REPRESENTATION OF CONSTITUTIONS

UNDER INCOMPLETE INFORMATION

By

BEZALEL PELEG and SHMUEL ZAMIR

Discussion Paper # 634 Jan 2013

האוניברסיטה העברית בירושליםTHE HEBREW UNIVERSITY OF JERUSALEM

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Representation of constitutions under incomplete information

Bezalel Peleg and Shmuel Zamir

Center for the Study of RationalityThe Hebrew University, Jerusalem, Israel

January 15, 2013

Abstract

We model constitutions by effectivity functions. We assumethat the constitutionis common knowledge among the members of the society. However, the preferencesof the citizen are private information. We investigate whether there exist decisionschemes (i. e., functions that map profiles of (dichotomous)preferences on the setof outcomes to lotteries on the set of social states), with the following properties:i) The distribution of power induced by the decision scheme is identical to the ef-fectivity function under consideration; and ii) the (incomplete information) gameassociated with the decision scheme has a Bayesian Nash equilibrium in pure strate-gies. If the effectivity function is monotonic and superadditive, then we find a classof decision schemes with the foregoing properties. When applied to n-person gamesin strategic form, a decision schemed is a mapping from profiles of (dichotomous)preferences on the set of pure strategy vectors to probability distributions over out-comes (or equivalently, over pure strategy vectors). We prove that for any feasibleand individually rational payoff vector of a strategic game, there exists a decisionscheme that yields that payoff vector as a (pure) Nash equilibrium payoff in the gameinduced by the strategic game and the decision scheme. This can be viewed as a kindof purification result.

introduction

Following Gardenfors (1981) and Peleg (1998) we model constitutions by effectivity func-tions. Formally, aneffectivity functionis the coalitional function of a game form, that is, acoalitional game form (see Abdou and Keiding (1991, p. 28)).We assume that the consti-tution is common knowledge among the members of the society.However, the preferencesof the members of the society over the set of social states areprivate information. In orderto enable the citizen to exercise their rights and comply with their obligations accordingto the constitution, we need some kind of a game form or mechanism to represent it (seePeleg (1998) and Peleg and Peters (2010)). In this paper we represent constitutions bydecision schemes, that is, functions from profiles of preferences of citizen to probabilitydistributions over the set of social states. A decision scheme is a representation of a con-stitution if the power distribution (among coalitions of players) induced by it coincideswith the constitution. A representation induces a Bayesiangame whose players are themembers of the society as we shall see in Section 2. We shall investigate various kinds

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of representations for which the induced incomplete information game possesses a pureBayesian Nash equilibria.

The following simple example may help the reader to become familiar with the fore-going concepts. It is a modification of a famous example of Gibbard (1974).

Example 1. Let N = {1,2} be a society. Assume that each member has two shirts,one white and one blue, and he must wear exactly one of them. Then there are foursocial states: ww,wb,bw and bb, where ww means that they both wear white shirtsetc. Assume further that each citizen can freely choose the color of his shirt. Thenthe constitution, that is the associated effectivity function E, is given by: E({1}) ={{ww,wb}+,{bw,bb}+}; E({2}) = {{ww,bw}+,{wb,bb}+}; and E(N) is the set of allnonempty subsets of A, where A= {ww,wb,bw,bb} is the set of all social states and forany B⊆ A, we denote by B+ the collection of all supersets of B. Assume now that player1 has two types1w and1b, and player2 believes that they have the same probability. LetW be the set of all complete and transitive (weak) orderings of A. A decision scheme isa function d: WN → ∆(A) where∆(A) is the set of all probability distributions on A. InSubsection 1.1 we shall find a representation for E. To complete our example we need tospecify von Neumann Morgenstern utility functions for1w, 1bb, and2. We shall do this inSection 2 and then compute a pure Bayesian Nash equilibrium for the induced Bayesiangame.

From a broad perspective our study belongs to the vast literature on the tension be-tween social welfare and the distribution of rights (see Suzumura (2011) for a compre-hensive survey of this field). On the one hand our approach allows for any reasonabledistribution of group rights. Thus we can avoid dictatorialdecision schemes. On the otherhand, because we insist on precise representation of group rights we might lose incentivecompatibility of some of our Bayesian Nash equilibria. (We prove existence of BayesianNash equilibria in pure strategies.) Thus, although our representing decision schemes areex-post Pareto optimal we might only obtain Pareto optimality with respect to reportedpreferences. Hence, we do not fully avoid Sen’s Paradox of the Paretian Liberal. Compar-ing with the results for the case of complete information where there is always at least onePareto optimal Nash equilibrium (for each profile of preferences), we conclude that thereis a price to pay for generalizing the model of Peleg (1998) and Peleg , Peters, and Stor-cken (2002) to allow incomplete information, namely, we might lose Pareto optimality ofthe resulting social state.

Our work is not the first one that investigates representations of power structures underincomplete information; d’Aspremont and Peleg (1988) studies representations of simplegames by decision schemes under the same assumptions of incomplete information..(Asimple game is an example of a neutral effectivity function.) There are two significantdifferences between the two papers: i) d’Aspremont and Peleg use a slightly strongernotion of representation; and ii) they focus on a stronger notion of equilibrium, namely,ordinal Bayesian (incentive compatible) equilibria.

We now review briefly the contents of the paper. Our model and some basic definitionsare introduced in the first half of Section 1. The rest of Section 1, Subsection 1.1, is

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devoted to recalling some results on the uniform core (due toAbdou and Keiding (1991)).The uniform core of an effectivity function plays an important role throughout our paper.In section 2 we consider a society whose members have incomplete information on eachother’s preferences. We prove, under mild restrictions, that the constitution of the societycan be represented by a decision scheme such that the resulting game (of incompleteinformation) has a Bayesian Nash equilibrium in pure strategies. We further show thatthe decision scheme may be chosen to be ex-post Pareto optimal and that we may restrictourselves to dichotomous preferences. Section 3 is devotedto various examples, mainlywith complete information. For neutral effectivity functions we give a simple algorithmfrom Peleg and Peters (2010). We also discuss in some detail the class of finite strategicgames. In particular, the solution of the Prisoners’ Dilemma is displayed in Figure 1. Ourmethod that yields equilibria in pure strategies provides anew approach to the problem ofpurification in game theory.

1 The model

• Let N = {1,2, . . . ,n} be the set ofplayers(voters).

• Let A= {a1,a2, . . . ,am} be the set ofalternatives, m≥ 2.

• For a finite setD let P(D) = {D′|D′ ⊆ D} andP0(D) = P(D)\{ /0}.

An effectivity function(EF) is a functionE : P(N)→ P(P0(A)) satisfying:(i) A∈ E(S) for all S∈ P0(N), (ii) E( /0) = /0, and (iii)E(N) = P0(A).

An effectivity functionE is monotonicif:

[S∈ P0(N), S′ ⊇ S, andB′ ⊇ B, B∈ E(S)]⇒ B′ ∈ E(S′).

An effectivity functionE is superadditiveif:

[Bi ∈ E(Si), i = 1,2, andS1∩S2 = /0]⇒ B1∩B2 ∈ E(S1∪S2).

A social choice correspondence(SCC) is a functionH : WN → P0(A) whereW is theset ofweak(i.e., complete and transitive ) orderings ofA.

Let H : WN → P0(A) be an SCC. A coalitionS∈ P0(N) is effectivefor B ∈ P0(A) ifthere existsQS ∈ WS such that for allRN\S ∈ WN\S, H(QS,RN\S) ⊆ B. The EF ofH,denoted byEH , is given byEH( /0) = /0 and forS∈ P0(N),

EH(S) = {B∈ P0(A)|S is effective forB}.

