Bezalel Peleg

• Ph.D.
• Professor Emeritus at Hebrew University of Jerusalem

A social choice correspondence is self-implementable in strong equilibrium if it is implementable in strong equilibrium by a social choice function selecting from the correspondence itself as a game form. We characterize all social choice correspondences implementable this way by an anonymous social choice function satisfying no veto power, given t...

We survey the results on representations of committees and constitutions by game forms that possess some kind of equilibrium strategies for each profile of preferences of the players. The survey is restricted to discrete models, that is, we deal with finitely many players and alternatives. No prior knowledge of social choice is assumed: As far as d...

Feasible elimination procedures (Peleg, 1978) play a central role in constructing social choice functions which have the following property: in the associated game form, for any preference profile there exists a strong Nash equilibrium resulting in the sincere outcome. In this paper we provide an axiomatic characterization of the social choice corr...

We show that feasible elimination procedures (Peleg, 1978) can be used to select k from m alternatives. An important advantage of this method is the core property: no coalition can guarantee an outcome that is preferred by all its members. We also show that the problem of determining whether a specific k-tuple can result from a feasible elimination...

We show that the Aumann-Davis-Maschler bargaining set and the Mas-Colell bargaining set of a non-leveled NTU game that is either ordinal convex or coalition merge convex coincides with the core of the game. Moreover, we show by means of an example that the foregoing statement may not be valid if the NTU game is marginal convex.

We model constitutions by effectivity functions. We assume that the constitution is common knowledge among the members of the society. However, the preferences of the citizens are private information. We investigate whether there exist decision schemes (i.e., functions that map profiles of (dichotomous) preferences on the set of outcomes to lotteri...

In this paper we o⁄er an axiomatic approach for the investigation of rights by means of game forms. We give a new definition of constitution which consists of three components: the set of rights, the assignment of rights to groups of members of the society, and the distribution of power in the society (as a function of the distribution of rights)....

We investigate necessary and sufficient conditions for the existence of Bayesian-Nash equilibria that satisfy the Condorcet Jury Theorem (CJT). In the Bayesian game G n among n jurors, we allow for arbitrary distribution on the types of jurors. In particular, any kind of dependency is possible. If each juror i has a “constant strategy”, σ i (that...

On several classes of n-person NTU games that have at least one Shapley NTU value, Aumann characterized this solution by six axioms: Non-emptiness, efficiency, unanimity, scale covariance, conditional additivity, and independence of irrelevant alternatives (IIA). Each of the first five axioms is logically independent of the remaining axioms, and th...

In this chapter we first recall the Gibbard-Satterthwaite Theorem and review some of its implications. This is done in Section 8.2. The rest of the chapter is devoted to consideration of the problem of preference distortion as a consequence of manipulation of non-dictatorial voting rules. First we observe that the Gibbard-Satterthwaite Theorem does...

In this chapter we extend the model of Chapters 9 and 10 to a classical voting system with still finitely many alternatives (candidates) but with very many voters. Such a system is representative of political elections on the local or national level. As an, in our view, best approximation we model voters as elements of a non-atomic measure space. I...

In Chapter 2 we have seen how a constitution of a society or more formally, an effectivity function, and a set of rules that enable the members of the society to exercise their rights simultaneously, i.e., a game form representing the effectivity function, govern the behavior of the members of a (civilized) state. In this chapter we introduce a new...

As argued in the introductory section of the previous chapter it is important to find robust voting procedures for constitutions. In our approach, ‘robust’ means ‘exactly and strongly consistent’ (ESC): the game induced by the social choice function should have a strong equilibrium for each profile of preferences which, moreover, results in the sam...

In this chapter we expound on Gärdenfors’s (1981) theory of rights-systems or constitutions. Gärdenfors formalizes rights-systems as follows. If S is a coalition (a group of individuals, members, players,...) and B is a set of social states (outcomes, alternatives,...), then B is a ‘right’ of S in the sense of Gärdenfors if S is legally entitled to...

In the preceding chapters we have studied representations of constitutions (effectivity functions) under a minimal stability condition, namely Nash consistency: for every admissible profile of preferences there should be at least one Nash equilibrium in the representing game. In Chapter 4 we have studied acceptable representations, meaning that all...

