Social Choice Correspondences Research Articles

A characterization of dictatorial social choice correspondences with continuous preferences

In this paper, under the assumption that all preferences are continuous and have unique top-ranked alternatives, we establish the equivalence between strict monotonicity and dictatorship for social choice correspondences.

Nash implementation in pure public good economies

I study Nash implementation in a model of pure public good economies without the controversial no veto power condition. I prove that any social choice correspondence that is monotonic and satisfies unanimity and minimal liberty can be implemented in Nash equilibrium.

Expected Utility Consistent Extensions of Preferences

We consider the problem of extending a (complete) order over a set to its power set. The extension axioms we consider generate orderings over sets according to their expected utilities induced by some assignment of utilities over alternatives and probability distributions over sets. The model we propose gives a general and unified exposition of expected utility consistent extensions whilst it allows to emphasize various subtleties, the effects of which seem to be underestimated – particularly in the literature on strategy-proof social choice correspondences.

Monotonicity, implementation and generalized strategy-proofness

In this note, we show that Maskin monotone social choice correspondences satisfy a generalized strategy-proofness property. This result strengthens that of Nehring by dropping the requirement of finite number of alternatives and linear preference. We also show that Nash implementation implies generalized strategy-proof.

Strategy-proof resolute social choice correspondences

We qualify a social choice correspondence as resolute when its set valued outcomes are interpreted as mutually compatible alternatives which are altogether chosen. We refer to such sets as “committees” and analyze the manipulability of resolute social choice correspondences which pick fixed size committees. When the domain of preferences over committees is unrestricted, the Gibbard–Satterthwaite theorem—naturally—applies. We show that in case we wish to “reasonably” relate preferences over committees to preferences over committee members, there is no domain restriction which allows escaping Gibbard–Satterthwaite type of impossibilities. We also consider a more general model where the range of the social choice rule is determined by imposing a lower and an upper bound on the cardinalities of the committees. The results are again of the Gibbard–Satterthwaite taste, though under more restrictive extension axioms.

Coalitionally Strategy-Proof Social Choice Correspondences and the Pareto Rule

Coalitionally Strategy-Proof Social Choice Correspondences and the Pareto Rule

On the manipulation of social choice correspondences

Duggan and Schwartz (Soc Choice and Welfare 17: 85–93, 2000) have proposed a generalization of the Gibbard–Satterthwaite Theorem to multivalued social choice rules. They show that only dictatorial rules are strategy-proof and satisfy citizens sovereignty and residual resoluteness. Citizens sovereignty requires that each alternative is chosen at some preference profile. Residual resoluteness compels the election to be single-valued when the preferences of the voters are “similar”. We propose an alternative proof to the Duggan and Schwartz’s Theorem. Our proof highlights the crucial role of residual resoluteness. In addition, we prove that every strategy-proof and onto social choice correspondence concentrates the social decision power in the hands of an arbitrary group of voters. Finally, we show that this result still holds in a more general framework in which voters report their preferences over sets of alternatives.

On the equivalence of the HEX game theorem and the Duggan–Schwartz theorem for strategy-proof social choice correspondences

Gale [D. Gale, The game of HEX and the Brouwer fixed-point theorem, American Mathematical Monthly 86 (1979) 818–827] has shown that the so called HEX game theorem that any HEX game has one winner is equivalent to the Brouwer fixed point theorem. In this paper we will show that under some assumptions about marking rules of HEX games, the HEX game theorem for a 6 × 6 HEX game is equivalent to the Duggan–Schwartz theorem for strategy-proof social choice correspondences [J. Duggan, T. Schwartz, Strategic manipulability without resoluteness or shared beliefs: Gibbard–Satterthwaite generalized, Social Choice and Welfare 17 (2000) 85–93] that there exists no social choice correspondence which satisfies the conditions of strategy-proofness, non-imposition, residual resoluteness, and has no dictator.

Binary effectivity rules

A social choice rule (SCR) 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 SCR. A SCR is binary if it is rationalized by an acyclic binary relation. The foregoing result motivates our definition of a binary effectivity rule as the effectivity rule of some binary SCR. A binary SCR is regular if it satisfies unanimity, monotonicity, and independence of infeasible alternatives. A binary effectivity rule is regular if it is the effectivity rule of some regular binary SCR. We characterize completely the family of regular binary effectivity rules. Quite surprisingly, intrinsically defined von Neumann–Morgenstern solutions play an important role in this characterization.

Hierarchies of power in non-binary social choice

In the context of fixed agenda social choice correspondences, and by imposing usual axioms, we prove that power is distributed among either a non-complete hierarchy of dictators or of individuals with veto power in the society, which allows us to explicitly describe the social choice set as an alternative maximization process.

Nash Implementation with Lottery Mechanisms

Consider the problem of exact Nash Implementation of social choice correspondences. Define a lottery mechanism as a mechanism in which the planner can randomize on alternatives out of equilibrium while pure alternatives are always chosen in equilibrium. When preferences over alternatives are strict, we show that Maskin monotonicity (Maskin, 1999) is both necessary and sufficient for a social choice correspondence to be Nash implementable. We discuss how to relax the assumption of strict preferences. Next, we examine social choice correspondences with private components. Finally, we apply our method to the issue of voluntary implementation (Jackson and Palfrey, 2001).