We assume thatH satisfies: For allx∈A there existsRN ∈WN such thatH(RN) = {x}.Thus ,EH is indeed an EF.

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Remark 1. The definition of EH is valid for any restricted domain SCC, H: W̃N → P0(A)whereW̃ is any nonempty subset of W (that satisfies some mild conditions).

Definition 1. A social choice correspondence H is arepresentationof the effectivity func-tion E if EH = E.

For a finite setD denote by∆(D) the set of all probability distributions onD.

A decision scheme(DS) is a functiond : WN → ∆(A). With a decision schemed weassociate an SCC which we denote byHd and define by:

Hd(RN) = {x∈ A|d(x;RN)> 0}.

A decision schemed is said to be arepresentationof the effectivity functionE ifEHd =E. A decision schemed is said to bederivedfrom the social choice correspondenceH if Hd(RN) = H(RN) for all RN ∈WN.

1.1 The uniform core

The notion ofuniform corewill play an important role in our analysis. LetE : P(N) →P(P0(A)) be a monotonic and superadditive EF. For any weak preferencerelation onA,R∈W, we denote the strict preference byP, that is,xPyholds forx,y∈ A if xRyand notyRx, and the indifference relation by 1, that is,xIy holds forx,y∈ A if xRyandyRx. Givena vector of preference relationsRN and a coalitionS⊆ N, we writeB PS(A\B) if xPiy forall x∈ B, y∈ A\B andi ∈ S.

ForRN ∈WN and an effectivity functionE we define theuniform coreof E andRN asfollows.

Definition 2. Given an effectivity function E and a vector of preference relations RN,

• An alternative x∈ A is uniformly dominatedby B⊆ A, x 6∈ B via the coalitionS∈ P0(N), if B∈ E(S) and B PS(A\B).

• An alternative x∈ A isnot uniformly dominatedat (E,RN) if there is no pair(S,B)of coalition S∈ P0(N) and a set of states B not containing x that uniformly domi-nates x via the coalition S.

• Theuniform coreof (E,RN), denoted by Cu f (E,RN), is the set of all alternatives inA that are not uniformly dominated at(E,RN).

When the underlying effectivity functionE is fixed, we write shortlyCu f(RN) insteadof Cu f(E,RN).

This notion is to be compared to the notion of thecore of an effectivity functiondefined as follows,

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Definition 3. Given an effectivity function E and a vector of preference relations RN,

• An alternative x∈ A is dominatedby B⊆ A, x 6∈ B via the coalition S∈ P0(N), ifB∈ E(S) and B PS{x}.

• An alternative x∈ A is not dominatedat (E,RN) if there is no pair(S,B) of acoalition S∈ P0(N) and a set of states B not containing x that dominates x via thecoalition S.

• Thecoreof (E,RN), denoted by C(E,RN), is the set of all alternatives in A that arenot dominated at(E,RN).

It follows from the definitions that the core is a subset of theuniform core. In thefollowing example, based on the Condorcet Paradox, the coreis empty while the uniformcore is not.

Example 2. Let N= {1,2,3}, A= {x,y,z} and the effectivity function E given by:

E(S) =

{P0(A) if |S|> 1{A} if |S|= 1

For the vector of preference relations:

RN =

1 2 3x z yy x zz y x

At (E,RN) every alternative is dominated but not uniformly dominated. Hence,C(E,RN) = /0 while Cu f(E,RN) = A.

Given a preference profileRN = (R1, . . . ,Rn) and a coalitionS⊆ N we denote byQN = (RS, IN\S) the preference profile in whichQi = Ri for i ∈ SandRi = I for i ∈ N\S,whereI is the total indifference relation onA, that isxIy for all x,y∈ A.

Remark 2. For any RN and for any S⊆ N we have Cu f(RN)⊆Cu f(RS, IN\S).

Indeed, since uniform domination is defined via strict preference, replacing a strictpreference of a player by indifference reduces (weakly) uniform dominance and henceincreases (weakly) the uniform core.

As stated in the following theorem, for a monotone and superadditive effectivity func-tion, the uniform core is always non-empty.

Theorem 1. (Abdou and Keiding (1991)). Let E be a monotonic and superadditive EFand let RN ∈WN. Then the uniform core Cu f(E,RN) is non-empty.

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Corollary 2. For any monotonic and superadditive effectivity function E, the uniform coreCu f(E, ·) : WN → P0(A) is a social choice correspondence.

The following result is strongly used in this paper.

Theorem 3. (Peleg and Peters (2010)). Let E be a monotonic and superadditive EF.Then the social choice correspondence Cu f(E, ·) is a representation of E that is ECu f = E.

Corollary 4. Given a monotonic and superadditive effectivity function E, then any deci-sion scheme d whose support is the uniform core (i.e.{x∈ A|d(x;RN)> 0}=Cu f(E,RN)for all RN ∈WN), is a representation of E. In particular, any monotonic andsuperadditiveeffectivity function has a representation by a decision scheme. For example, the decisionscheme denoted by du f and defined by du f (x;RN) = 1/|Cu f(E,RN)| for x ∈ Cu f(E,RN)and du f(x;RN) = 0 otherwise, which will be calledthe uniform representationof E.

1.2 Example 1 continued

Using Theorem 3, we know thatCu f(E, ·) is a representation ofE by a social choicecorrespondence. This can be easily converted into a representation by a decision scheme(of the same effectivity function) by assigning the uniformdistribution onCu f (E,RN) foreachRN ∈ WN. For example, letR1 = (ww,wb,bw,bb) andR2 = (bw,wb,ww,bb). Asit can be easily verifiedCu f(E,RN) = {ww,wb} and hence the uniform decision schemerepresentingE satisfiesd(ww,RN) = d(wb,RN) = 1/2.

2 Bayes-Nash equilibrium representation

A decision schemed applied to a situation of a collective choice of a social state, inducesa game in which each member of the society (player), endowed with a von-NeumannMorgenstern utility function on∆(A), chooses a preference relation and the final state ischosen (randomly) according to the decision schemed. When a player may have incom-plete information about the preferences of the other players, this is a game of incompleteinformation. The question addressed in this paper is:

Given a monotone and superadditive effectivity function E,can it be represented bya decision scheme so that the induced game of incomplete information has a Bayes Nashequilibrium in pure strategies ?

We provide an affirmative answer to this question. For the sake of the presentation wewill first state and prove the result for the situation of complete information and then stateand prove the more general result for the incomplete information situation.

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2.1 The complete information case

Given a societyN= {1,2, . . . ,n}, a set of social statesA= {a1,a2, . . . ,am}, and effectivityfunctionE, a utility function of playeri is a von-Neumann Morgenstern utility functionon ∆(A), induced byui : A → R. For any decision schemed : WN → ∆(A) consider thestrategic form gameΓd = (N;W, . . . ,W;u1, . . . ,un;d). Our objective is to find a decisionschemed representing the effectivity functionE such that the gameΓd has a NE in purestrategies. We illustrate this in the following example.

Example 3(Neutral effectivity functions).

Aveto functionis a function v: P(N)→{−1,0, . . .,m−1} such that v( /0)=−1, v(S)≥0 if S 6= /0 and v(N) = m− 1. The interpretation is that a nonempty set of players S,can veto any set of at most v(S) alternatives. A veto function v defines a neutral EF,Ev : P(N)→ P(P0(A)) by Ev( /0) = /0 and Ev(S) = {B|v(S)≥ m−|B|} for S 6= /0. That is,B∈ Ev(S) if the coalition S can veto all the alternatives in A\B. This effectivity functionis neutral (with respect to the alternatives) as Ev(S) does not depend on the names of thealternatives. We remark that Ev is monotonic (superadditive) if and only if v is monotonic(superadditive). We assume complete information. Thus, the specification of the model iscompleted by an n-tuple of utility functions for the players: ui : A→R, i = 1, . . . ,n.