In Chapter 3 we have studied the existence of Nash consistent representations of effectivity functions. We have, in fact, shown that the same conditions that guarantee existence of Nash consistent representations also guarantee the existence of weakly acceptable representations, that is, representations that always admit also Pareto optimal Nash eq...

We have seen in Chapter 8 that a possible way to avoid the consequences of the Gibbard-Satterthwaite Theorem is to construct exactly and strongly consistent social choice functions. We recall that such functions ensure that the sincere outcome is an outcome of a strong Nash equilibrium for each profile of preferences of the voters. In this chapter...

In Chapter 3 we have seen that – under the usual assumptions of monotonicity and superadditivity, and for a finite set of alternatives (social states) – effectivity functions (constitutions) can be represented by Nash consistent game forms if and only if the intersection condition on individual polar sets (3.6) is satisfied. This condition is quite...

In the previous chapters we have disregarded the topological properties of the strategy sets and the outcome functions of representations of topological effectivity functions. In this chapter we enquire about the existence of representations of which the strategy sets and outcomes space are compact metric spaces and where the outcome function is co...

In this chapter we explain why we adopt Gärdenfors’s (1981) model of a constitution rather than Arrow’s model of a (‘well behaved’) social welfare function. We start, in Section 1.2, with the definition of a social welfare function and recall some of the questions that it invoked. Then we proceed to formulate Arrow’s Impossibility Theorem in Sectio...

On several classes of n-person NTU games that have at least one Shapley NTU value, Aumann characterized this solution by six axioms: Non-emptiness, efficiency, unanimity, scale covariance, conditional additivity, and independence of irrelevant alternatives (IIA). Each of the first five axioms is logically independent of the remaining axioms, and th...

Effectivity functions for finitely many players and alternatives are considered. It is shown that every monotonic and superadditive effectivity function can be augmented with equal chance lotteries to a finite lottery model--i.e., an effectivity function that preserves the original effectivity in terms of supports of lotteries--which has a Nash con...

We investigate sufficient conditions for the existence of Bayesian-Nash equilibria that satisfy the Condorcet Jury Theorem (CJT). In the Bayesian game Gn among n jurors, we allow for arbitrary distribution on the types of jurors. In particular, any kind of dependency is possible. If each juror i has a “constant strategyâ€, si (that is, a strategy...

We investigate sufficient conditions for the existence of Bayesian-Nash equilibria that satisfy the Condorcet Jury Theorem (CJT). In the Bayesian game Gn among n jurors, we allow for arbitrary distribution on the types of jurors. In particular, any kind of dependency is possible. If each juror i has a “constant strategyâ€, si (that is, a strategy...

Let A be a finite set of m alternatives, let N be a finite set of n players and let RN be a profile of linear preference orderings on A of the players. Let uN be a profile of utility functions for RN. We define the NTU game VuN that corresponds to simple majority voting, and investigate its Aumann-Davis-Maschler and Mas-Colell bargaining sets. The...

Implementation theory tackles the following problem: given a social choice correspondence, find a decentralized mechanism such that for every constellation of the individuals' preferences, the set of outcomes in equilibrium is exactly the set of socially optimal alternatives (as specified by the correspondence). In this paper we are concerned with...

The Gibbard-Satterthwaite Theorem asserts the impossibility of designing a non-dictatorial social choice function (SCF) under which agents can never benefit by lying, assuming complete information. We show that in voting games of complete information where a mediator is on hand, the implications of this troubling impossibility result can be allevia...

We consider the class of all abstract economies with convex and compact strategy sets, continuous and quasi-concave payoff functions, and continuous and convex-valued feasibility correspondences. We prove that the Nash correspondence is the unique solution on the foregoing class of abstract economies that satisfies nonemptiness, rationality in one-...

Voting problems with a continuum of voters and finitely many alternatives are considered. The classical Arrow and Gibbard-Satterthwaite theorems are shown to persist in this model, not for single voters but for coalitions of positive size. The emphasis of the study is on strategic considerations, relaxing the nonmanipulability requirement: are ther...

