Elsevier

Journal of Economic Theory

Volume 148, Issue 5, September 2013, Pages 2183-2193
Journal of Economic Theory

Notes
A simple sufficient condition for strong implementation

https://doi.org/10.1016/j.jet.2013.04.013Get rights and content

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.

Introduction

The goal of implementation theory is to characterize those social choice correspondences that can be realized in a decentralized way when information is dispersed in society and individuals cannot be trusted to act sincerely. In a seminal contribution, Maskin [12]1 showed that a property called monotonicity is necessary for Nash implementation, and furthermore, together with a weak property called no veto power it forms a sufficient condition. Monotonicity says, roughly speaking, that when an alternative is selected under one preference profile and its ranking has not dropped under another, then it should be selected also under this second profile. No veto power, on the other hand, says that when an alternative is top ranked by all individuals except possibly one, it should be selected. However, it was not until [16] and [5] that the gap between the necessary and sufficient conditions was finally closed.2

The logical next step after solving the Nash implementation problem, which can be interpreted as the competitive case, was to study a cooperative solution concept. An obvious candidate was the strong equilibrium. This is a Nash equilibrium that is not resistant only to individual deviations, but also to deviations by coalitions, all coalitions being possible.3 It was discovered quite early in the literature that monotonicity is necessary also for implementation in strong equilibrium, but it does not form a sufficient condition when taken together with no veto power [13]. However, a full characterization (a necessary and sufficient condition) exists today also in the cooperative case. Dutta and Sen [4] and Suh [21] gave a full characterization that is based on the idea laid down by Moore and Repullo [16] and Dutta and Sen [5] in the case of Nash implementation, while Fristrup and Keiding [6] gave a full characterization that is based on the idea of effectivity functions developed for example by Moulin and Peleg [17] and Holzman [8].

Despite the considerable progress, there is a notable shortcoming in the literature on strong implementation. Namely, there does not exist a condition comparable to the sufficient condition of Maskin [12] in the case of Nash implementation. This is a substantial handicap. The original sufficient condition of Maskin has two big advantages in comparison over the vast literature that followed: First, it is often easy to verify, and second, it has a nice and intuitive interpretation. All the characterizations that are based on the idea of Moore and Repullo [16], while informative, are almost as hard to verify as it is to find an implementing game form itself. All the characterizations that are based on the effectivity function, on the other hand, apply only to comprehensive preference domains.

In this note we improve the situation by giving a simple and intuitive sufficient condition for strong implementation. We dub the central property of this condition as the axiom of sufficient reason. Letʼs say that a pair (b,i), where b is an alternative and i is an individual, is a reason to select alternative a if individual i prefers alternative a to alternative b. A social choice correspondence satisfies the axiom of sufficient reason if, whenever an alternative a is selected under one preference profile and every reason to select it is also a reason to select c (possibly different from a) under another profile, then alternative c should be selected under this second profile. However, this property is not sufficient by itself. Two other conditions, the existence of a default alternative and strong Pareto optimality (which is itself an “almost” necessary condition for strong implementation), are also needed. Although our sufficient condition is considerably stronger than monotonicity, as a sufficient condition should be by the results of Maskin [13], it seems to be applicable in many interesting cases.

The rest of this note is organized as follows: In Section 2 we introduce all the basic definitions needed in every implementation paper. A reader familiar with the field can easily skip this section, although one should take a look at our solution concept, which is a bit different than usual. In Section 3 we show that when there are only two individuals, strong implementation does not much differ from Nash implementation beyond the additional requirement of strong Pareto optimality. In Section 4 we move to the more difficult case of at least three individuals. Here we finally introduce the axiom of sufficient reason and prove the main theorem (Theorem 2). Then, in Section 5, we show that there are many well-known examples to which our sufficient condition can be applied. In particular, we show that it is useful in the context of matching problems. This is potentially significant, since the original result of Maskin [12] does not seem to work in the context of matching problems. Usually, the result of Danilov [2] or Moore and Repullo [16] have to be used instead (see Kara and Sönmez [9], [10]). Finally, Section 6 concludes the paper with a brief discussion.

Section snippets

Notation and preliminaries

Let N={1,,n} be the set of individuals (the society) and A the set of (social) alternatives. We denote the set of all complete and transitive preference relations over A by RA. This is the unrestricted domain or the full domain. A typical element of RA is denoted by R and the preference relation of individual i in this profile by Ri, with a strict part Pi and indifference part Ii respectively. As standard, Ri denotes an (n1)-dimensional profile that specifies the preference relation of every

The relatively simple two player case

In Nash implementation theory the case of two individuals (n=2) is much harder than the case of three or more individuals (n3) (see, for example, Moore and Repullo [16], or Dutta and Sen [5]). Roughly speaking, this is because an additional intersection condition must hold for lower contour sets when there are only two individuals. In contrast, the case of three or more individuals turns out to be much harder than two individual cases if we want to implement in strong equilibrium. In fact, as

The considerably more difficult case of at least three players

What exactly makes the case of three or more individuals more difficult than the case of two individuals? When there are only two individuals, essentially two things can happen. These individuals can either compete, that is, they can play Nash equilibrium, or cooperate against the planner, in other words form a grand coalition. What cannot happen is that a set of individuals cooperate to compete against the rest of the individuals, which is something that can happen only when there are at least

A few important examples

We conclude this paper by giving three examples in which our new sufficient condition can be applied.

Example 1 Individually rational and Pareto optimal SCCs

Fix an alternative a0A and define IRi(a0,R)={aA|aRia0}. This is the set of all alternatives that are at least as good as a0 for individual i. The setIR(a0,R)iNIRi(a0,R) is called the individually rational correspondence with respect to a0. This correspondence satisfies DA, since a0sPO(R) implies IR(a0,R)={a0}. It is easy to verify that this correspondence satisfies also ASR. However, it

Concluding comments

We have derived a new sufficient condition for a social choice correspondence to be implementable in strong equilibrium. The main property of this condition, called the axiom of sufficient reason (ASR), is often easy to verify and has a nice and intuitive interpretation. In addition, it is appealing also as a normative criterion, and therefore one may speculate whether a good SCC should satisfy it quite irrespective of the implementation goal. In other words, one might claim that the goal of a

Acknowledgments

I wish to thank all the participants of the weekly seminar of the Public Choice Research Centre (PCRC) at University of Turku for their helpful comments. In particular, Hannu Salonen and Marko Ahteensuu. I also would like to thank an anonymous referee for useful comments that have improved the quality of this paper considerably. Part of this paper was written while I was visiting Harvard Economics Department, and I am grateful for their hospitality. The financial support from the Academy of

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