Let E : P(N) → P(P0(A)) be a monotonic , superadditive, and neutral EF. Letv :P(N)→ {−1,0, . . . ,m−1} be the veto function ofE and letRN ∈ WN. Sincere vetoingwith respect tov andRN (in the natural ordering of the players) is as follows: Player 1vetoesv(1) of his worst alternatives; next, player 2 vetoesv(2) of his worst alternativesin the remaining set of alternatives and so forth. By superadditivity, v(1)+ . . .+ v(n) ≤v(N) = m−1 hence there is always a non-empty set of remaining alternatives. Clearly,this could be done with any other ordering of the players.

By Peleg and Peters (2010, Theorem 6.4.4), for any preference ordering profileRN,sincere vetoing (with respect to any ordering of the players) is a Nash equilibrium for theuniform decision schemedu f (see Corollary 4). The following is an explicit example.

Let N = {1,2,3} and A= {a,b,c,d,e}. Let the veto functionv be specified byv(i) =1, v(i, j) = 2 andv(N) = 4. Let the preferences be defined through the utility functions:

u1(a)> u1(b)> u1(c)> u1(d)> u1(e),u2(e)> u2(d)> u2(c)> u2(b)> u2(a), andu3(a)> u3(e)> u3(b)> u3(d)> u3(c).

Vetoing sincerely amounts to presenting the following dichotomous preferences:

R1 =a b c d

eR2 =

b c d ea

R3 =d b e a

c

The vector of preference orderingsRN = (R1,R2,R3) is indeed the desired NE (notethatCu f(RN) = {b,d}.)

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The general result of this kind is given by the following theorem.

Theorem 5. Given a monotonic and superadditive effectivity function E, and vNM utilityfunctions(u1, . . . ,un), then there is a decision scheme d such that,

• The decision scheme d is a representation of the effectivityfunction E.

• The gameΓd = (N;W, . . . ,W;u1, . . . ,un;d) has a Nash equilibrium in pure strate-gies.

Proof. Let du f : WN → ∆(A) be uniform core representation ofE defined in Corollary 4.Letq=(q(s))s∈Sbe correlated equilibrium of the gameΓdu f =(N;W, . . . ,W;u1, . . . ,un;du f)(whereS=W×, . . . ,×W denotes the set of pure strategy vectors in this game)). Then,

∑s∈S

q(s) ∑x∈A

ui(x)du f(x;s)≥ ∑s∈S

q(s) ∑x∈A

ui(x)du f(x;(s−i ,Ri)), (1)

holds for alli ∈ N andRi ∈W.

Define now a new decision schemed by:

1. d(x; I , . . . , I) = ∑s∈Sq(s)du f(x;s), ∀x∈ A.

2. d(x;

i−1︷ ︸︸ ︷

I , . . . , I ,Ri , I , . . . , I) = ∑s∈Sq(s)du f(x;(s−i,Ri)),

∀x∈ A, ∀i ∈ N, ∀Ri ∈W.

3. Otherwise, for any other vector of preferencesRN ∈ WN and anyx ∈ A defined(x;RN) = du f(x;RN).

Corollary 4 and part 3 in the definition ofd guarantee that for each coalitionS, theeffectivity function ofd containsE(S). In order to prove thatd is a representation ofE allwe have to show is that part 2 in the definition ofd does not give extra power (w.r.t.d) toN \ {i} for any playeri. SupposeN \ {i} is not effective forB (according toE), then bypart 3 of the definition ofd, N\{i} is not effective forB via any strategy vector differentform I−i . It remains to see thatN \ {i} cannot guarantee an outcome inB by the strategyvectorI−i. Indeed, choose a strategy vectors such thatq(s) > 0, then choosex 6∈ B andRi ∈W such thatdu f(x;(s−i ,Ri))> 0 (suchx andRi exist sinceN\{i} is not effective forB according toE). Then, part 2 in the definition ofd implies thatd(x;(I−i,Ri)) > 0 andthus,N\{i} is not effective forB w.r.t. d via I−i.

Inequalities (1) imply that the pure strategy vector(I , . . . , I) is a Nash equilibrium inthe gameΓd. Indeed, for any deviationRi ∈W of playeri,

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∑x∈A

ui(x)d(x; I , . . . , I) = ∑x∈A

ui(x)∑s∈S

q(s)d1(x;s)

= ∑s∈S

q(s) ∑x∈A

ui(x)d1(x;s)

≥ ∑s∈S

q(s) ∑x∈A

ui(x)d1(x;s−i ,Ri)

= ∑x∈A

ui(x)∑s∈S

q(s)d1(x;s−i ,Ri)

= ∑x∈A

ui(x)d(x;

i−1︷ ︸︸ ︷

I , . . . , I ,Ri, I , . . . , I)

2.2 The incomplete Information case: Main result

An information structure(IS) is a 2n-tupleI = (T1, . . . ,Tn; p1, . . . , pn) where for eachi ∈ N, T i is the (finite) set oftypesof player i, and pi is a probability distribution onT =×i∈NT i such thatpi(t i = t i

0)> 0 for all t i0 ∈ T i . This is theprior distributionof player

i on the set of typsT, which induces the conditional probability distributionpi(t−i|t i) onT−i = × j 6=iT j (the beliefs of playeri of type t i on the types of the other players). In aHarsanyi-consistent information structure there is acommon priornamely,pi = p, for alli ∈ N.

We now modify the notion of decision scheme so as to adapt it tothe context ofincomplete information.

Definition 4.1. A generalized decision scheme (GDS) is a function d: WN ×T → ∆(A).

2. A strategy of player i (with respect to a GDS) is a pair(si,π i) where si : T i → W(denote by Si the set of all such mappings, let S= S1×, · · · ,×Sn) andπ i : T i → T i .Equivalently, a strategy of player i is a mappings̃i : T i →W×T i . Denote byS̃i theset of pure strategies of player i and byS̃= S̃1×·· ·× S̃n the set of vectors of purestrategies. A vector̃s∈ S̃ will also be written as̃s= (s,π) where s= (s1, . . . ,sn) ∈ Sandπ = (π1, . . . ,πn).

The idea behind this definition is that in a situation of incomplete information, eachplayer is asked to input to the (generalized) decision scheme, both his preferences andhis type. As a result, the (Bayes Nash) equilibrium of the induced game will exhibit the‘spirit’ of the revelation principle, as each player will input his true type.

Any generalized decision schemes (GDS) induces an effectivity functions in a similarway that a DS does. Letd : WN×T → ∆(A) be a GDS. The associated (generalized) SCC,H : WN ×T → P0(A) is defined byH(RN, t) = {x∈ A|d(x;RN, t)> 0}.

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A coalitionSis effective for a non-empty subsetB of A (w.r.t. H) if there existRS∈WS

andtS∈ TS such thatH(RS,QN\S, tS, rN\S)⊆B for all QN\S∈WN\S andrN\S∈ TN\S. Theeffectivity function ofH is defined byEH(S) = {B|S is effective forB}. The effectivityfunction of d is defined to be that ofH. The generalized decision schemed is a repre-sentation of a given (monotonic and superadditive) effectivity functionE if the effectivityfunction ofd equalsE.

Let I = (T1, . . . ,Tn; p1, . . . , pn) be an IS and let(ui)i∈N whereui : A×T →R, be thevector of utility functions of the players. Then, a generalized decision schemed defines aBayesian game of incomplete information:

Γd = (N;W, . . . ,W;I ;u1, . . . ,un;d).