In standard rational choice modelling decisions are made according to given information and preferences. In the model presented here the 'information technology' of individual decision makers as well as their preferences evolve in a dynamic process. In this process decisions are made rationally by players who differ in their informational as well a...

Certain voting bodies can be modeled as a simple game where a coalition's winning depends on whether it wins, blocks or loses in two smaller simple games. There are essentially five such ways to combine two proper games into a proper game. The most decisive is the lexicographic rule, where a coalition must either win in G1, or block in G1 and win i...

A social choice rule is a collection of social choice correspondences, one for each agenda. An effectivity rule is a collection of effectivity functions, one for each agenda. We prove that every monotonic and superadditive effectivity rule is the effectivity rule of some social choice rule. A social choice rule is binary if it is rationalized by an...

Voting problems with a continuum of voters and finitely many alternatives are considered. Since the Gibbard–Satterthwaite theorem persists in this model, we relax the non-manipulability requirement as follows: are there social choice functions (SCFs) such that for every profile of preferences there exists a strong Nash equilibrium resulting in the...

An effectivity function assigns to each coalition of individuals in a society a family of subsets of alternatives such that the coalition can force the outcome of society’s choice to be a member of each of the subsets separately. A representation of an effectivity function is a game form with the same power structure as that speci?ed by the effecti...

A game form constitutionally implements a social choice correspondence if it implements it in Nash equilibrium and, moreover, the associated effectivity functions coincide. This paper presents necessary and sufficient conditions for a unanimous social choice correspondence to be constitutionally implementable, and sufficient and almost necessary co...

This paper considers generalized Lorenz-maximal solutions in the core of a convex TU-game and demonstrates that such solutions satisfy coalitional monotonicity and population monotonicity.

We introduce an extension of the Mas-Colell bargaining set and construct, by an elaboration on a voting paradox, a superadditive four-person non-transferable utility game whose extended bargaining set is empty. It is shown that this extension constitutes an upper hemicontinuous correspondence. We conclude that the Mas-Colell bargaining set of a non...

In voting theory, the result of a paired comparison method such as the one suggested by Condorcet can be represented by a tournament, i.e.,a complete asymmetric directed graph. When there is no Condorcet winner, i.e.,a candidate preferred to any other candidate by a majority of voters, it is not always easy to decide who is the winner of the electi...

Let A be a finite set of m ³ 3 alternatives, let N be a finite set of n ³ 3 players and let R<SUP>n</SUP> be a profile of linear preference orderings on A of the players. Throughout most of the paper the considered voting system is the majority rule. Let u<SUP>N</SUP> be a profile of utility functions for R<SUP>N</SUP>. Using a -effectiveness we de...

Acceptable game forms were introduced in Hurwicz and Schmeidler [Econometrica 46 (1978) 1447]. Dutta [Econometrica 52 (1984) 1151] considered effectivity functions of acceptable game forms. This paper unifies and extends the foregoing two papers. We obtain the following characterization of the effectivity functions of acceptable game forms: An effe...

A game form constitutionally implements a social choice correspondence if it implements it in Nash equilibrium and, moreover, the associated effectivity functions coincide. This paper presents necessary and sufficient conditions for a unanimous social choice correspondence to be constitutionally implementable, and sufficient and almost necessary co...

In this chapter we adopt the axiomatic approach in order to find (new) voting procedures to committees that are immune against deviations by coalitions of voters. We shall now describe our approach. Let G be a committee and let A be a finite set of m alternatives, m ⩾ 2. Our problem is to find a social choice function F that will enable the members...

The concept of coalition proof Nash equilibrium was introduced by Bernheim et al. [5]. In the present paper, we consider the representation problem for coalition proof Nash equilibrium: For a given effectivity function, describing the power structure or the system of rights of coalitions in society, it is investigated whether there is a game form w...

As shown by Peleg (1993), the core of market games is characterized by nonemptiness, individual rationality, superadditivity, the weak reduced game property, the converse reduced game property, and weak symmetry. It was not known whether weak symmetry was logically independent. With the help of a certain transitive 4-person TU game, it is shown tha...

Survival of payoff maximization is the usual as if-justification for assuming rational economic agents. An indirect evolutionary analysis allows for stimuli which are not directly related to reproductive success although they affect behavior. One first determines the solution for all possible constellations of stimuli, and then the evolutionarily s...

