Social Choice Correspondences Research Articles

The finer structure of resolute, neutral, and anonymous social choice correspondences

Abstract When the number of alternatives exceeds the smallest prime dividing the number of individuals, all social choice functions satisfying resoluteness, neutrality, and anonymity (RNA rules) sometimes choose alternatives that everyone ranks near the bottom. When the number of individuals is divisible by two or more distinct primes, RNA rules only exist for a small (finite) number of values of the number of alternatives.

Preference domains and the monotonicity of condorcet extensions

An alternative is a Condorcet winner if it beats all other alternatives in a pairwise majority vote. A social choice correspondence is a Condorcet extension if it selects the Condorcet winners–and nothing else–whenever a Condorcet winner exists. It is well known that Condorcet extensions are not monotonic (hence, not Nash implementable) when all preferences are admissible, but are implementable when restricted to a domain in which Condorcet winners always exist. We fill the gap by studying the intermediate domains and find that monotonicity is violated on all such domains.

Voting games of resolute social choice correspondences

A resolute social choice correspondence is a social choice rule which maps preference profiles into sets of mutually compatible outcomes. We consider a fairly large class of resolute social choice correspondences and characterize the strong Nash equilibrium outcomes of their voting games in terms of a generalization of the Condorcet principle. Our findings generalize those of Sertel and Sanver (Soc Choice Welf 22:331–347, 2004) who address the same question in a more restricted framework.

Nash implementation in private good economies with single-plateaued preferences and in matching problems

In this paper we explore the problem of Nash implementation providing two new properties called I-monotonicity and I-weak no-veto power. These properties form together with unanimity a new sufficient condition for a social choice correspondence to be implementable in Nash equilibria with at least three agents. As applications, we prove that: (1) In private good economies with single-plateaued preferences, each solution of the problem of fair division that has a full range is Nash implementable if and only if it satisfies Maskin monotonicity, (2) In matching problems, Maskin monotonicity is a necessary and sufficient condition for implementation.

Nash Implementation in Rationing Problems with Single-Crossing Preferences

In this paper we study the rationing problems in using the issue of Nash implementation in an environment of single-crossing preferences. We show that strict monotonicity ( i ) implies strict weak no-veto power and unanimity and ( ii ) is equivalent to Maskin monotonicity, which is vacuously checked in this domain. We show that any social choice correspondence that has full range can be implemented in Nash equilibria.

A fundamental structure of strategy-proof social choice correspondences with restricted preferences over alternatives

I prove that under each strategy-proof and unanimous social choice correspondence, there is at least one agent who is decisive. Because the result is established on a weak requirement on preferences over sets, the existence of a decisive agent is an underlying feature of most strategy-proof and unanimous social choice correspondences. Moreover, I consider a restriction on the space of preferences over alternatives. I prove that circular sets of preferences over alternatives are sufficient for the existence of a decisive agent.

Natural Implementation with Partially Honest Agents

In this paper, we introduce the weak and the strong notions of partially honest agents (Dutta and Sen, 2012), and then study implementation by natural price-quantity mechanisms (Saijo et al., 1996, 1999) in pure exchange economies with three or more agents in which pure-consequentialistically rational agents and partially honest agents coexist. Firstly, assuming that there exists at least one partially honest agent in either the weak notion or the strong notion, the class of efficient social choice correspondences which are Nash-implementable by such mechanisms is characterized. Secondly, the (unconstrained) Walrasian correspondence is shown to be implementable by such a mechanism when there is at least one partially honest agent of the strong type, which may provide a behavioral foundation for decentralized implementation of the Walrasian equilibrium. Finally, in this set-up, the effects of honesty on the implementation of more equitable Pareto optimal allocations can be viewed as negligible.

A simple sufficient condition for strong implementation

Abstract In an important step forward Maskin [E. Maskin, Nash equilibrium and welfare optimality, Rev. Econ. Stud. 66 (1999) 23–38] showed that two properties – monotonicity and no veto power – are together sufficient for Nash implementation. In contrast to the vast literature that followed, this characterization has two major advantages: First, it is often easy to verify, and second, it has an elegant and simple interpretation. However, there does not exist a similar condition for social choice correspondences that are implementable in strong equilibrium. All existing characterizations are either hard to verify or apply only to comprehensive preference domains. In this paper we improve the situation by giving one such condition. Moreover, using well-known examples we show that this is a practical tool.

A unifying impossibility theorem

This paper identifies and illuminates a common impossibility principle underlying a number of impossibility theorems in social choice. We consider social choice correspondences assigning a choice set to each non-empty subset of social alternatives. Three simple axioms are imposed as follows: unanimity, independence of preferences over infeasible alternatives, and choice consistency with respect to choices out of all possible alternatives. With more than three social alternatives and the universal preference domain, any social choice correspondence that satisfies our axioms is serially dictatorial. A number of known impossibility theorems—including Arrow’s Impossibility Theorem, the Muller–Satterthwaite Theorem, and the impossibility theorem under strategic candidacy—follow as corollaries.

Nash Implementation in an Allocation Problem with Single-Dipped Preferences

In this paper, we study the Nash implementation in an allocation problem with single-dipped preferences. We show that, with at least three agents, Maskin monotonicity is necessary and sufficient for implementation. We examine the implementability of various social choice correspondences (SCCs) in this environment, and prove that some well-known SCCs are Maskin monotonic ( but they do not satisfy no-veto power) and hence Nash implementable.

Nash Implementation in Private Good Economies when Preferences are Single-Dipped with Best Indifferent Allocations

In this paper we study the problem of the Nash implementation in private good economies when preferences are single-dipped by allowing multiple best indifferent elements. We give a full characterization in this domain by showing that a social choice correspondence (SCC) is Nash implementable if and only if satisfies Maskin monotonicity. To support this result, we give examples of some well-known SCCs that satisfying or not satisfying Maskin's proprieties.

