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 in Rev Econ stud 66: 23–38, 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 (Jackon and Palfrey in J Econ Theory 98: 1–25, 2001).I thank Toyo Sakai for his comments on a previous draft. I also thank two anonymous referees and an editor of this journal for helpful comments that improved this paper. A previous version circulated as “A note on Maskin monotonicity”. After the results presented here were obtained, I became aware of a new unpublished paper by Benoit and Ok (2004). The result of Theorem 2 and the discussion that follows is partially similar to their Theorem 1.     

Strongly implementable social choice correspondences and the supernucleus

In this paper, we present a characterization of social choice correspondences which can be implemented in strong Nash equilibrium, stated in terms of the power structure implicit in the social choice rule. We extend the notion of an effectivity function to allow for simultaneous vetoing by several coalitions. This leads to the concept of a domination structure as a generalized effectivity function.  Using this concept and a solution known from the theory of effectivity functions, the supernucleus, we give a characterization of strongly implementable social choice correspondences as supernucleus correspondence relative to an appropriate domination structure. Received: 2 February 1996/Accepted: 2 February 1998     

Monotonicity implies generalized strategy-proofness for correspondences

We show that Maskin monotone social choice correspondences on sufficiently rich domains satisfy a generalized strategy-proofness property, thus generalizing Muller and Satterthwaite’s (1977) theorem to correspondences. The result is interpreted as a possibility theorem on the dominant-strategy implementability of monotone SCCs via set-valued mechanisms for agents who are completely ignorant about the finally selected outcome. Alternatively, the result yields a partial characterization of the restrictions entailed by Nash implementability of correspondences. Received: 3 November 1997/Accepted: 26 April 1999     

Existence of a coalitionally strategyproof social choice function: A constructive proof

This paper gives a concrete example of a nondictatorial, coalitionally strategyproof social choice function for countably infinite societies. The function is defined for those profiles such that for each alternative, the coalition that prefers it the most is “describable.” The “describable” coalitions are assumed to form a countable Boolean algebra. The paper discusses oligarchical characteristics of the function, employing a specific interpretation of an infinite society. The discussion clarifies within a single framework a connection between the negative result (the Gibbard-Satterthwaite theorem) for finite societies and the positive result for infinite ones. Received: 10 August 1998/Accepted: 29 February 2000     

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.     

Characterization of social choice sets in terms of individuals' maximal sets: The fixed agenda framework

In this paper we analyze the explicit representation of fixed agenda social choice correspondences under different rationality assumptions (independence, Pareto optimality, etc.). It is well known that, under some of these assumptions, the existence of dictators, oligarchies or individuals with veto power can be proven ([6] and [10]); but only a partial characterization of the social choice set is obtained. We establish a relationship between the social choice set and the individuals' maximal sets, which explicitly describes a fixed agenda social choice correspondence that satisfies these rationality assumptions. Received: 30 December 1997/Accepted: 20 December 1999     

The implementation of social choice functions via social choice correspondences: A general formulation and a limit result

A corollary of Maskin's characterization theorem for Nash implementable social choice correspondences is that only trivial social choice functions can be implemented. This paper explores the consequences of implementing non-trivial social choice functions by extending them minimally to social choice correspondences which are implementable. The concept of asymptotic monotonicity is introduced. The main result states that it is not possible to find social choice rules satisfying a mild condition on its range, which is asymptotically monotonic. The implication of this result is that the multiplicity of equilibria problem which is at the heart of Nash implementation theory persists even in the limit as the number of individuals in society tends to infinity. This is true even though the opportunities for an individual to manipulate the outcome disappears in the limit.This paper is an extensively revised version of a chapter of my Ph.D. dissertation submitted to Princeton University in June 1987. I wish to thank my advisor Hugo Sonnenschein for his valuable advice and constant encouragement. I am also grateful to Andrew Caplin, Vijay Krishna, William Thomson, Jean-Luc Vila and two anonymous referees of this journal for their numerous suggestions. All remaining errors are my own responsibility.     

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. An erratum to this article can be found at     

Strategic manipulability without resoluteness or shared beliefs: Gibbard-Satterthwaite generalized

The Gibbard-Satterthwaite Theorem on the manipulability of social-choice rules assumes resoluteness: there are no ties, no multi-member choice sets. Generalizations based on a familiar lottery idea allow ties but assume perfectly shared probabilistic beliefs about their resolution. We prove a more straightforward generalization that assumes almost no limit on ties or beliefs about them. Received: 15 December 1997/Accepted: 16 November 1998     

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.     

Nash implementation via hyperfunctions

Hyperfunctions are social choice rules which assign sets of alternatives to preference profiles over sets. Therefore, they are more general objects compared to standard (social choice) correspondences. In fact, every correspondence can be expressed in terms of an equivalent hyperfunction. We give a partial characterization of Nash-implementable hyperfunctions and explore the conditions under which correspondences have Nash-implementable equivalent hyperfunctions. While the strength of these conditions depends on the axioms used to extend preferences over alternatives to sets, they are at most as strong as the conjunction of Maskin monotonicity with the no veto power condition. Thus, our approach expands the set of Nash-implementable social choice rules. In fact, social choice rules such as the majority rule and the top cycle are Nash-implementable through their equivalent hyperfunctions, while they are not Maskin-monotonic, and thus, not Nash-implementable in the standard framework.