Voting problems with a continuum of voters and finitely many alternatives are considered. The classical Arrow and Gibbard-Satterthwaite theorems are shown to persist in this model, not for single voters but for coalitions of positive size. The emphasis of the study is on strategic considerations, relaxing the nonmanipulability requirement: are ther...

Abstract The concept of an effectivity function is adopted as a formal model,of a constitution. A game form models,the actions available and permissible to individuals in a society. As a representation of the constitution such a game,form should endow,each group in society with the same,power as it has under the constitution. Another desirable prop...

Acceptable game forms were introduced in Hurwicz and Schmeidler (1978). Dutta (1984) considered effectivity functions of acceptable game forms. This paper unifies and extends the foregoing two papers. We obtain the following characterization of the effectivity functions of acceptable game forms: An effectivity function belongs to some acceptable ga...

We consider the problem of implementing a social choice correspondence H in Nash equilibrium when the constitution of the society is given by an effectivity function E. It is assumed that the effectivity function of , is a sub-correspondence of E. We found necessary and efficient conditions for a game form to implement H (in Nash equilibria), and t...

We consider the problem of implementing a social choice correspondence H in Nash equilibrium when the constitution of the society is given by an effectivity function E. It is assumed that the effectivity function of $ H,E^{H}$, is a sub-correspondence of E. We found necessary and efficient conditions for a game form $\Gamma $ to implement H (in Nas...

This paper considers the set of Lorenz-maximal imputations in the core of a balanced cooperative game as a solution concept. It is shown that the Lorenz-solution concept satisfies a number of suitable properties such as desirability, continuity and the reduced game property. Moreover, the paper consideres alternative characterizations where it is s...

A strong representation of a committee, formalized as a simple game, on a convex and closed set of alternatives is a game form with the members of the committee as players such that (i) the winning coalitions of the simple game are exactly those coalitions, which can get any given alternative independent of the strategies of the complement, and (ii...

As shown by Peleg, the core of market games is characterized by nonemptiness, individual rationality, superadditivity, the weak reduced game property, the converse reduced game property, and weak symmetry. It was not known whether weak symmetry was logically independent. With the help of a certain transitive 4-person TU game it is shown that weak s...

The positive prekernel, a solution of cooperative transferable utility games, is introduced. We show that this solution inherits many properties of the prekernel and of the core, which are both sub-solutions. It coincides with its individually rational variant, the positive kernel, when applied to any zero-monotonic game. The positive (pre)kernel i...

Let G m,n be the class of strategic games with n players, where each player has m≥2 pure strategies. We are interested in the structure of the set of correlated equilibria of games in G m,n when n→∞. As the number of equilibrium constraints grows slower than the number of pure strategy profiles, it might be conjectured that the set of correlated eq...

This paper exhibits to any noncooperative game in strategic or normal form a `canonical' game in extensive form that preserves all symmetries of the former one. The operation defined this way respects the restriction of games to subgames and yields a minimal total rank of the tree involved. Moreover, by the above requirements the `canonical extensi...

In standard rational choice modelling decisions are made according to given information and preferences. In the model presented here the `information technology' of individual decision-makers as well as their preferences evolve in a dynamic process. In this process decisions are made rationally by players who differ in their informational as well a...

We prove that multidimensional generalized median voter schemes are coalition-proof.

This book is a theoretical and completely rigorous analysis of voting in committees that provides mathematical proof of the existence of democratic voting systems, which are immune to the manipulation of preferences of coalitions of voters. The author begins by determining the power distribution among voters that is induced by a voting rule, giving...

We prove that almost all equilibria in dominant strategies of finite strategic games are coalition-proof. Also, we investigate existence of coalition-proof equilibria of separable games and strategyproof mechanisms. In particular, we give an example of a strategyproof mechanism, which is defined for strict orderings, and which is coalition-proof bu...

The MC-value is introduced as a new single-valued solution concept for monotonic NTU-games. The MC-value is based on marginal vectors, which are extensions of the well-known marginal vectors for TU-games and hyperplane games. As a result of the definition it follows that the MC-value coincides with the Shapley value for TU-games and with the consis...

