedom in a game form Γ with respect to her own perception of the social statesΩ(Γ) that are the possible outcomes of Γ. For individual i the set of outcomes of Γ is vi(Ω(Γ))pXi (I will denote the set vi(Ω(Γ)) byΩi(Γ)). I start with the assumption that the perception function vi does not depend on the game form Γ. Thus, I assume that thefunction vi is exogenously given. This enables us to compare the procedural freedom of the same person in different game formswithout modeling that the perception of the same state might be different in different games.

A very special example of a quite coarse perception functionwould be a utility function. However, I do not assume the existenceof preference orderings on social states at the moment.

2.3. Aspects of freedom

There are two aspects of freedom an individual may face when she is involved in a procedure defined by a game form Γ.First there may exist some single outcomes in Ωi(Γ) which the individual can enforce. That means that she is able to determinethe development of the game by choosing a strategy that leads to a certain perceived social state no matter what the otherindividuals do. This idea reflects a very strong kind of positive freedom. The individual is free to choose one of her perceived socialalternatives.

Definition 1. For any given procedure Γ and each individual i we define a set Di to be the set of all perceived social states in Ωi(Γ)individual i can determine.

Di = fz∈ΩiðΓÞ j∃si∈Si such that ∀sk∈Sk with k≠i viðωðs1;…; snÞÞ = zg:

Note that the definition of Di is independent of the perceptions of the other individuals, it depends on the game form, on thepossible social outcomes and on i's perception of social outcomes. It is intuitively clear and it will be clear from the examples thatDi can be empty. In case every individual perceives the social states in injectively (different states are perceived differently; thefinest possible perception), the following holds: If Di contains more than one element for individual i then Dk is empty for all otherindividuals k. If Di contains exactly one element for individual i then Dk\Di is empty.

This fact makes it necessary to consider a second kind of freedomwhich is generated by the possibility to exclude certain singleperceived social alternatives in Ωi(Γ) from being realized. Here an individual has the power to prevent the realization of a certainperceived outcome by choosing a certain strategy, independent of the strategies of the other players.

Definition 2. For any given procedure Γ and each individual i we define a set Ei to be the set of social states individual i canexclude. Ei={z∈Ωi(Γ)/∃si∈Si such that ∀sk∈Sk with k≠ i vi(ω(s1,…,sn))≠z}.

Remark 1. For each game Γ and each individual i there is a pair of sets (Di,Ei) that characterizes the freedom to determineand the freedom to exclude certain single outcomes in Ωi(Γ). For this pair (Di,Ei) the following holds: z∈Di⇒ [Ei=Ωi(Γ) or Ei=Ωi(Γ)\{z}].

In Remark 2 I consider the special case of a situation where individuals perceive the outcomes of a social interaction as fine aspossible. This means that the individual freedom will be strongly restricted by the decisions of other individuals.

Remark 2. In case of vi(x)=x for all i=1,…,n and all x∈ (i.e. every individual perceives the social states in as they “are”) thefollowing relations hold:

x∈Di⇒ ∀k≠iDk = ∅ or Dk = fxg

x; y∈Di such that x≠y ⇒ ∀k≠i Dk = ∅

x∈Di⇒ ∀k≠i x ∉ Ek:

y∈Ei⇒ ∀k≠i y ∉Dk:

In the general case, however, where the perception of states by individuals will be concentrated on different aspects of socialstates there will be less mutual restriction.

In the model I keep perceptions fixed. The framework, however, allows to analyse how the two aspects of an individual'sfreedom are affected if her perception of outcomes changes, e.g. by refining or coarsening.

3. Applications to examples of procedures

3.1. A simple version of a two person dictator game Γ1

The proposer (player 1) can propose one of two social states, namely a fair state F and an unfair state U, U≠F. The responder(player 2) has no decision to make. The outcome of the game is what the proposer chooses.

I assume that the perception of the state F is F and of state U is U for both players.

Determinations and exclusions in Γ1: D1={F,U}, E1={F,U}, D2=∅, E2=∅.

Fig. 2. Extensive form of Γ2.



The proposer is free to determine each state and to exclude each state, whereas the responder has no power to determine orexclude any alternative (Fig. 1).

3.2. A simple version of the ultimatum bargaining game Γ2

The proposer (player 1) can propose one of two social states, namely a fair proposal F and an unfair proposal U. The responder(player 2) can accept the proposal or reject it. If the responder accepts the proposal the outcome of the game is the alternative theproposer has chosen. If the responder rejects the proposal the outcome is a state called B.

Again I assume that the perception of both players is identical to the described states.

Determinations and exclusions in Γ2: D1=∅, E1={F,U}, D2={B}, E2={F,U,B}.