This is the strategic form game in which:

• The set of actions of playeri ∈ N of any possible typet i is W×T i . The set of purestrategies of playeri is S̃i.

• The payoff to typet i when the players play the pure strategy vector ˜s=(s̃1, . . . , s̃n)∈ S̃isU i(s̃|t i) given by:

U id(s̃|t

i) = ∑t−i∈T−i

pi(t−i|t i) ∑x∈A

ui(x; t)d(x; s̃1(t1), . . . , s̃n(tn)). (2)

Whend(·;RN, t) does not depend onπ , the expected utilityU id(s̃|t

i) also does notdepend onπ and we write:

U id(s|t

i) = ∑t−i∈T−i

pi(t−i|t i) ∑x∈A

ui(x; t)d(x;s1(t1), . . . ,sn(tn)). (3)

As expected, the dependence ofU id(s|t

i) onsi is only viasi(t i) and it does not dependonsi(t̂ i) for t̂ i 6= t i.

An n-tuple of strategies ˜s is aBayesian Nash equilibrium(BNE) if for all i ∈ N, all t i ∈ T i

and all(Ri, t̂ i) ∈W×T i ,

∑t−i∈T−i

pi(t−i|t i) ∑x∈A

ui(x; t)d(x; s̃(t))≥ ∑t−i∈T−i

pi(t−i|t i) ∑x∈A

ui(x; t)d((x; s̃−i(t−i),(Ri, t̂ i))),

(4)wheres̃(t) is the vector(s̃i(t i))i∈N ands̃−i(t−i) is the vector(s̃j(t j)) j 6=i .

Theorem 6. Let E : P(N) → P(P0(A)) be a monotonic and superadditive EF. LetI =(T1, . . . ,Tn; p1, . . . , pn) be an IS, and let(u1, . . . ,un), ui : A×T →R, be a vector of vNMutilities for the players. Then E has a representation by a generalized decision schemed : WN ×T → ∆(A) such that the gameΓd = (N;W, . . . ,W;I ;(ui)i∈N;d) has a BNE inpure strategies in which each player reports his true type.

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Proof. Define the generalized decision schemed1 : WN ×T → ∆(A) by

d1(RN, t) = du f(R

N), ∀(RN, t) ∈WN ×T.

As d1(RN, t) depends only onRN, by slight abuse of notation, we shall also writed1(RN)instead ofd1(RN, t). Consider now the ex-ante game:

Gd1 = (N;S1, . . . ,Sn;h1, . . . ,hn;d1) (5)

in which the payoff functions are:

hi(s1, . . . ,sn) = ∑t∈T

pi(t) ∑x∈A

ui(x, t)d1(x;s(t)), (6)

which can be written as (by (3) asd1 does not depend onπ):

hi(s1, . . . ,sn) = ∑t i∈T i

pi(t i) ∑t−i∈T−i

pi(t−i|t i) ∑x∈A

ui(x, t)d1(x;s(t)) = ∑t i∈T i

pi(t i)U id(s|t

i). (7)

Note that in this game, the strategy sets areSi rather thanS̃i sinced1(RN, t) doesnot depend ont. This is a finite game with complete information, so it has a correlatedequilibrium (CE). Let(q(s))s∈S be a CE of the gameGd1, then the equilibrium conditionis:

∑s∈S

q(s)hi(s)≥ ∑s∈S

q(s)hi(s−i ,δ i(si)), (8)

which holds for alli ∈ N and for allδ i : Si → Si . Substitutinghi from (7) we have:

∑s∈S

q(s) ∑t̂ i∈T i

pi(t̂ i)U id1(s|t̂ i)≥ ∑

s∈S

q(s) ∑t̂ i∈T i

pi(t̂ i)U id1(s−i,δ i(si)|t̂ i),

which we rewrite as:

∑t̂ i∈T i

pi(t̂ i)∑s∈S

q(s)U id1(s|t̂ i)≥ ∑

t̂ i∈T i

pi(t̂ i)∑s∈S

q(s)U id1(s−i,δ i(si)|t̂ i). (9)

For t i ∈ T i let δ i : Si → Si be defined as follows:

• δ i(si)(t i) = Ri ∈W, ∀si ∈ Si .

• δ i(si)(t̂ i) = si(t̂ i), if t̂ i 6= t i, for all si ∈ Si .

Inserting thisδ i in (9) all terms witht̂ i 6= t i will be the same on both sides of theinequality and will cancel, dividing the remaining term bypi(t i) (which is positive) weobtain that:

∑s∈S

q(s)U id1(s|t i)≥ ∑

s∈S

q(s)U id1(s−i ,Ri|t i) (10)

holds for alli ∈ N, t i ∈ T i andRi ∈W.

For t i ∈ T i andt̃ i ∈ T i let δ̃ i : Si → Si be defined as follows:

11

• δ̃ i(si)(t i) = si(t̃ i), ∀si ∈ Si.

• δ̃ i(si)(t̂ i) = si(t̂ i), if t̂ i 6= t i, for all si ∈ Si .

Inserting thisδ̃ i in (9) all terms witht̂ i 6= t i will be the same on both sides of theinequality and will cancel, dividing the remaining term bypi(t i) (which is positive) weobtain that:

∑s∈S

q(s)U id1(s|t i)≥ ∑

s∈S

q(s)U id1(s−i ,si(t̃ i)|t i) (11)

holds for alli ∈ N and for allt i andt̃ i in T i .

Define now a generalized decision schemed by:

d(x; IN, t) = ∑s∈S

q(s)d1(x;s(t)), ∀x∈ A, ∀t ∈ T. (12)

d(x;(I−i,Ri), t) = ∑s∈S

q(s)d1(x;s−i(t−i),Ri), (13)

∀i ∈ N, Ri ∈W, Ri 6= I , t ∈ T, x∈ A.

d(x;RN, t) = du f (x;RN) otherwise. (14)

We first claim thatd is a representation of the effectivity functionE. The idea of theproof is the same as in the proof of Theorem 5: By Corollary 4 and (14) of the definitionof d, the effectivity function ofd is at least as rich asE. We have to show that (13) doesnot give extra power, w.r.t.d, to N \ {i} for everyi ∈ N. SupposeN \ {i} is not effectivefor B (according toE), then by part (14) of the definition ofd, N\{i} is not effective forBvia any strategy vector different formI−i. It remains to see thatN \ {i} cannot guaranteean outcome inB by the strategy vectorI−i. Indeed, choose a strategy vectors such thatq(s)> 0, then for everyt ∈ T choosex 6∈ B andRi ∈W such thatdu f(x;(s−i(t−i),Ri))> 0(suchx andRi exist sinceN\{i} is not effective forB according toE). Then, part (13) inthe definition ofd implies thatd(x;(I−i,Ri , t)) > 0 and thus, at anyt ∈ T, N \ {i} is noteffective forB w.r.t. d via I−i .

We next claim that the pure strategy vector ˜s in which s̃i(t i) = (I , t i), for all i ∈ N andfor all t i ∈ T i , is a BNE of the gameΓd = (N;W, . . . ,W;I ;(ui)i∈N;d) that is, inequalities(4) are satisfied for anyt i ∈ T i , s̃i(t i) = (I , t i) and any deviation to(Ri , t̂ i). To do this weshall treat each of the three possible deviations:

(i) Deviation from(I , t i) to (Ri, t i) with Ri 6= I .

(ii) Deviation from(I , t i) to (I , t̃ i) with t̃ i 6= t i.

(iii) Deviation from (I , t i) to (Ri, t̃ i) with Ri 6= I andt̃ i 6= t i.

