Choice Correspondences Research Articles

Rationality in a General Model of Choice

In this paper we consider choice correspondences which may be empty-valued. We study conditions under which such choice correspondences are rational, transitively rational, partially rational, partially almost transitive rational, partially almost quasi-transitive rational.

Contraction Consistent Stochastic Choice Correspondence

We model a general choice environment via probabilistic choice correspondences, with (possibly) incomplete domain and infinite universal set of alternatives. We offer a consistency restriction regarding choice when the feasible set contracts. This condition, ‘contraction consistency’, subsumes earlier notions such as Chernoff’s Condition, Sen’s a and s, and regularity. We identify a restriction on the domain of the stochastic choice correspondence, under which contraction consistency is equivalent to the weak axiom of revealed preference in its most general form. When the universal set of alternatives is finite, this restriction is also necessary for such equivalence. Analogous domain restrictions are also identified for the special case where choice is deterministic but possibly multi-valued. Results due to Sen (Rev Econ Stud 38: 307-317, 1971) and Dasgupta and Pattanaik (Econ Theory 31: 35-50, 2007) fall out as corollaries. Thus, conditions are established, under which our notion of consistency, articulated only in reference to contractions of the feasible set, suffices as the axiomatic foundation for a general revealed preference theory of choice behaviour.

Implementation in Minimax Regret Equilibrium

This note studies the problem of implementing social choice correspondences in environments where individuals have doubts about the rationality of their opponents. We postulate the concept of e-minimax regret as our solution concept and show that social choice correspondences that are Maskin monotonic and satisfy the no-veto power condition are implementable in e-minimax regret equilibrium for all e ∈ [0,1).

Coalitionally strategy-proof social choice correspondences and the Pareto rule

This paper studies coalitional strategy-proofness of social choice correspondences that map preference profiles into sets of alternatives. In particular, we focus on the Pareto rule, which associates the set of Pareto optimal alternatives with each preference profile, and examine whether or not there is a necessary connection between coalitional strategy-proofness and Pareto optimality. The definition of coalitional strategy-proofness is given on the basis of a max–min criterion. We show that the Pareto rule is coalitionally strategy-proof in this sense. Moreover, we prove that given an arbitrary social choice correspondence satisfying the coalitional strategy-proofness and nonimposition, all alternatives selected by the correspondence are Pareto optimal. These two results imply that the Pareto rule is the maximal correspondence in the class of coalitionally strategy-proof and nonimposed social choice correspondences.

On Maskin monotonicity of solution based social choice rules

Howard (J Econ Theory 56:142–159, 1992) argues that the Nash bargaining solution is not Nash implementable, as it does not satisfy Maskin monotonicity. His arguments can be extended to other bargaining solutions as well. However, by defining a social choice correspondence that is based on the solution rather than on its realizations, one can overcome this shortcoming. We even show that such correspondences satisfy a stronger version of monotonicity that is even sufficient for Nash implementability.

Epsilon-Nash implementation

This paper provides necessary and sufficient conditions for a social choice correspondence to be implementable via epsilon-Nash equilibrium when agents' preferences can be represented by cardinal utilities. Our results also display robustness of Maskin's seminal result on Nash implementation.

Coalitionally Strategy-Proof Social Choice Correspondences and the Pareto Rule

Coalitionally Strategy-Proof Social Choice Correspondences and the Pareto Rule

A general impossibility result on strategy-proof social choice hyperfunctions

A social choice hyperfunction picks a non-empty set of alternatives at each admissible preference profile over sets of alternatives. We analyze the manipulability of social choice hyperfunctions. We identify a domain D λ of lexicographic orderings which exhibits an impossibility of the Gibbard–Satterthwaite type. Moreover, this impossibility is inherited by all well-known superdomains of D λ . As most of the standard extension axioms induce superdomains of D λ while social choice correspondences are particular social choice hyperfunctions, we are able to generalize many impossibility results in the literature.

Choice Problems with a Status-Quo Alternative

In this paper we consider choice correspondences which may be empty-valued. We study conditions under which such choice correspondences are rational, transitively rational, partially rational, partially triple-acyclic rational and partially acyclic rational.