If we compare the D- and E-sets for player 1 in the dictator and the ultimatum bargaining game we find that D1 decreases andthat E1 remains constant. This coincides with the intuition that the power of player 1 to ensure certain outcomes decreases, if in Γ2player 2 has the possibility to reject the proposal player 1 makes (Fig. 2).

In contrast, the chance of player 2 to determine the outcome B and also to exclude each of the three possible outcomes enlargesthe two types of freedom for player 2. Again, this is in line with the intuition that in a dictator game player 2 is “completely nonfree”, whereas he has got some positive freedomby the veto power to determine the conflict outcome and negative freedomby thepower to avoid each of the possible proposals and the conflict outcome. In an experimental study with monetary payoffs, Ahlertand Crueger (2004) observe howmuchmoney subjects that are in the role of player 2 arewilling to pay tomake sure that they playan ultimatum game instead of a dictator game.

3.3. A coordination game Γ3 with perfect information

Player 1 chooses L(eft) or R(ight). Player 2 observes the choice and then chooses L or R. If both choose L, state coordination on Lis the outcome, if both choose R, coordination on R is the outcome, in case of no coordination the outcome is M. Again, these maybe also the perceived outcomes of both players.

Determinations and exclusions in Γ3: D1=∅, E1={L,R}, D2={M}, E2={L,R,M}.

We observe that the structure of the D- and E-sets in Γ2 and Γ3 are equivalent. Nevertheless, it is not clear how to compare thedegree of freedom in both cases. Intuitively the degrees of freedommay be different because the evaluation of the perceived socialalternatives by the players in both games may be different (Fig. 3).




3.4. A coordination game Γ4 with imperfect information

Player 1 chooses L(eft) or R(ight). Player 2 does not observe the choice and chooses L or R. If both choose L, the state coordi-nation on L is the outcome, if both choose R, coordination on R is the outcome, in case of no coordination the outcome is M. Invariation I we assume that perception is identical for both players and coincides with the outcomes. (This example is discussed indifferent forms in many papers on rights and game forms, c.f. Analyse & Kritik 1/1996).

Determinations and exclusions in Γ4 — variation I: D1=D2=∅, E1=E2={L, R}.

Imperfect information changes the situation to a less free procedure for player 2 than Γ3. Nobody can determine any stateseparately. Both can exclude L or R.

Determinations and exclusions in Γ4— variation II: Let us assume different perceptions of states by the players. Now individual 1may perceive only her own choice (let us call these choices L1, or R1), whereas individual 2 may be interested in the coordinationand perceives successful coordination on L or on R or unsuccessful coordination M like above (Fig. 4).

In this second variation, individual 1 is free to chose from D1=Ω1(Γ4)={L1, R1} and can exclude each element from E1=Ω1(Γ4)={L1, R1}. Individual 2, however, because of her special interest can determine nothing D2=∅ and exclude E2={L, R},like in variation I.

The two variations of the same game form show the impact of perception on the degree of freedom of the persons. Here it isevenmeaningful to compare the degrees of freedom of both persons, though interpersonal comparisons are not in the focus of thispaper. In this example the individual whose perception of a state is focused on her own decision is as free as possible, whereas theindividual who cares about the decision of the other person too, is not completely free. The intuition behind this formulation is thata person i is completely free, if her determination and exclusion sets are equal to Ωi(Γ).

4. Rankings of procedures in terms of freedom

In this section I consider different models of how a person ranks procedures in terms of her freedom to determine and toexclude social states.

Since I wish to construct rankings of procedures I consider binary relations on G×G. Let a pair (Γ,Γ′)∈G×G be given withdetermination and inclusion sets Di, Ei, and D′i, E′i, respectively, for the individuals i=1,…,n. For each player i I define the ranking


ℜi between two procedures Γ and Γ′ in terms of procedural freedom by a ranking Ri between the two respective pairs of sets (Di,Ei)and (Di′,Ei′).

Thus, for all i=1,…,n Ri is a binary relation defined on pairs of subsets of Xi. Pi and Ii denote the strict ranking and indifference,respectively.

Definition 3. Let individual i be given. For any binary ranking Ri on Π(Xi)2×Π(Xi)

2, we derive a binary relation ℜi on G×G inthe following way: Let (Γ,Γ′)∈G×G be pair of procedures such that the determination and inclusion sets of individual i are Di, Ei,and Di′, Ei′, respectively. Then ΓℜiΓ′⇔(Di, Ei)Ri(D′i, E′i).