12

Case (i). Substituting ˜s= (IN, t) in the left hand side of (4) andd from(12) we have:

∑t−i∈T−i

pi(t−i|t i) ∑x∈A

ui(x; t)d(x; IN, t) = ∑t−i∈T−i

pi(t−i|t i) ∑x∈A

ui(x; t)∑s∈S

q(s)d1(x;s(t))

= ∑s∈S

q(s) ∑t−i∈T−i

pi(t−i|t i) ∑x∈A

ui(x; t)d1(x;s(t))

by (3) = ∑s∈S

q(s)U id1(s|t i)

by (10) ≥ ∑s∈S

q(s)U id1(s−i,Ri |t i)

by (3) = ∑s∈S

q(s) ∑t−i∈T−i

pi(t−i|t i) ∑x∈A

ui(x; t)d1(x;s−i(t−i),Ri)

by (13) = ∑t−i∈T−i

pi(t−i|t i) ∑x∈A

ui(x; t)d(x;(I−i,Ri), t).

Case (ii). Substituting ˜s= (IN, t) in the left hand side of (4) andd from(12) we have:

∑t−i∈T−i

pi(t−i|t i) ∑x∈A

ui(x; t)d(x; IN, t) = ∑t−i∈T−i

pi(t−i|t i) ∑x∈A

ui(x; t)∑s∈S

q(s)d1(x;s(t))

= ∑s∈S

q(s) ∑t−i∈T−i

pi(t−i|t i) ∑x∈A

ui(x; t)d1(x;s(t))

by (3) = ∑s∈S

q(s)U id1(s|t i)

by (11) ≥ ∑s∈S

q(s)U id1(s−i,si(t̃ i)|t i)

by (3) = ∑s∈S

q(s) ∑t−i∈T−i

pi(t−i|t i) ∑x∈A

ui(x; t)d1(x;s−i(t−i),si(t̃ i))

by (12) = ∑t−i∈T−i

pi(t−i|t i) ∑x∈A

ui(x; t)d(x; IN,(t−i, t̃ i)).

Case (iii). This case follows from case (i) since by (13), forRi 6= I :

d(x;(I−i,Ri),(t−i, t̃ i))=d(x;(I−i,Ri), t)= ∑s∈S

q(s)d1(x;s−i(t−i),Ri), ∀i ∈N, Ri ∈W, t ∈T, x∈A.

2.3 Bayesian incentive compatibility

In this section we reconsider the gameΓd =(N;W, , . . . ,W;T1, . . . ,Tn; p1, . . . , pn;u1, . . . ,un;d)introduced in the previous section. We assume, as in d’Aspremont and Peleg (1988), that

13

the types of the players include (explicitly) information on their ordinal preferences on thesetA of alternatives. More precisely, we assume that for every playeri in N, every typet i ∈ T i is of the formt i = (Ri,τ i), whereRi is the ordinal preferences oft i andτ i representsthe rest of the characteristics oft i. This imposes the following constraints on the utilityfunctions:ui(· ,(t i, t−i)) must be a faithful representation ofRi wheret i = (Ri ,τ i). Thuswe are able to define Bayesian incentive compatibility in ourmodel.

Definition 5. . Consider the gameΓ = (N;W, , . . . ,W;T1, . . . ,Tn; p1, . . . , pn;u1, . . . ,un;d).The generalized decision scheme d isBayesian incentive compatible(BIC) if truth-tellingis a BNE of the gameΓ. That is, the n-tuple of strategiess̃0 = (s̃1

0, . . . , s̃n0), wheres̃i

0(ti) =

(Ri, t i) when ti = (Ri,τ i), for all t i ∈ T i and i∈ N, is a BNE ofΓ.

Unfortunately, there exist robust examples that possess no‘nice’ (in a sense to beexplained later) BIC solutions even in the complete information case as the followingexample shows.

Example 2 continued.

To Example 2 add the utility functions: Forδ > 0 and23(1+δ )< 1 let

u1(x) = u2(z) = u3(y) = 1+δ . (15)

u1(y) = u2(x) = u3(z) = 1. (16)

u1(z) = u2(y) = u3(x) = 0. (17)

If we consider a decision schemed : WN → ∆(A) to be a ‘nice’ representation ofE ifit satisfies the Condorcet condition1(CC), then we claim thatE has no BIC representationthat satisfies CC. Assume, on the contrary thatd : WN → ∆(A) is a representation ofE satisfying CC andRN is a Nash equilibrium of the gameGd(N;W,W,W;u1,u2,u3;d).Without loss of generality assume thatd(z;RN)≥ 1

3. Then:

∑a∈A

u1(a)d(a;RN)≤23(1+δ ).

If player 1 deviates toQ1 = R3 thend(y;Q1,R2,R3) = 1 sinced is a representation ofEthat satisfies CC. Hence this is a profitable deviation from truth telling since

∑a∈A

u1(a)d(a;R3,R2,R3) = 1>23(1+δ ).

Continuing our discussion of Example 2 we consider now the well known Borda rule.This is an SCC in which the states inA are ranked by each player, in our case 2 (best), 1

1A representationd of E satisfies the CC if for allRN ∈WN, if c∈ A beats everyb∈ A\ {c} by simplemajority rule, thend(c,RN) = 1.

14

(middle) or 0 (worst), and the chosen states are those with the maximal total score. For theprofile of preferences in our example, each state scores 3 andhence all states are chosenaccording to the Borda rule. This is a representation ofE (the simple majority rule) sinceno player is effective for any proper subset of{x,y,z} and any two players can force anystate, sayx by submitting the preferences(x,y,z) and(x,z,y) (thus guaranteeing a scoreof at least 4 forx and at most 3 for each ofy andz).

The Borda rule does not satisfy the CC. To see this consider the profile of preferences:

RN =

1 2 3x x yy y zz z x

The Borda rule chooses{x,y} althoughx is the Condorcet winner. Thus, the possibilitythat a decision scheme representing the Borda rule is a BIC isnot excluded by the aboveproved claim. However, nevertheless, no decision scheme representing the Borna rule isnot BIC. To see this, consider the original profileRN in our example:

RN =

1 2 3x z yy x zz y x

As we saw, the Borda rule selects{x,y,z} and hence any decision schemed representingit is of the formd(x;RN) = p1, d(y;RN) = p2, d(z;RN) = p3, wherep1+ p2+ p3 = 1. Atleat one state is chosen with probability 1/3 sayd(z;RN)≥ 1/3. With the utility functionsgiven in (15)–(17), the utility of player 1 isp1(1+δ )+ p2. By presenting the preference(y,x,z), player 1 guarantees utility 1 and this is a profitable deviation since:

p1(1+δ )+ p2 < (p1+ p2)(1+δ)≤23(1+δ )< 1.

Finally, we remark that there exist BIC representations of the effectivity functionE in ourexample (which necessarily are not ‘nice’). Letd : WN → ∆(A) satisfyd(a;QN) = 1 forall QN of the form:

QN =

S N\Sa a QN\S

bc bc,

where|S|= 2, a∈ {x,y,z} and{b,c}= {x,y,z}\{a}, andd(·;QN) = (13,

13,

13) otherwise.

Then,d is a representation ofE and the true preference profileRN is a NE of the gameΓ = (N;W,W,W;u1,u2,u3;d). This does not contradict the result of d’Aspremont andPeleg (1988) as their definition of representation is stronger than ours. Note also that thisdecision scheme is not ‘nice’: First, it clearly does not satisfies the CC; it does not choose

(with certainty) the Condorcet winnerx in the profileRN =

1 2 3x x yy z zz y x

15

Second, it is not monotonic: By improving the position of z,

from

1 2 3x x yy yz zz x

to

1 2 3x x yyz yz z

x

,

the probability ofz decreases from 1/3 to 0.