Implementation in mixed Nash equilibrium

A mechanism implements a social choice correspondence f in mixed Nash equilibrium if, at any preference profile, the set of all (pure and mixed) Nash equilibrium outcomes coincides with the set of f-optimal alternatives for all cardinal representations of the preference profile. Unlike Maskinʼs definition, our definition does not require each optimal alternative to be the outcome of a pure equilibrium. We show that set-monotonicity, a weakening of Maskinʼs monotonicity, is necessary for mixed Nash implementation. With at least three players, set-monotonicity and no-veto power are sufficient. Important correspondences that are not Maskin monotonic can be implemented in mixed Nash equilibrium.

A full characterization of nash implementation with strategy space reduction

The paper fully characterizes the class of Nash implementable social choice correspondences (SCCs) by mechanisms endowed with Saijo’s message space specification—\(s\)-mechanisms. This class of SCCs is equivalent to the class of Nash implementable SCCs, though any game form involving ‘one less’ preference announcements breaks this equivalence relationship down.

Nash implementation via simple stochastic mechanisms: strategy space reduction

Benoit and Ok (Games Econ Behav 64:51–67, 2008) show that in a society with at least three agents any weakly unanimous social choice correspondence (SCC) is Maskin’s monotonic if and only if it is Nash-implementable via a simple stochastic mechanism (Benoit-Ok’s Theorem). This paper fully identifies the class of weakly unanimous SCCs that are Nash-implementable via a simple stochastic mechanism endowed with Saijo’s message space specification (Saijo in Econometrica 56:693–700, 1988). It is shown that this class of SCCs is equivalent to the class of SCCs that are Nash-implementable via Benoit-Ok’s Theorem.

Secure Implementation in Queueing Problems

This paper studies secure implementability (T. Saijo, T. Sjostrom and T. Yamato, “Secure Implementation,” Theoretical Economics, Vol. 2, No. 3, 2007, pp. 203-229) in queueing problems. Our main result shows that the social choice function satisfies strategy-proofness and strong non-bossiness (Z. Ritz, “Restricted Domains, Arrow-Social Welfare Functions and Noncorruptible and Non-Manipulable Social Choice Correspondences: The Case of Private Alternatives,” Mathematical Social Science, Vol. 4, No. 2, 1983, pp. 155-179), both of which are necessary for secure imple- mentation, if and only if it is constant on the domains that satisfy weak indifference introduced in this paper. Weak in- difference is weaker than minimal richness (Y. Fujinaka and T. Wakayama, “Secure Implementation in Economies with Indivisible Objects and Money,” Economics Letters, Vol. 100, No. 1, 2008, pp. 91-95). Our main result illustrates that secure implementation is too difficult in queueing problems since many reasonable domains satisfy weak indifference, for example, convex domains.

Internal slackening scoring methods

We deal with the ranking problem of the nodes in a directed graph. The bilateral relationships specified by a directed graph may reflect the outcomes of a sport competition, the mutual reference structure between websites, or a group preference structure over alternatives. We introduce a class of scoring methods for directed graphs, indexed by a single nonnegative parameter α. This parameter reflects the internal slackening of a node within an underlying iterative process. The class of so-called internal slackening scoring methods, denoted by λα, consists of the limits of these processes. It is seen that λ0 extends the invariant scoring method, while λ∞ extends the fair bets scoring method. Method λ1 corresponds with the existing λ-scoring method of Borm et al. (Ann Oper Res 109(1):61–75, 2002) and can be seen as a compromise between λ0 and λ∞. In particular, an explicit proportionality relation between λα and λ1 is derived. Moreover, the internal slackening scoring methods are applied to the setting of social choice situations where they give rise to a class of social choice correspondences that refine both the Top cycle correspondence and the Uncovered set correspondence.

Natural Implementation with Partially-Honest Agents

The paper proposes necessary and su¢ cient conditions for the natural implementation of (e¢ cient) social choice correspondences (SCCs) in pure …nite exchange economies when some of the agents are partially honest. A partially honest agent is an agent who strictly prefers to tell the truth when lying has no better material consequences for her. Firstly, it is shown that if there is even one partially honest agent in the economy (and the planner does not know her identity), then any SCC is Nash implementable by a natural price-allocation mechanism. Secondly, and in sharp contrast with the results of conventional models of natural implementation, it is shown that the equivalence relationship between natural price-allocation mechanisms and natural price-quantity 2 mechanisms no longer holds. Finally, and even more strikingly, the paper reports that the class of implementable SCCs by natural price-quantity mechanisms is signi…cantly enlarged.

A Full Characterization of Nash Implementation with Strategy Space Reduction

Noting that a full characterization of Nash-implementation is given using a canonical-mechanism and Maskin's theorem (Maskin, 1999) is shown using a mechanism with Saijo's type of strategy space reduction (Saijo, 1988), this paper fully characterizes the class of Nash-implementable social choice correspondences (SCCs) by mechanisms endowed with Saijo's message space specification - s-mechanisms. This class of SCCs is equivalent to the class of Nash-implementable SCCs, though any further `strategy space reduction' mechanism breaks this equivalent relationship down

Implementation with renegotiation when preferences and feasible sets are state dependent

In this paper, we present a model of implementation where infeasible allocations are converted into feasible ones through a process of renegotiation that is represented by a reversion function. We describe the maximal set of Social Choice Correspondences that can be implemented in Nash Equilibrium in a class of reversion functions that punish agents for infeasibilities. This is used to study the implementation of the Walrasian Correspondence and several axiomatic solutions to problems of bargaining and taxation.

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 ε-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 ε-minimax regret equilibrium for all ε ∈ [ 0 , 1 ) .