In this case we say that individual i prefers procedure Γ to procedure Γ′ in terms of procedural freedom.The only procedural information my approach incorporates into the ranking with respect to procedural freedom are the sets

of perceived outcomes an individual can determine or exclude. Here I implicitly assume an independence of the evaluation ofthe pairs of sets from the underlying game form. This is in contrast to the model of Gaertner and Xu (2004) who focus on thedependence of choices on procedures. Arlegi and Dimitrov (2004), too, choose the procedure as one component of therepresentation of the choice situation. However, different from my model they do not construct an endogenous relation betweenprocedure and opportunity sets.

I will have to rank pairs of sets in order to construct a ranking in terms of freedom of choice. Van Hees (1998) also ranks pairs ofsets, where one set describes the opportunity set and the other set represents the set of feasible alternatives capturing van Hees'interpretation of negative freedom. In his case both sets are related by inclusion, since opportunities have to be feasible. Thisinclusion does not hold in my framework, and in addition both sets capture different opportunity aspects of freedom. Therefore, Ineed new considerations to construct rankings in terms of freedom in my framework.

4.1. Some canonical rankings

In the literature on freedom of choice there are many suggestions how to rank opportunity sets in terms of freedom of choice(see e.g. Sugden (2003) for a critical overview). In this paper I deal with rankings of pairs of sets. There are, however, certainproperties I will import from the rankings of single opportunity sets.

From now on we consider a fixed individual i that compares different procedures. Therefore, we omit the subscript for theindividual i in this section. In the following section e.g. X denotes the set of all perceived outcomes and Ω(Γ) denotes the set ofperceived outcomes in a game form Γ for individual i. The general concept presented here is prepared to compare the situations ofdifferent individuals in a given game form or in different game forms. However, this will be a project for a new paper.

One basic assumption in the literature on freedom of choice is that larger opportunity sets imply more freedom. We want theranking R to have the analogous properties in both dimensions.

Axiom weak monotonicity: Let (D,E) and (D′,E) be pairs in Π(X)2 such that D′pD. Then (D,E)R(D′,E) holds. Let (D,E) and (D,E′)be pairs in Π(X)2 such that E′pE. Then (D,E)R(D,E′) holds.

The larger the set of perceived alternatives an individual can determine in a procedure the larger her freedom, and the largerthe set of perceived alternatives she can exclude from being realized the larger her freedom.

We can interpret the ranking R as a ranking of procedures with respect to two criteria, the first criterion being the freedom todetermine outcomes, the second being the freedom to exclude outcomes. This enables us to apply the literature on freedom ofchoice to each of these two dimensions and to construct a ranking R by aggregating rankings with respect to the two criteria. Asimple way of aggregation is to define a lexicographic ordering of the two criteria (see e.g. Klemisch-Ahlert (1993) for examples ofdifferent lexicographic rankings in terms of freedom of choice). There are two choices. We can look at the ranking of sets D firstand in case of indifference decide with respect to the ranking of the sets E, or the other way round.

Definition 4. (Lexicographic rankings): Let R1 be any ranking in terms of freedom of choice for sets of states that can be deter-mined (short D-sets), and let R2 be any ranking in terms of freedom of choice for sets of states that can be excluded (short E-sets),with strict preference and indifference defined as usual. We define a lexicographic ranking Rlex 1,2 by sufficient conditions for strictpreference and indifference:

(D,E) Plex 1,2 (D′,E′), if D P1 D′ or (D I1 D′ and E P2 E′)

(D,E) Ilex 1,2 (D′,E′), if D I1 D′ and E I2 E′.

Rlex 2,1 is defined analogously to give priority to the ranking of the E-sets.Another thought exercise would be to use the cardinality based ordering of opportunity sets (Pattanaik and Xu (1990))) in both

dimensions D and E and then aggregate the two results. Of course, this can be done in a lexicographic manner like above. However,one could also simply add both cardinalities. Let us call this ranking the cardinality sum ranking. The sum of the cardinalities #Dand #E can be axiomatically characterized as follows.

Definition 5. For a given pair (D,E) we define a set D⨁E by D⨁E={(x,λ)|(x∈D and λ=1) or (x∈E and λ=2)}.

The λ in Definition 5 makes it possible to distinguish whether an element x is in D or in E or in both sets. Therefore if wecalculate the cardinality of D⨁E, an element that is in both sets D and E is counted twice.


Remark 3. #D⨁E=#D+#E. The cardinality of D⨁E is the sum of the cardinalities of both sets. If an element x happens to be in Dand in E it is counted twice, first as (x,1) and secondly as (x,2).

Definition 6. (Cardinality sum ranking): For given pairs (D,E) and (D′,E′)we define a ranking R# by (D,E) R# (D′,E′)⇔#D⨁E≥#D′⨁E′.