2.4 Ex-post Pareto optimality of representations by decision schemes

We now investigate the possibility that our construction isPareto optimal in some sense.First we need the following definition.

Definition 6. A generalized decision scheme d: WN×T → ∆(A) is Pareto optimal ex-postif the following condition is satisfied:

[RN ∈WN andx∈ A is not Pareto optimal w. r. t.RN]⇒ d(x;RN) = 0.

It is possible to strengthen Theorem 6 by demanding that the solutiond is also Paretooptimal ex-post. More precisely, the following result is true.

Theorem 7. Let E be a monotonic and supperadditive effectivity function, let I be aninformation structure and let u1, . . . ,un be the utility functions of the players. Then E hasa representation by a Pareto optimal ex-post generalized decision scheme d, such thatthe gameΓ = (N;W, . . . ,W;I ;u1, . . . ,un;d) has a BNE in pure strategies in which eachplayer reports his true type.

Proof. We begin the proof of the theorem with some preliminary remarks. Let E bea monotonic and superadditive effectivity function. Then for everyRN ∈ WN the setH(RN) = PAR(RN)∩Cu f (E,RN) is nonempty (herePAR(RN) is the set of Pareto opti-mal alternatives inA w.r.t. RN). Indeed, ifx∈Cu f (E,RN) andy∈ A satisfiesyPix for alli ∈ N, theny∈Cu f(E,RN).Our second claim is thatEH = E, which we deduce from Theorem 3 as follows: SinceH(RN)⊆Cu f(RN) for all RN ∈WN, it follows from Theorem 3 thatEH(S)⊇ E(S) for allsubsetsS⊆ N. To prove the converse inclusion letS∈ P0(N) andB ∈ EH(S) . Thenthere existsRS ∈ WS such thatH(RS,QN\S) ⊆ B for all QN\S ∈ WN\S. In particular,H(RS, IN\S) ⊆ B. By definition,EH(N) = E(N), hence we may assume thatS 6= N. Thisimplies thatPAR(RS, IN\S) = A. Therefore,H(RS, IN\S) = Cu f (RS, IN\S) ⊆ B. This im-plies, by Theorem 3 thatB ∈ E(S). In order to prove our theorem, it remains now torepeat the proof of Theorem 6 withCu f(E,RN) replaced byH(RN).

Remark that if the decision scheme of the last theorem is alsoBIC, then the finaloutcome is always Pareto optimal.

16

2.5 Dichotomous preferences

In this subsection we prove a variant of Theorem 6. For that, we first define a subset ofWas follows:

Definition 7. A preference relation R∈ W is dichotomousif there exist B1,B2 ∈ P(A)such that B1 6= /0,B1∩B2 = /0 and B1∪B2 = A such that xIy if x,y∈ Bi, i = 1,2 and xPyif x ∈ B1, y∈ B2. The set of all dichotomous preferences in W is denoted by Wδ .

Since a dichotomous preference relation is determined by a single subsetB⊆A, the set

of most preferred alternatives, we use the notationR=B

A\Bfor a generic dichotomous

preference relation.

Remark 3. Let RN = (R1, . . . ,Rn) ∈WNδ and let A\Cu f(E,RN) = {x1, . . . ,xk}.

A profile of preference orderings RN is regularif (i) There exist disjoint coalitions S1, . . . ,Sk

and sets B1, . . . ,Bk ∈ P0(A) such that xj is uniformly dominated by Bj via coalition Sj atRN for j = 1, . . . ,k and(ii) Ri = I for all i ∈ N\

⋃kj=1Sj .

Denote the set of regular dichotomous preferences profiles by WNδ . When Ri =

BA\B

we say that player (voter) iapprovesof B over A\B. If RN ∈ WNδ then, in the foregoing

notations, Cu f (E,RN) =⋂k

j=1B j and, thus the uniform core Cu f(E,RN) coincides with theset of alternatives with maximum approval score. This may beuseful for the computationof uniform core in some cases.

Lemma 1. The social choice correspondence H: WNδ → P0(A) defined by H(RN) =

Cu f(E,RN) for all RN ∈WNδ , is a representation of E.

Proof. We first prove the following claim: IfRN ∈ WN then there existsRN1 ∈ WN

δ suchthatCu f(E,RN) =Cu f(E,RN

1 ) = H(RN1 ). That is, for any profile of weak preferences onA

there exists a profile of dichotomous preferences with the same uniform core.

To see that, letA\Cu f(RN)= {x1, . . . ,xk}. By Abdou and Keiding (1991, p. 145) thereexist disjoint coalitionsS1, . . . ,Sk and setsB1, . . . ,Bk ∈ P0(A)such that forj = 1, . . . ,k, theoutcomex j is uniformly dominated byB j via Sj at RN. Define nowRN

1 as follows:

• For j = 1, . . . ,k and fori ∈ Sj ; xI i1y if x,y∈ B j or x,y∈ A\B j .

• For j = 1, . . . ,k let B jPSj1 A\B j .

• For i ∈ N\∪ jSj let xI i1y for all x,y∈ A.

It follows readily from the definition thatRN1 ∈WN

δ and thatCu f(E,RN) =Cu f(E,RN1 ) =

H(RN1 ).

17

We now prove thatEH = ECu f . By Theorem 3 this will complete the proof of thelemma. LetS∈P0(N) andB∈EH(S). Then there existsRS∈WS

δ such thatH(RS,QN\S)⊆

B for all QN\S∈WN\Sδ . In particularH(RS, IN\S)⊆ B and, by Remark 2,H(RS,QN\S)⊆ B

for all QN\S ∈ WN\S, implying B ∈ ECu f (S). Thus,EH(S) ⊆ ECu f (S) (in the usual setinclusion sense:B∈ EH(S)⇒ B∈ ECu f (S)), for all S∈ P0(N).

In the other direction, letS∈ P0(N) andB∈ ECu f (S). Then there existsRS∈WS suchthatCu f(RS,QN\S) ⊆ B for all QN\S∈WN\S, in particularCu f (RS, IN\S) ⊆ B. By the firststep of the proof, there existsRS

1 ∈WSδ such that

H(RS1, I

N\S) =Cu f(RS1, I

N\S) =Cu f(RS, IN\S)⊆ B.

By Remark 2 again,H(RS1,Q

N\S) =Cu f(RS1,Q

N\S)⊆ B for all QN\S∈WN\Sδ implying that

B ∈ EH(S) and henceECu f (S) ⊆ EH(S). As this holds for allS∈ P0(N), this completesthe proof of Lemma 1.

Using Lemma 1, we can repeat the proof of Theorem 6 to the game in which theplayers are restricted to dichotomous preference relations that is, replacingW by Wδ toobtain:

Theorem 8. Let E : P(N) → P(P0(A)) be a monotonic and superadditive EF. LetI =(T1, . . . ,Tn; p1, . . . , pn) be an IS, and let(u1, . . . ,un) be a vector of utilities for the players.Then E has a representation by a generalized decision schemed : WN

δ ×T → ∆(A) suchthat the gameΓδ = (N;Wδ , . . . ,Wδ ;I ;(ui)i∈N;d) has a BNE in pure strategies in whicheach player reports his true type.

2.6 Example 1 continued.

Omitting the singleton type set of player 2 (and the trivial beliefs of player 1 on thistype set) our information structure isI = (T1, p2) whereT1 = {1w,1b} and p2(1w) =p2(1b) = 1/2. We now define the utility functions of the agents:

• u1(ww,1w) = 2, u1(bb,1w) = 1 andu1(bw,1w) = u1(wb,1w) = 0 (1w likes ‘confor-mity’ with preference to white shirts).

