Arunava Sen

VIRTUAL IMPLEMENTATION IN NASH EQUILIBRIUM

Consider a social choice correspondence as a mapping from preference profiles to lotteries over some finite set of alternatives. A virtually implementable social choice function in Nash equilibrium is defined, under mild domain restrictions it is shown that in societies with at least three individuals all social choice correspondences are virtually implementable in Nash equilibrium. This contrasts with Maskins classic characterization, which requires monotonicity as a necessary condition for exact implementation in Nash equilibrium. The two person case is considered seperately. While not all two-person social choice functions are virtually implementable, our necessary and sufficient condition is simple, which contrasts with the complex necessary and sufficient conditions for exact implementation. Copyright 1991 by The Econometric Society.

A Necessary and Sufficient Condition for Two-Person Nash Implementation

The main result of this paper is to characterize the class of two-person social choice correspondences which are Nash-implementable. The characterization result is used to formulate domain restrictions which allow the construction of non-dictatorial and Pareto-efficient social choice correspondences which are Nash-implementable.

Subgame perfect implementation: A necessary and almost sufficient condition

We present a necessary and almost sufficient condition for subgame perfect implementation of social choice correspondences. In societies with at least three individuals, any social choice correspondence which satisfies no veto power and our necessary Condition α is subgame perfect implementable. Thus Condition α is analogous to monotonicity which, by Maskins celebrated characterization, is necessary and, in a similar way, almost sufficient for Nash implementation.

Nash implementation through elementary mechanisms in economic environments

This paper identifies a class of mechanisms, called elementary mechanisms, which are (in a precisely defined sense) the “simplest” mechanisms that can implement efficient outcomes in economic environments. The class of social choice correspondences that can be implemented by elementary mechanisms is completely characterized in a variety of different economic contexts.

Separable preferences, strategyproofness, and decomposability

We consider strategyproof social choice functions defined over product domains. If preferences are strict orderings and separable, then strategyproof social choice functions must be decomposable provided that the domain of preferences is rich. We provide several characterization results in the case where preferences are separable only with respect to the elements of some partition of the set of components and these partitions vary across individuals. We characterize the libertarian social choice function and show that no superset of the tops separable domain admits strategyproof nondictatorial social choice functions.

Another direct proof of the Gibbard-Satterthwaite Theorem

Abstract This paper provides a new and direct proof of the Gibbard–Satterthwaite Theorem based on induction on the number of individuals.

Implementation under strong equilibrium : A complete characterization

Abstract This paper provides a complete characterization of the class of social choice correspondences which are implementable under strong Nash equilibrium. The paper departs from earlier literature on this topic by discarding the use of effectivity functions and uses instead a variant of the game forms used in the context of Nash implementation. It is also shown that the results derived earlier with the use of effectivity functions are implied by the characterization result presented here.

Strategy-proof Cardinal Decision Schemes

This paper analyses strategy-proof mechanisms or decision schemes which map profiles of cardinal utility functions to lotteries over a finite set of outcomes. We provide a new proof of Hylland’s theorem which shows that the only strategy-proof cardinal decision scheme satisfying a weak unanimity property is the random dictatorship. Our proof technique assumes a framework where individuals can discern utility differences only if the difference is at least some fixed number which we call the grid size. We also prove a limit random dictatorship result which shows that any sequence of strategy-proof and unanimous decision schemes defined on a sequence of decreasing grid sizes approaching zero must converge to a random dictatorship.

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

A corollary of Maskins 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.

Random Dictatorship Domains

A domain of preference orderings is a random dictatorship domain if every strategy-proof random social choice function satisfying unanimity defined on the domain is a random dictatorship. Gibbard (1977) showed that the universal domain is a random dictatorship domain. We ask whether an arbitrary dictatorial domain is a random dictatorship domain and show that the answer is negative by constructing dictatorial domains that admit anonymous, unanimous, strategy-proof random social choice functions which are not random dictatorships. Our result applies to the constrained voting model. Lastly, we show that substantial strengthenings of linked domains (a class of dictatorial domains introduced in Aswal et al., 2003) are needed to restore random dictatorship and such strengthenings are “almost necessary”.