If we consider sets D⨁E as opportunity sets, the axioms defined by Pattanaik and Xu (1990) can be applied to all pairs of nonempty sets. Since in our approach we will also have to face cases where D and/or E are empty we have to adjust the indifferenceaxiom. We also have to rewrite the axioms such that joining opportunities are represented as states that are added to D-sets or E-sets. We denote the ranking between sets D⨁E and D′⨁E′ by R⨁ and translate the axioms of Pattanaik and Xu to our framework asfollows.

Axiom relations between pairs if sets are empty: Let pairs (D,E) and (D′,E′) inΠ(X)2 be given. If # D⨁E=1 (i.e. one of the sets D,E is empty and the other set is a singleton) and # D′⨁E′=1 then D⨁E I⨁ D′⨁E′ (i.e. (D,E) I (D′,E′)). If D⨁E=∅ and # D′⨁E′=1,then D′⨁E′ P⨁ D⨁E.

This axiom claims the indifference between singletons. This means that the individual is indifferent between the following fourcases: she can determine a certain alternative x or she can exclude x or she can determine or exclude another alternative y. Theaxiom also states that a singleton set D′⨁E′ is strictly preferred to having no influence on perceived outcomes in cases whereD⨁E=∅. This relation defines the new starting point zero-cardinality of the cardinality based ranking.

Axiom strict monotonicity: Let pairs (D,E) and (D′,E′) inΠ(X)2 be given. If D⨁E={z} and D′⨁E′={z,w} with z≠w (i.e. z=(x, λ)with λ=1, if x∈D and x∈D′, or λ=2, if x∈E and x∈E′, w=(y, μ) with μ=1 if y∈D′ or μ=2 if y∈E′, and (x≠y or λ≠μ)),then D′⨁E′ P⨁ D⨁E (i.e. (D′,E′)P(D,E)).

Strict monotonicity compares a situation where an individual has some influence on a single alternative x to a situation wherethe individual has two possibilities of influence. This could be a situation where y is different from x. It could also be a situationwhere she can now determine x and exclude x (i.e. x=y), whereas in D⨁E she can only either determine x or exclude x (λ≠μ).She strictly prefers to have two possibilities of influence to having only one.

Axiom independence: For all pairs (D,E) and (D′,E′) inΠ(X)2 and all z=(x, λ) with x∈X and (λ=1 or λ=2) such that z∉(D⨁E)∪(D′⨁E′) the following holds: D⨁E R⨁ D′⨁E′⇔(D⨁E)∪{z} R⨁ (D′⨁E′)∪{z}. This means the following:

case λ=1 x∉∪D′ (D,E) R (D′,E′)⇔(D∪{x},E) R (D′∪{x},E′)

case λ=2 x∉E∪E′ (D,E) R (D′,E′)⇔(D,E∪{x}) R (D′,E′∪{x}).

Independence describes a situation where a new alternative joins the sets D⨁E and D′⨁E′ in the form of (x, λ). In case λ=1 itjoins the D-sets and in case λ=2 it joins the E-sets. Adding the new alternative does not change the rankings R⨁ (between therespective sets) and R (between the respective pairs).

Applying the theorem proven by Pattanaik and Xu (1990) to this framework leads to the characterization of the ranking R⨁ asthe cardinality ranking on pairs of sets D⨁E. With the definition of R# on pairs of situations (D,E) by comparing the cardinalities ofthe sets D⨁E the characterization theorem follows:

Theorem. There is a unique reflexive and transitive relation in terms of procedural freedom on pairs in Π(X)2 that satisfies the axiomsof relations between pairs if sets are empty, strict monotonicity and independence. This relation is the cardinality sum ranking R#.

The axioms above do not take into consideration whether the new alternative joins the D-set or the E-set. If x is a state theindividual likes, however, it might be the case that the individual prefers the situation where she can determine x to the situationwhere she can exclude x. The consequence of the theorem above is that the cardinality sum ranking does not discriminate betweenthe two dimensions of freedom.

The general form of the lexicographic rankings and the cardinality sum ranking do not capture the intuition that different statesmay have different values in terms of procedural freedom to the individual in case they can be determined or in case they can beexcluded. Tomodel this intuition in the analysis it is necessary to consider the trade off in freedom between both dimensions— thedetermination and the exclusion of alternatives.

4.2. A concept for rankings with trade offs in freedom

In this section I deal with the problem that an individual might prefer to be able to determine a certain perceived outcome xthan to exclude x or vice versa.

Multi criteria decisions by lexicographic rankings of criteria have the disadvantage of not paying any attention to trade offsbetween the criteria. A lexicographic ordering as described in the section above would first focus on the determination of a certainstate x or first on the exclusion of a certain state x. No compensation between dimensions is possible. The cardinality sum rankinghas a constant trade off of 1 in terms of numbers of states and would be indifferent between determining x or excluding x.