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Rational budgeters in the theory of social choice

The purpose of this paper is to explore duality in the theory of social choice. As application Arrow's Impossibility Theorem and another impossibility theorem using the notion of positive responsiveness are chosen. It will be seen that we can establish notions and theorems which are symmetric to the original ones. However, if we establish impossibility theorems when rational behaviour is described by budget correspondences and not by choice correspondences, we need not assume that every subset of X (a family of alternatives) with cardinality 2 is a budget set. Therefore the dual theorems also may hold for families of competitive budget sets. It will also be shown that although the underlying preferences on X need not be acyclic, local decisiveness on budget sets may lead to global decisiveness on these sets.     

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.     

Manipulating an aggregation rule under ordinally fuzzy preferences

It is well known that many aggregation rules are manipulable through strategic behaviour. Typically, the aggregation rules considered in the literature are social choice correspondences. In this paper the aggregation rules of interest are social welfare functions (SWFs). We investigate the problem of constructing a SWF that is non-manipulable. In this context, individuals attempt to manipulate a social ordering as opposed to a social choice. Using techniques from an ordinal version of fuzzy set theory, we introduce a class of ordinally 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 SWFs are dictatorial, provided that they are not constant. This theorem uses a weaker transitivity condition than the one in Perote-Peña and Piggins (J Math Econ 43:564–580, 2007), and the ordinal framework we employ is more general than the cardinal setting used there. We conclude by considering several ways of circumventing this impossibility theorem.     

Characterization of self-selective social choice functions on the tops-only domain

Self-selectivity is a new kind of consistency pertaining to social choice rules. It deals with the problem of whether a social choice rule selects itself from among other rival such rules when a society is also to choose the choice rule that it will employ in making its choice from a given set of alternatives. Koray [3] shows that a neutral and unanimous social choice function is universally self-selective if and only if it is dictatorial. In this paper, we confine the available social choice functions to the tops-only domain and examine whether such restriction allow us to escape the dictatoriality result. A neutral, unanimous, and tops-only social choice function, however, turns out to be self-selective relative to the tops-only domain if and only if it is top-monotonic, and thus again dictatorial. Received: 8 October 2001/Accepted: 4 June 2002     

On strategy-proof social choice correspondences: a comment

In a recent paper, Sato (Soc Choice Welf doi:) 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 Rodríguez-álvarez (Soc Choice Welf 29:175–199, 2007) to directly prove stronger versions of Sato’s theorems. The author wishes to thank Shin Sato and John Weymark for their comments and suggestions. Financial support from the Ministerio de Educación y Ciencia through the Programa Ramón y Cajal 2006 and grant SEJ-2005-04805, the Consejería de Innovación, Ciencia y Empresa, Junta de Andalucía through grant SEJ-01645, and the Fundación Ramón Areces is gratefully acknowledged.     

Veto in fixed agenda social choice correspondences

In this paper we analyze the relationship between social decision functions (SDF) and fixed agenda social choice correspondences (fixed agenda SCC), satisfying some rationality conditions (such as Pareto, Independence, Monotonicity, Neutrality or Anonimity), with the aim of translating known results on the existence of individuals with veto power into the fixed agenda framework (e.g., Blair and Pollak [2], Blau and Deb [3], …). In order to do this, a new independence condition (which we call Pseudo-Independence), as well as a notion of veto power, are introduced in the context of fixed agenda SCC. The conclusion is that, by fixing the agenda, under Independence and Pareto assumptions, the existence of individuals with veto power cannot be avoided. Received: 6 November 1995 / Accepted: 9 September 1997     

Anonymity, ordinal preference proximity and imposed social choices

Extending on an impossibility result by Baigent [1], it is shown that an anonymous social choice procedure which preserves preference proximity cannot satisfy the weakest possible form of non-imposition. Received: 20 September 1999/Accepted: 11 May 2001     

The structure of global centrality measures

The aim of this article is to identify and analyse the logic and structure of centrality measures applied to social networks. On the basis of the article by Borgatti and Everett, identifying the latent functions of centrality, we first use a survey of personal networks with 450 cases to perform an empirical study of the differences and correspondences between degree, closeness and betweenness centrality in personal networks. Then, we examine the correspondences between the three global indicators in each type of centrality: the maximum value, the mean value and the hierarchy or centralization. The results provide a better understanding of the centrality indicators of networks and the reality that they express in an empirical context.     

Comparison functions and choice correspondences

In this paper, we introduce the concept of a comparison function, which is a mapping g that assigns numbers to ordered pairs of alternatives (x,y) with the property that g(x,y)=−g(y,x). The paper discusses how some well-known choice correspondences on tournaments such as the uncovered set, the minimal covering set and the bipartisan set can be extended to this general framework. Axiomatic characterizations and properties are studied for these correspondences. Received: 4 November 1996/Accepted: 13 April 1998