• u1(ww,1b) = 1, u1(bb,1b) = 2 andu1(bw,1b) = u1(wb,1b) = 0 (1b likes ‘confor-mity’ with preference to blue shirts).

• u2(a,1w) =−u1(a,1w) andu2(a,1b) =−u1(a,1b) for all a∈ A(the utility of player 2 is ‘opposed’ to that of player 1 whatever his type is).

Consider the Bayesian game in which the players submit dichotomous preferences:Γδ = (N;Wδ ,Wδ ;I ;u1,u2;du f). As a game in strategic form this is a game in whichplayer 2 has 16 pure strategies (indexed by the subsets ofA) and player 1 has 162 pure

18

❅❅❅

12

(ww,wb)

(bw,bb)

(ww,bw) (wb,bb)

ww2, −2

wb0,0

bw0,0

bb1, −1

������

���

Nature

1w

1/2

❅❅❅

12

(ww,wb)

(bw,bb)

(ww,bw) (wb,bb)

ww1,−1

wb0,0

bw0,0

bb2,−2

❅❅

❅❅

❅❅

❅❅❅

1b

1/2

Figure 1 The restricted game ofΓδ .

strategies. In order to find a BNE, and hence a CE of this game, we focus on the followingsubmatrix ofΓδ described in Figure 1 which we shall refer to as the ‘restricted game’.

Here, the pure strategies are denoted by the upper-set in thedichotomous preference

that is:(ww,wb)≡ww,wbbw,bb

etc. Note that since player 1 is effective for the set{ww,wb},

simply by wearing a white shirt, playing the pure strategy(ww,wb) guarantees an outcomein {ww,wb}. Therefore, this strategy can be abbreviated asw (wearing a white shirt). Sim-ilarly for the other strategies in the reduced game. Thus, the reduced game is equivalentto the game with incomplete information on one side (on the side of player 2 regardingthe type of player 1) in which the actions set of each player is{w,b}, wearing a white ora blue shirt.

A BNE of this restricted game is(s1,s2) where

s1(1c) =ww,wbbw,bb

, s1(1n) =bw,bbbw,bb

,

and

s2 =12

ww,bwwb,bb

+12

wb,bbww,bw

.

It can be shown that this is also a BNE of the gameΓδ , and as far as we can see,Γδ hasno BNE in pure strategies.

The strategic form (i.e. the ex-ante Harsanyi game) of the reduced game is thus givenin Figure 2. The strategies of player 1 are to be read in the natural way: (w, Iw),(b, Ib)means to playw when his type isIw and playb when his type isIb etc. A correlated

19

equilibrium of this game is given in Figure 3. The generalized decision scheme can nowbe defined by inserting this correlated equilibrium in equations (12)–(13)–(14).

❅❅1

2

(w, Iw),(w, Ib)

(w, Iw),(b, Ib)

(b, Iw),(w, Ib)

(b, Iw),(b, Ib)

w b

32,−

32 0,0

1,−1 1,−1

12,−

12

12,−

12

0,0 32,−

32

Figure 2 The restricted gamein strategic form.

❅❅1

2

(w, Iw),(w, Ib)

(w, Iw),(b, Ib)

(b, Iw),(w, Ib)

(b, Iw),(b, Ib)

w b

0 0

23

13

0 0

0 0

Figure 3 A correlated equilib-rium in the restrictedgame.

3 Potential application to purification of mixed strategies

The introduction of the decision scheme as a randomizing mechanism choosing the out-come enabled us to obtain pure strategy Nash equilibria. This can be thought of as apurification device of mixed strategies: Rather than letting the players randomize (usemixed strategies) we leave the randomization to the mechanism device which chooses theoutcome with probability distribution determined by the pure strategies submitted by theplayers. In this section we explore this idea by consideringstrategic games.

3.1 Two-person2×2 games

Consider the two-person gameG= ({1,2};C1,C2;u1,u2) in which the players are 1 and2, their pure strategy sets areC1 andC2 respectively satisfying|Ci| = 2, i = 1,2 andui :C1×C2 → R, i = 1,2 are the utility functions. Relating to our general model, the set ofstates isA :=C1×C2.

Any gameG in strategic form with the the set of playersN is naturally associated withan effectivity functionE : P(N) → P(P0(C)) defined as follows: A coalitionS∈ P0(N)is effective forB ∈ P0(C) if there existscS

0 ∈ CS such thatB⊇ {cS0}×CN\S. That is, the

coalitionScan guarantee that the outcome will be inB. The effectivity functionEG(S) isthen defined by:

EG(S) = E(S) := {B∈ P0(C)|S is effective forB},

20

for S 6= /0 andE( /0) = /0.

A correlated strategyin the two-person gameG is a probability distributionp onC :=C1×C2. The corresponding payoffs to a correlated strategyp is

ui(p) = ∑c1∈C1

∑c2∈C2

p(c)ui(c1,c2), i = 1,2.

Recall thatWδ =Wδ (C) is the set of dichotomous preferences onC. We ask the fol-lowing question: Given a correlated strategyp, find a necessary and sufficient conditionsfor the existence of a DSd : WN

δ → ∆(C) such that(i) d is a representation ofEG, the ef-fectivity function ofG; and(ii) the gameΓ= (N;Wδ ,Wδ ;u1,u2;d) has a Nash equilibrium(R1,R2) ∈WN

δ such thatd(· ,(R1,R2)) = p.

Quite surprisingly, there is a simple answer to this question. To formulate it we needthe following notations and definitions. Fori ∈ N we denote byσ i ∈ ∆(Ci) a mixedstrategy of playeri in the gameG. For notational simplicity we denote the linear extensionof the payoff functionui to correlated strategies also byui : ∆(C)→ R, and thus we writeui(p) for p ∈ ∆(C), or ui(σ1,σ2) (where the argument stands forσ1×σ2 ∈ ∆(C)), orui(d(RN)) (whereRN ∈WN or RN ∈WN

δ ).Let v1 =maxσ1∈∆(C1)minc2∈C2 u1(σ1,c2) andv2 =maxσ2∈∆(C2)minc1∈C1 u2(c1,σ2) be thesecurity levels (in mixed strategies) of player 1 and player2 respectively.

Definition 8. A decision scheme d: WNδ → ∆(C) is individually rational(IR) (w.r.t. the

game G) if each player i∈ N has a strategy Vi ∈Wδ such that ui(d(V i ,RN\{i})) ≥ vi for

all RN\{i} ∈WN\{i}δ .

Proposition 1. Let p∈ ∆(C). Then ui(p) ≥ vi for i = 1,2, if and only if there exists adecision scheme d: WN

δ → ∆(C) such that,

(i) The decision scheme d is a representation of EG, the EF of G.

(ii) The gameΓ = (N;Wδ ,Wδ ;u1,u2;d) has a Nash equilibrium(R1,R2) ∈ WNδ such

that d(· ,(R1,R2)) = p.

(iii) The decision scheme d is individually rational.

In other words, individual rationality (in mixed strategies) of the induced payoffs is anecessary and sufficient condition for the existence of sucha DS.

Proof. Sufficiency: Givenp ∈ ∆(C) for which ui(p) ≥ vi for i = 1,2, we construct a de-cision schemed so that(i),(ii) and (iii ) are satisfied. Recall thatI ∈ Wδ is the totalindifference ordering onC defined bycIc′,∀c,c′ ∈C.

Step 1. Defined(· ,(I , I)) = p.