If we take into account that an individual will have or at least is able to develop some kind of preference over possible outcomesof a procedure and therefore over the perceived elements in X, it seems natural that an individual might be interested in having thefreedom to determine outcomes she likes (or assumes she will like in the future) and to exclude outcomes she does not like. Thisevaluation may depend on the outcomes of a game form and especially on the comparison of different outcomes.

Since in this analysis I do not want to assume the existence of given, fixed preferences on outcomes, I can try to induce from theindividual's ranking of procedureswhether an outcome is seen by her as being one that should better be selectable than excludableor vice versa. This concept will depend on the outcome sets that are induced by the procedure.

For a start of the analysis I assume that the individual has a complete ordering of pairs (D, E).I consider three cases dependent on the cardinality of set D.

Case 1. #D=0.

The individual compares pairs of sets (∅, E) and (∅, E′) such that E, E′pX.

Analogously to Puppe (1996) I use the terminology of essential alternatives in this case.

Definition 7. An outcome x∈E is essential for exclusion in E if and only if (∅, E) P (∅, E\{x}). B(E) denotes the set of outcomesessential for exclusion in E.

If x is not essential for exclusion given the other excludable outcomes in E, then the individual is indifferent between having theopportunity of excluding x or not having it.

Axiom independence of non-essential alternatives in exclusion sets (cf. Puppe 1996): (∅, E) I (∅, B(E)).

By the axiom of independence of non-essential alternatives I can reduce the information that is used to compare two pro-cedures with exclusion sets E and E′ in Case 1 of empty D-sets to comparing (∅, B(E)) and (∅, B(E′)). This means that I have tocompare the sets of outcomes that can be excluded and that have a certain value of being excludable. In complex interactions thiswill be a likely case.

Puppe (1996)gives examples how to compare sets of essential alternatives.Which typeof comparisonmaybeuseddependson thefiner informational structure that is assumed to exist, especially whether I assume the individual to have a preference ordering on X.

Case 2. #D=1.

Assume D={x} with x∈X. This implies E=Ω(Γ) (Case 2.1) or E=Ω(Γ)\{x} (Case 2.2).

Case 2.1. #D=1 and E=Ω(Γ).

If the individual compares two procedures Γ and Γ′ with the same perceived outcome sets Ω(Γ)=Ω(Γ′) and pairs of determi-nation and exclusion sets ({x},Ω(Γ)) and (∅,Ω(Γ)) respectively, then monotonicity implies ({x},Ω(Γ)) R (∅,Ω(Γ)).

Definition 8. In case of ({x},Ω(Γ)) P(∅,Ω(Γ)), x is called wanted. The set of all wanted outcomes in Ω(Γ) is called W(Ω(Γ)).

An outcome x is wanted, if the individual prefers to have the opportunity of determining x to not having this opportunity, givenshe can exclude every single outcome in Ω(Γ).

This implies that the individual ranks all procedures with outcome setΩ(Γ), exclusion setΩ(Γ) and determination sets with notwanted outcomes as being indifferent to a situation where only the outcomes in Ω(Γ) that are essential for exclusion can beexcluded and nothing can be determined. This follows from the definition of the indifference relation I (R∧¬P holds) and theassumption of transitivity of relation I.

Remark 4. For all x,y∈Ω(Γ)\W(Ω(Γ)): ({x},Ω(Γ))I({y},Ω(Γ))I(∅,Ω(Γ))I(∅,B(Ω(Γ))).

In cases where some single wanted outcome can be determined and every single outcome in Ω(Γ) can be excluded the aboveassumptions do not prescribe a special ranking between the procedures. For instance, if we take any preference ordering≥⁎ on X,we may consider the following induced ranking of such cases:

For all x,y∈W(Ω(Γ)): xN⁎y⇒({x},Ω(Γ))P({y},Ω(Γ))P(∅,Ω(Γ)) and x=⁎y⇒({x},Ω(Γ))I({y},Ω(Γ))P(∅,Ω(Γ)).This special ranking means that for wanted alternatives their in-between-ordering “counts”whereas not wanted outcomes do

not have an impact on the positive aspect of freedom, independent on how they are ranked among each other.

Case 2.2. #D=1 and #E=#Ω(Γ)−1.

Here we deal with procedures such that the set of outcomes is Ω(Γ) and the determination and exclusion sets are D={x} andE=Ω(Γ)\{x}, respectively. The following four subcases are defined by assumptions on x beingwanted or not and being essential or not.