Reason-based choice correspondences

A reason-based choice correspondence rationalizes choice behaviour in terms of a two-stage choice procedure. Given a feasible set S , the individual eliminates from it all of the dominated alternatives according to her fixed (not necessarily complete) strict preference relation, in the first step. In the second step, first she constructs for each maximal alternative identified in the first step its lower contour set (i.e. the set of alternatives which are dominated by it in S ), and then she eliminates from the maximal set all of those alternatives so that the following justification holds: there exists another maximal alternative whose lower contour set strictly contains that of another maximal alternative. This procedural model captures the basic idea behind the experimental finding known as “attraction effect”. We study the rationalizability of reason-based choice correspondences axiomatically. We relate our choice-consistency conditions to standard consistency proprieties. Our characterization result offers testable restrictions on this ‘choice anomaly’ for large (but finite) set of alternatives.

Non-manipulable Social Welfare Functions when Preferences are Fuzzy

It is well known that many social decision procedures are manipulable through strategic behaviour. Typically, the decision procedures considered in the literature have been social choice correspondences. In this article, we investigate the problem of constructing a social welfare function that is non-manipulable. In this context, individuals attempt to manipulate a social ordering as opposed to a social choice. Using techniques from fuzzy set theory, we introduce a class of fuzzy binary relations of which exact binary relations are a special case. Operating within this family enables us to prove an impossibility theorem. This theorem states that all non-manipulable social welfare functions are dictatorial, provided that they are not constant. A proof of this theorem first appeared in Perote-Pena and Piggins (2007, J. Math. Econ., 43, 564–580). This article contains a new proof of this theorem which is considerably simpler than the original. Moreover, we also consider a possibility result which this earlier article neglects.

A theory of reference-dependent behavior

Extensive field and experimental evidence in a variety of environments show that behavior depends on a reference point. This paper provides an axiomatic characterization of this dependence. We proceed by imposing gradually more structure on both choice correspondences and preference relations, requiring increasingly higher levels of rationality, and freeing the decision-maker from certain types of inconsistencies. The appropriate degree of behavioral structure will depend on the phenomenon that is to be modeled. Last, we provide two applications of our work: one to model the status-quo bias, and another to model addictive behavior.

A theory of kindness, reluctance, and shame for social preferences

Recent experimental evidence from dictator games suggests that proposers take money from receivers when taking is an option, and that many proposers are reluctant to play the game. This paper proposes a behavioral model with two components: a choice correspondence that depends on the endowed allocation and the menu of allocations available, and a preference ordering over endowment/menu pairs. The choice correspondence governs behavior when the proposer actually plays a game, and the preference ordering governs the proposer's willingness to play a particular game. The model is then used to characterize notions of proposer kindness, reluctance, and shame.

On strategy-proof social choice correspondences: a comment

In a recent paper, Sato (Soc Choice Welf doi:10.1007/s00355-007-0285-5) has provided two alternative definitions of strategy-proofness for social choice correspondences and proved that they are incompatible with the joint requirements of anonymity, neutrality, and a minimal range condition. In this note, we use the results in Rodriguez-Alvarez (Soc Choice Welf 29:175–199, 2007) to directly prove stronger versions of Sato’s theorems.

On strategy-proof social choice correspondences

We introduce two new concepts of strategy-proofness for social choice correspondences based on the theory of preferences over sets of alternatives under complete uncertainty. One is based on Pattanaik and Peleg (Soc Choice Welf 1:113–122, 1984) and the other is based on Bossert et al. (Econ Theory 16:295–312, 2000). We prove that there is no social choice correspondence satisfying anonymity, neutrality, a range condition, and either of our concepts of strategy-proofness.

Reexamination of Maskin's Theorem on Nash implementability

We reexamine Maskin's Theorem by proposing slight modifications on Maskin's conditions of monotonicity and no veto power. We show that any social choice correspondence (SCC) that satisfies strict monotonicity, strict no veto power and unanimity can be implemented in Nash equilibria.

A characterization of social choice correspondences that implement the core of simple games

In a coalitional voting game, an alternative is said to be in the core when no majority is willing to replace it with another alternative. A social choice correspondence is a mapping which associates any profile of voters’ preferences with a non-empty subset of the set of alternatives, which is understood as the set of selected outcomes. We characterize the social choice correspondences which always select the core of a simple game for any profile of preferences.

On strategy-proof social choice correspondences

We introduce two new concepts of strategy-proofness for social choice correspondences based on the theory of preferences over sets of alternatives under complete uncertainty. One is based on Pattanaik and Peleg (Soc Choice Welf 1:113–122, 1984) and the other is based on Bossert et al. (Econ Theory 16:295–312, 2000). We prove that there is no social choice correspondence satisfying anonymity, neutrality, a range condition, and either of our concepts of strategy-proofness.

Intersection of Nash implementable social choice correspondences

We show that there exist two Nash implementable social choice correspondences defined on an environment with strict preferences for which the intersection is not Nash implementable.

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.