Two preimputations of a given TU game can be compared via the Lorenz order applied to the vectors of satisfactions. One preimputation is `socially more desirable' than the other, if its corresponding vector of satisfactions Lorenz dominates the satisfaction vector with respect to the second preimputation. It is shown that the prenucleolus, the anti...

In this paper we offer an axiomatic approach for the investigation of rights by means of game forms. We give a new definition of constitution which consists of three components: the set of rights, the assignment of rights to groups of members of the society, and the distribution of power in the society (as a function of the distribution of rights)....

This paper presents a generalization of earlier results on the axiomatization of the Walras correspondence to generalized (pure exchange) economies, where the commodity space is the positive cone in an ordered locally convex topological vector space. The authors' main result characterizes the Walras correspondence completely over an 'acceptable' cl...

Let g be a cooperative game and let N be the set of players of g. According to Nash's Program N can find a noncooperative game G such that some Nash equilibrium of G may serve as a solution to g. We show that the implementation of Nash's Program might face some difficulties. In this paper we restrict ourselves to finite games. However, we proved in...

We investigate the ESS’s of payoff-irrelevant asymmetric animal conflicts in Selten’s (1980) model. We show that these are determined by the symmetric and strict correlated equilibria of the underlying (symmetric) two-person game. More precisely, the set of distributions (on the strategy space) of ESS’s coincides with the set of strict and symmetri...

A generalized (pure exchange) economy consists of a list of agents, utility functions and initial endowments for each agent, and a net trade vector that describes the trade between the economy and the outside world. In this paper we extend the definition of Walras equilibria to generalized economies without free disposal of commodities. An existenc...

This paper constructs a two-country (Home and Foreign) general equilibrium model of Schumpeterian growth without scale effects. The scale effects property is removed by introducing two distinct specifications in the knowledge production function: the permanent effect on growth (PEG) specification, which allows policy effects on long-run growth; and...

Bayesian equilibria are characterized by means of consistency and one-person rationality in combination with non-emptiness or converse consistency. Moreover, strong and coalition-proof Bayesian equilibria of extended Bayesian games are introduced and it is seen that these notions can be characterized by means of consistency, one-person rationality,...

It is assumed that first ring members bid in a preauction or knockout and that then only the rings representative actively bids in the subsequent main auction. The rules must specify for every vector of bids who represents the cartel in the main auction, which transfers are paid to other ring members, who wins the main auction, and finally which pr...

We provide a continuous and feasible double implementation of the Walras equilibrium. In our game form the set of traders is partitioned intok subsets,k>-2, and for each member of the partition there is an (outside) auctioneer. Also, each agent announces a price-allocation pair so that all agents become price takers. The outcome allocation is defin...

Solutions defined on classes of strategic games, satisfying One-Person Rationality (OPR), Non-emptiness (NEM) and Consistency (CONS) are considered. The main question to be answered is whether these conditions characterize the Nash Equilibrium solution NE for the given class of games. Depending on the structure of the class of games positive as wel...

We start with giving an axiomatic characterization of the Nash equilibrium (NE) correspondence in terms of consistency, converse consistency, and one-person rationality. Then axiomatizations are given of the strong NE correspondence, the coalition-proof NE correspondence and the semi-strong NE. In all these characterizations consistency and suitabl...

At the beginning the author formulates the implementation problem (with complete information). Then he gives a short historical survey of implementation theory. Naturally, the classical works of Hurwicz (1960, 1972) and the seminal work of Maskin (1977) are mentioned as the cornerstone of implementation theory. The description of the present state...

This paper is the first in a series of papers in which we plan to exhibit to any noncooperative game in strategic or normal form a “canonical” representation in extensive form that preserves all symmetries of the game. The operation defined this way will respect the restriction of games to subgames and yield a minimal total rank of the tree involve...

Solutions defined on classes of strategic games, satisfying One-Person Rationality (OPR), Non-emptiness (NEM) and Consistency (CONS) are considered. The main question to be answered is whether these conditions characterize the Nash Equilibrium solution NE for the given class of games. Depending on the structure of the class of games positive as wel...