Case 2.2.1. x is wanted in Ω(Γ) and essential for exclusion in Ω(Γ).

In this case we know that ({x}, Ω(Γ))P(∅,Ω(Γ))P(∅,Ω(Γ)\{x}) holds. The question is where the pair ({x}, Ω(Γ)\{x}) is located inthis chain. Monotonicity implies ({x}, Ω(Γ))R({x}, Ω(Γ)\{x}). If we assume that it is more important to determine a wanted x than


to exclude it (cf. axiom trade off below) we come up with the following ranking: ({x}, Ω(Γ))R({x}, Ω(Γ)\{x})P(∅,Ω(Γ))P(∅,Ω(Γ)\{x}). Since I think that x being essential for exclusion in Ω(Γ) also implies being essential if x can be determined (cf. axiom pre-serving of essentiality), I replace the R in the first comparison by a P, i.e. ({x}, Ω(Γ))P({x}, Ω(Γ)\{x})P(∅,Ω(Γ))P(∅,Ω(Γ)\{x}).

Case 2.2.2. x is wanted in Ω(Γ) and non-essential for exclusion in Ω(Γ).

In this case we know that ({x}, Ω(Γ))P(∅,Ω(Γ))I(∅,Ω(Γ)\{x}) holds. If we again assume that being wanted is more importantthan exclusion (axiom trade off) and being non-essential for exclusion also holds if x can be determined, we get the ranking: ({x},Ω(Γ))I({x}, Ω(Γ)\{x})P(∅,Ω(Γ))I(∅,Ω(Γ)\{x}).

Case 2.2.3. x is not wanted in Ω(Γ) and essential for exclusion in Ω(Γ).

In this case we know that ({x}, Ω(Γ))I(∅,Ω(Γ))P(∅,Ω(Γ)\{x}) holds. If we assume that in case of a not wanted outcome itis important to be able to exclude it (axiom trade off) and applying monotonicity we get the ranking: ({x}, Ω(Γ))I(∅,Ω(Γ))P({x},Ω(Γ)\{x})I(∅,Ω(Γ)\{x}).

Case 2.2.4. x is not wanted in Ω(Γ) and non-essential for exclusion in Ω(Γ).

With the same assumptions as above we receive ({x}, Ω(Γ))I({x}, Ω(Γ)\{x})I(∅,Ω(Γ))I(∅,Ω(Γ)\{x}).In the constructions of the rankings in Case 2.2 I have used the following axiom claiming for a state x that essentiality and non-

essentiality for exclusion is preserved if the element x can be determined.

Axiom preserving essentiality: If x is essential for exclusion in Ω(Γ), ({x}, Ω(Γ))P({x}, Ω(Γ)\{x}) holds. If x is non-essential forexclusion in Ω(Γ), ({x}, Ω(Γ))I({x}, Ω(Γ)\{x}) holds.

The other assumptions I used in the intuitive construction of the rankings are related to the comparison of two situations of thekind ({x}, Ω(Γ)\{x}) and (∅,Ω(Γ)). In the first one the outcome x can be determined and not excluded and vice versa in the secondsituation. I have incorporated different decisions on this trade off in Case 2.2 dependent on characteristics of x. They are sum-marized in the following axiom.

Axiom trade off:

(i) If x is wanted and non-essential for exclusion from Ω(Γ), ({x}, Ω(Γ)\{x})P(∅,Ω(Γ)) holds.(ii) If x is wanted and essential for exclusion from Ω(Γ), ({x}, Ω(Γ)\{x})P(∅,Ω(Γ)) holds.(iii) If x is not wanted and non-essential for exclusion from Ω(Γ), ({x}, Ω(Γ)\{x})I(∅,Ω(Γ)) holds.(iv) If x is not wanted, but essential for exclusion from Ω(Γ), (∅,Ω(Γ))P({x}, Ω(Γ)\{x}) holds.

In cases (i), (iii), and (iv) the ranking seems intuitively clear, whereas in (ii) priority is given to being able to determine awanted outcome. It would also be possible to give priority to the exclusion of x or to define indifference. The ranking may alsodepend on the situation and would in this case have to be revealed by the individual. However, I assume that being able todetermine a wanted outcome x is a very valuable opportunity and that x cannot be excluded is not a high price in this case. Thepossibility to determine x under an irrelevant restriction is more valuable than having nothing to determine, but being able toexclude x. Implicitly, in the axiom I introduce an aggregation of the criteria “wanted” and “essential for exclusion” to decide on thetrade off: If only one criterion is positive decide the trade off in favour of that. If both are negative, decide on indifference. If bothare positive, decide in favour of determination.

Case 3. No restriction on the cardinality of D.