Virtual Nash Implementation with Admissible Support

A social choice correspondence (SCC) is virtually implementable if it is E-close (in the probability simplex) to some (exactly) implementable correspondence. Abreu and Sen (1991) proved that, without restriction on the set of alternatives receiving strictly positive probability at equilibrium, every SCC is virtually implementable in Nash Equilibrium. We study virtual implementation when the supports of equilibrium lotteries are restricted. We provide a necessary and sufficient condition, imposing joint restrictions on SCCs and admissible supports. Then, we discuss how to construct supports. Finally, we study virtual implementation when the support is restricted to the efficient or individually rational alternatives.

Implementation with State Dependent Feasible Sets and Preferences: A Renegotiation Approach

In this paper we present a model of implementation based on the idea that agents renegotiate unfeasible allocations. We characterize the maximal set of Social Choice Correspondences that can be implemented in Nash Equilibrium with a class of renegotiation functions that do not reward agents for unfeasibilities. This result is used to study the possibility of implementing the Walrasian Correspondence in exchange economies and several axiomatic solutions to problems of bargaining and bankruptcy.

Locating public facilities by majority: Stability, consistency and group formation

We consider the following allocation problem: A fixed number of public facilities must be located on a line. Society is composed of N agents, who must be allocated to one and only one of these facilities. Agents have single peaked preferences over the possible location of the facilities they are assigned to, and do not care about the location of the rest of facilities. There is no congestion. We show that there exist social choice correspondences that choose locations and assign agents to them in such a way that: (1) these decisions are Condorcet winners whenever one exists, (2) the majority of the users of each facility supports the choice of its location, and (3) no agent wishes to become a user of another facility, even if that could induce a change of its present location by majority voting.

Constitutional implementation of social choice correspondences

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 conditions for an arbitrary (but surjective) social choice correspondence to be constitutionally implementable. It is shown that the results apply to interesting classes of scoring and veto social choice correspondences.

Nash Implementation with Lottery Mechanisms

Consider the problem of exact Nash Implementation of social choice correspondences. Define a lottery mechanism as a mechanism in which the planner can randomize on alternatives out of equilibrium while pure alternatives are always chosen in equilibrium. When preferences over alternatives are strict, we show that Maskin monotonicity (Maskin, 1999) is both necessary and sufficient for a social choice correspondence to be Nash implementable. We discuss how to relax the assumption of strict preferences. Next, we examine social choice correspondences with private components. Finally, we apply our method to the issue of voluntary implementation (Jackson and Palfrey, 2001).

Exact Nash Implementation with Lottery Mechanisms

Consider the problem of exact Nash implementation of social choice correspondences. Define a mechanism in which the planner can randomize on alternatives out of equilibrium while pure alternatives are always chosen in equilibrium. We call such a game form a lottery mechanism. When preferences over alternatives are strict, we show that Maskin Monotonicity (Maskin, 1999) is both necessary and sufficient for the Nash implementation of social choice correspondences satisfying unanimity. We discuss how to dispense with unanimity and relax the assumption of strict preferences by modifying the canonical mechanism we consider. We then study some examples of Maskin monotonic social choice correspondences violating no-veto power. Finally, we apply our method to the issue of voluntary implementation (Jackson-Palfrey, 2001).

On a compromise social choice correspondence

This paper analyzes the compromise social choice correspondence derived from the τ-value of digraph games. Monotonicity of this correspondence is shown. A connection between several properties of social choice correspondences based on game theoretical solutions and game theoretical properties of the underlying solutions is given. Applications to several game theoretic solutions are provided.

Garchy for Social Choice Correspondences and Strategy-Proofness

We study the existence of a group of individuals which has some decisive power for social choice correspondences that satisfy a monotonicity property which we call modified monotonicity. And we examine the relation between modified monotonicity and strategy-proofness of social choice correspondences according to the definition by Duggan and Schwartz (2000). We will show mainly the following two results. (1) Modified monotonicity implies the existence of an oligarchy. An oligarchy is a group of individuals such that it has some decisive power (semi-decisiveness), and at least one of the most preferred alternatives of every its member is always chosen by any social choice correspondence. (2) Strategy-proofness of social choice correspondences is equivalent to modified monotonicity.

On two new social choice correspondences

We introduce two new majoritarian social choice correspondences. We initially assign to every alternative in a preference profile a weight equal to one. Using (cooperative) game theoretic considerations we redistribute these weights, and use these redistributed weights to define a social choice correspondence. Then we apply this procedure iteratively by taking in each step as weights the redistributed weights obtained in the previous step. The resulting limit is used to define a second social choice correspondence. Both social choice correspondences are Pareto optimal and refinements of the Top cycle correspondence. Properties and comparisons with related social choice correspondences are discussed.