Step 2. If R1 =B

C\Band B 6= C contains a row, say{(c1

1,c21),(c

11,c

22)}, then let

c2j1 ∈ argminc2

ju1(c1

1,c2j ) and defined((c1

1,c2j1),(R

1, I)) = 1. If B does not contain a row,

21

let B0 = {(c11,c

21)} and distinguish the two subcases: IfB 6= B0, defined(· ,(R1, I)) =

d(· ,(I , I)). If B=B0, thenR1 =R1v :=

B0

C\B0. Letσ1 be the maxmin strategy of player 1

and letc2j ∈C2 be such thatu1(σ1,c2

j )= v1. Defined(·,(R1v, I)) to be the following product

of probability distributions in∆(C): d(·,(R1v, I)) = σ1×c2

j .

Step 3. Repeat the procedure of Step 2 with respect to player 2 (withthe appropriatechange of notations).

Step 4. Defined(·,(R1v,R

2v)) = σ1×σ2, and to defined(·,(R1

v,R2)) for R2 6= R2

v, recallthatd(· ,(R1

v, I))was defined in Step 2. ForR2 6= I we distinguish the two cases:(i) if R2=B

C\BwhereB contains a column, say{(c1

1,c21),(c

12,c

21)}, defined(·,(R1

v,R2)) = σ1×c2

1.

If R2 =B

C\BwhereB does not contains a column, letσ̂2 be the uniform distribution on

C2 and defined(·,(R1v,R

2)) = σ1× σ̂2.

We similarly defined(·,(R1,R2v)) for R1 6= R1

v, R1 6= I by interchanging the roles of theplayers.

Step 5. In all other cases defined(c,(R1,R2)) = 1b wheneverc∈Cu f(EG,RN) =B(RN)

andd(c,(R1,R2)) = 0 otherwise, whereb= |B(RN)|.

As the reader may verify, the decision schemed so defined satisfies conditions (i), (ii )and (iii ) of the proposition.

Necessity: Givenp∈∆(C) for which there exists a decision schemed such that (i), (ii )and (iii ) are satisfied. We have to show thatp is individually rational, that is,ui(p)≥ vi fori = 1,2. By (ii ) there is a Nash equilibrium(R1,R2)∈WN

δ such thatd(· ,(R1,R2)) = p. By(iii ) the decision schemed is individually rational; thus, fori = 1,2 there existsV i ∈ Wδsuch thatui(d(V i ,R−i))≥ vi for all R−i ∈W−i

δ . Therefore,

ui(p) = ui(d(· ,(R1,R2))≥ ui(d(· ,(V i ,R−i))≥ vi ,

where the first equality follows from (ii ), the second inequality follows since(R1,R2) is aNash equilibrium and the last inequality follows from the definition of V i. This completesthe proof of the proposition.

3.1.1 The prisoners’ dilemma

Consider the prisoners’ dilemma given in the following game:

Herev1 = v2 = 0 and the set of NE payoffs is given in Figure 1:

22

❅❅1

2

C

D

c d

2,2 −6,3

3,−6 0,0

G =

0 1 2 3 4 u1

01234

u2

(−6, 3)

(−3, 6)

(2, 2)

Figure 1: The NE payoffs in the prisoners’ dilemma .

The shaded area in this figure is the set of NE payoffs for some decision schemedconstructed in Proposition 1.

3.2 Generalization of Proposition 1

Let G = (N;C1, . . . ,Cn;u1, . . . ,un) be ann-person game in strategic form. Fori ∈ N letvi = maxσ i∈∆(Ci)minc−i∈C−i ui(σ i ,c−i) the security level of playeri (in mixed strategies) .DenoteC=C1×, . . . ,×Cn (Assume|Ci| ≥ 2,∀i ∈ N).

Theorem 9. Let p∈ ∆(C). Then ui(p) ≥ vi for all i ∈ N, if and only if there exists adecision scheme d: WN

δ → ∆(C) such that

(i) The decision scheme d is a representation of EG.

(ii) The gameΓ = (N;Wδ , . . . ,Wδ ;I ;(ui)i∈N;d) has a Nash equilibrium RN ∈WNδ such

that d(·,RN) = p.

(iii) The decision scheme d is individually rational.

Proof. Sufficiency: Givenp∈ ∆(C) for whichui(p)≥ vi for i = 1,2, we construct a deci-sion schemed so that(i),(ii) and(iii ) are satisfied.

Step 1. Defined(· ; IN) = p.

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Step 2. If R1 =B

C\BandB 6=C contains a row, say{c1

∗}×nj=2C j ,

let c−1∗ ∈ argminc−1∈C−1 u1(c1

∗,c−1) and defined((c1

∗,c−1∗ ),(R1,

n−1︷ ︸︸ ︷

I , . . . , I)) = 1. If B does

not contain a row, andB 6= {(c11, . . . ,c

n1)}, then defined(· ,(R1,

n−1︷ ︸︸ ︷

I , . . . , I)) = d(· ,(I , . . . , I)).

Finally if B= {(c11, . . . ,c

n1)} :=B0, denoteR1

v :=B0

C\B0, and choosec−1

v ∈C−1 such that

u1(σ1,c−1v ) = v1 whereσ1 is the maxmin strategy of player 1. Defined(·,(R1

v,

n−1︷ ︸︸ ︷

I , . . . , I))to be the product probability distributionsσ1×{c−1

v } in ∆(C).Steps 3 ton+ 1. Repeat step 2 for players 2, . . . ,n with the appropriate change of

notations.

Stepn+2. If RN ∈WNδ and there exists at least one playeri ∈ N such thatRi = Ri

v andR−i 6= (I , . . . , I), w.l.g. order the players so that:

RN = (R1v, . . . ,R

kv,R

k+1, . . . ,Rℓ,Rℓ+1, . . . ,Rn),

where 1≤ k ≤ n, ℓ ≥ k, andRj =B j

C\B j for j = k+1, . . . , ℓ whereB j contains a row

{c j∗}×C− j . Define

d(·,RN) = σ1×·· ·×σk,×{ck+1∗ }×· · ·{cℓ∗}×q,

whereσ i is the maxmin strategy of playeri, 1 ≤ i ≤ k, andq is the uniform probabilitydistribution onCℓ+1× . . .×Cn.

Stepn+3. In all other cases defined(c,RN) = 1b wheneverc∈Cu f(EG,RN) = B(RN)

andb= |B(RN)|. (Clearlyd(c,RN) = 0 if c 6∈ B(RN)) .

It is readily verified that the decision schemed so defined satisfies conditions (i), (ii )and (iii ) of Theorem 9. Finally, necessity follows from (ii ) and (iii ).

To see the consequences of this result in the payoff space, wedenote the set of feasiblepayoffs byF = Conv{u(c) = (u1(c), . . . ,un(c)|c ∈ C} and byIR the set of individuallyrational payoffs with respect to mixed strategies, that is,IR= {u ∈ R

N|ui ≥ vi ,∀i ∈ N},wherevi = maxσ i∈∆(Ci)minc−i∈C−i ui(σ i ,c−i). The set of correlated equilibrium payoffsin the original gameG which we denote byCEP satisfiesCEP⊆ F ∩ IR. Since clearlyIR⊆ IR, we conclude:

Corollary 10. (i) CEP⊆ NEP, that is, any correlated equilibrium payoff of the gameG can be obtained, using an appropriate decision scheme as a NE payoff in purestrategies of the induced game.

(ii) For any x∈CEP there is y∈ NEP that (weakly) Pareto dominates it.

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Indeed(i) follow from our previous discussion, and As we saw in the example of theprisoners’ dilemma, the inclusion in(i) can be strict. To see(ii), let y be the maximalpoint on the line fromv to x in the setNEP.

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