Here we consider the general case and develop properties of the comparison of arbitrary situations (D, Ω(Γ)).First we consider the ranking the individual reveals comparing the situations (D, Ω(Γ)) and (D\{x}, Ω(Γ)) with a set D con-

taining at least an outcome x.

Definition 9. If (D, Ω(Γ))P(D\{x}, Ω(Γ)) we call x essential in D. A(D) is the set of all essential outcomes if D can be determined.

Remark 5. From the definitions it follows that a wanted outcome x is essential in the determination set{x}.

What is the relation between the property of an outcome x to be wanted (x∈W(Ω(Γ))) and to be essential in some deter-mination set D (x∈A(D))? Property (α) (Sen (1970), Puppe (1996)) claims consistency in cases of set restriction. In this frame-work it means A(D)pW(Ω(Γ)).

If an outcome x is essential for determination given any set of determinable elements D then it is also essential if only x isdeterminable, i.e. it is wanted. This implies that only wanted elements in Ω(Γ) can be essential for determination.

Axiom independence of non-essential alternatives in determination sets: (D, Ω(Γ))I(A(D), Ω(Γ)).

The information to rank these situations is reduced to the essential outcomes.


Nowwe consider the comparison of two situations (D,Ω(Γ)) and (D′,Ω(Γ)) where D and D′ could be empty or any other subsetof Ω(Γ). We define the ranking for situations (D, Ω(Γ)) and (D′, Ω(Γ)) by A(D∪D′)pD⇔(D, Ω(Γ))R(D′, Ω(Γ)). This is equivalentto ¬[A(D∪D′)pD]⇔(D′, Ω(Γ))P(D, Ω(Γ)).

The definition of these rankings is consistent with the ranking of situations where D=∅ and D′ contains only one element x,which we have already defined in Case 2. If x is wanted, A(∅∪{x})={x} and therefore ¬[A(D∪D′)pD] holds. Thus the definitionof the ranking above implies ({x}, Ω(Γ))P(∅, Ω(Γ)) which is exactly what we wanted in Case 2 to hold for wanted outcomes x. If xis not wanted, i.e. non-essential in {x}, A(∅∪{x})=∅ and the definition of the ranking above implies (∅, Ω(Γ))I({x}, Ω(Γ)) whichagain is the definition of x being not wanted.

In the beginning of this section I have assumed that the individual has a complete preference ordering on pairs (D, E). Theaxioms above restrict the set of possible orderings. From the remarks it follows that certain relations will necessarily hold for anyof these orderings. However, in general they do not restrict the set of orderings to a unique one.

4.3. The example of trichotomous perceptions

Here is a simple example. Let us assume that the individual has trichotomous perceptions and distinguishes three classes ofstates: good states, neutral states and bad states. Let the perception be very coarse such that all good states are perceived asidentical, all neutral states and all bad states, too. Then I can model the perception such that there is only one good state a, oneneutral state m, and one bad state z. It is a plausible example to assume that the good state a is wanted and is essential fordetermination in any set of states. The bad state z is not wanted, not essential for determination but essential for exclusion. Theneutral statem is somehow in the middle between a and z, it is not wanted, not essential for determination in any subset and notessential for exclusion. (Another example would be that e.g. the neutral state is essential for determination if only the bad state ispresent.) The even simpler example of dichotomous perceptions of good and bad states does not show all features of the concept,since in that case being able to determine one state is logically equivalent to being able to exclude the other, so that there are notenough different pairs of sets left to apply the framework meaningfully.

I consider only game forms Γ such that Ω(Γ)={a,m,z}, i.e. each state can happen. The result below is derived from the appli-cation of Case 1 to pairs with D=∅, Case 2 to pairs with #D=1 and Case 3 to pairs with #D=2 or #D=3. The arrangement of thereceived relations leads to the following indifference classes of pairs (D,E).

The best class consists of ({a,m,z},{a,m,z})I({a,z},{a,m,z})I({a,m},{a,m,z})I({a},{a,m,z})I({a},{m,z}). Each element of this class isstrictly better (P) than each one of the next indifference class: ({m,z},{a,m,z})I({m},{a,m,z})I({z},{a,m,z})I({m},{a,z})I(∅,{a,m,z})I(∅,{a,z})I(∅,{m,z})I(∅,{z}). These elements are each better (P) than each element of the last indifference class: (∅,{a,m})I({z},{a,m})I(∅,{a})I(∅,{m})I(∅,∅).

The procedures in the best class can be described by the fact that the good state a can be determined and that the bad state zcan be excluded, too. The second best class has the property that the good state a cannot be determined, but the bad state z can beexcluded. The procedures in the lowest ranked group have the property that the good state a cannot be determined and the badstate z cannot be excluded. The situation such that the good state can be determined and the bad state cannot be excluded does notexist, since being able to determine the good state a implies the possibility to exclude state z.