Using the approach of bounded rationality and myopic learning, we attempt to explain why bees (as examples of a forager animal) do the right (optimal) thing in an environment of many foragers, namely to adopt the Ideal Free Distribution, but do the wrong thing when they are alone, namely stick to the Matching Law. We discuss two types of simple for...

The class of homogeneous n-person constant sum games was introduced by von Neumann-Morgenstern [20]; as a solution concept they treated the v.N.-M.-solution (or main simple solution). Peleg [9] discussed the kernel and nucleolus within this framework. The general theory of non-constant sum homogeneous games was developed by Ostmann [8], Rosenmüller...

The proof of Lemma 4.7 in Peleg (Peleg, B. 1989. An axiomatization of the core of market games. Math. Oper. Res. 14 448–456.) is incorrect. However, if we add the following innocent axiom, the proof of the lemma becomes straightforward.

The core is, the most intuitive solution concept in cooperative game theory. An intuitively acceptable axiom system for the core might reinforce its position as the most natural" solution. An axiomatization of the core may serve two other, more important goals: (1) by obtaining axioms for the core, those important properties of solutions are single...

We present a theorem on foraging graphs which yields bounds on the number of generalists and specialists in a two-sided natural market. With the biological assumption of the Ideal Free Distribution for each forager which visits different flower types, we use our previous game theory approach of matching between bees and flowers (Peleg & Shmida, 199...

We address the problem of short-run correspondence between been pollinators and plant flowers in nature. We prove the existence of stable matchings and provide necessary conditions which allow computation of equilibria. The model predicts, when enforced by some long-run equilibrium conditions, the following two important biological phenomena: 1.1....

Homogeneous weighted majority games were already introduced by von Neumann and Morgenstern; they discussed uniqueness of the representation and the “main simple solution” for constant-sum games. For the same class of games Peleg studied the kernel and the nucleolus. Ostmann and Rosenmüller described the nature of representations of general homogene...

This paper is devoted to a game theoretic analysis of a voting system which was used by some Jewish communities in Europe in the seventeenth century. The system is described in M.M. Krochmal [1675, Responsum No. 1]. We now present an abridged translation of its description:

The study by Mitra (1990) provides a convenient point of departure for reviewing and extending some of the analytical results proved by Gale (1967) in his masterly paper on optimal growth programmes in multi-sector economies. The main result of his paper was a proof of the existence of a Ramsey-Weizsacker optimal programme. But, as he noted, ‘to ob...

The most prominent solution in the theory of noncooperative games is the Nash equilibrium. However, there are some fundamental problems with the interpretation and application of it that led to many refinements (see, e.g., Selten, 1975). The following difficulty concerning the application of Nash equilibria has been pointed out in Bernheim, Peleg,...

We provide a self-contained axiomatization of linear inequality measures for coalitional games. Adding one axiom to those of Weymark (Math. Soc. Sci.1 (1981), 409–430), we prove that all such measures must be generalized Gini functions of the Shapley value. Also, we generalize the foregoing result to the spaces pNA(μ) of non-atomic games.

We shall review a solution to the problem of mass elections which has been developed during the past ten years. I have both existence and uniqueness results. Our approach is axiomatic in the following sense. We consider the following properties of social choice functions: (i) anonymity; (ii) Pareto-optimality; (iii) monotonicity; and (iv) exact and...

This chapter reviews the current axiomatizations of the core, the nucleolus, and the prekernel. The main axiom for all the foregoing solutions are the reduced game property (RGP). It has been investigated that the pseudokernel has RGP. The core, the pseudonucleolus, and the pseudobargaining set also have RGP. An axiomatization of the prenucleolus i...

We prove that all nondictatorial voting schemes whose range has more than two alternatives will be manipulable, when their domain is restricted to the set of all continuous preferences over alternatives. Our result neither implies nor is implied by the original Gibbard-Satterthwaite theorem, except if the number of alternatives is finite, when they...

We provide an axiomatization of the core of market games (totally balanced coalitional games). We prove that the core is fully characterized by the following five independent properties: (i) nonemptiness; (ii) individual rationality; (iii) superadditivity; (iv) the weak reduced game property; and (v) the converse reduced game property. Also, we con...