Theorem. If an individual has a trichotomous perception function (good, neutral, bad) such that the good state a is the one that isessential for determination in each subset and the bad state z is the one that is essential for exclusion then there is a unique reflexive andtransitive relation on the set of all game forms with Ω(Γ)={a,m,z} that fulfils the axioms of independence of non-essential alternativesin exclusion sets, preserving essentiality, trade off, and independence of non-essential alternatives in determination sets. This is theranking defined above. This ranking fulfils the axiom of weak monotonicity.

The four axioms independence of non-essential alternatives in exclusion sets, preserving essentiality, trade off, and inde-pendence of non-essential alternatives in determination sets are independent, since they define rankings on disjoint subsets ofpairs of determination and exclusion sets. There is no contradiction between these axioms and the axiom weak monotonicity.Quite to the contrary, the four axioms split weak monotonicity into indifference or strict monotonicity for certain comparisons.In special cases — e.g. in the case of the trichotomous preferences above — it can happen that the ranking in terms of freedom isalready defined by the four specific axioms and that weak monotonicity is redundant.

Different assumptions than made in the theorem— e.g. about the essentiality of the neutral state—would change the ranking.If there are more than three states, the restrictions derived from the proposed axioms may be less binding, so that there maybe more than one ranking fulfilling the axioms. If there are outcomes that have the property of being essential for determinationin certain subsets and simultaneously being essential for exclusion, the analysis becomes more complex. I did not formulaterequirements for these cases. Nevertheless, the conditions above may serve as a set of necessary properties of any aggregation offreedom to determine and freedom to exclude.

5. Concluding remarks

I have sketched how the freedom an individual experiences in different procedurally modelled interactions with otherindividuals may be conceptualized. The concept of freedom that has been modelled is based on the notion of an individual'sperception of outcomes. I have discussed several possibilities for ranking procedures in terms of freedom to determine and


freedom to exclude certain perceived outcomes. These two aspects of freedom are modelled from the perspective of a givenindividual and do not take into account the consequences of interactions for other individuals or the other individuals' perceptionsof outcomes. Thus I do not measure power over or influence on others. I capture however the control aspect of the broaderHobbesian power concept in terms of freedom.

The problem of how to aggregate the two different aspects of freedom can be solved in many ways. Dependent on the rankingswith respect to each of the two aspects, a lexicographic combination of both criteria can be used. The sum of the cardinality of thedetermination set and the exclusion set forms a measure of procedural freedom, too, which I have axiomatically characterized.However, I have argued that these methods of aggregation suffer from the fact that they do not explicitly consider trade offsbetween the freedom to determine and the freedom to exclude. I have suggested necessary conditions for any aggregation thatsolves this problem by taking the value of outcomes into account when they can be determined (or not) or when they can beexcluded (or not). To describe the importance of an outcome I use the notion of essentiality and define the notion of a “wantedoutcome”. I show, that for the case of trichotomous perceptions the axioms I propose uniquely characterize a ranking in terms ofprocedural freedom.

The general concept suggested here also enables us for some game forms and given perception functions to compare the degreeof procedural freedom of two individuals. For instance in a strategically symmetric game form, this comparison of freedom willdepend on the differences of perception of the outcomes and on interpersonal comparisons of perceived outcomes. The question,who is more free under a given procedure, if it is presented in the framework proposed here, is an interesting problem for furtherresearch. For a special class of perceptions that depend on certain guaranteedwelfare levels a solution to the problem of intra- andinterpersonal comparison is proposed in Ahlert (2008).

There are conceptual relations between the notion of procedural freedom in game forms, the notion of rights in game forms andthe notion of power in both theWeberian and the Hobbesian sense. To explore the connections between these notions seems to bea promising research prospect.

Acknowledgments

I thank Christian Aumann, Wulf Gaertner, Hartmut Kiemt, Bernd Lahno, Lars Schwettmann and the participants of theWorkshop on Freedom at the University of Bayreuth, February 2005, the Public Choice Meeting in New Orleans, March 2005 andthe LGS 4 Conference in Caen, June 2005, for helpful comments on earlier versions of this paper. I also thank two anonymousreferees for their helpful comments on the 2008 version of this paper.

References

Ahlert, M., 2008. Guarantees in game forms. In: Braham, M., Steffen, F. (Eds.), Power, Freedom and Voting: Conceptual. Formal and Applied Dimensions. SpringerBerlin, Heidelberg, pp. 316–332.

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A new approach to procedural freedom in game forms