Double implementation of Lindahl allocations by a pure mechanism

This paper considers the issue of designing mechanisms whose Nash allocations and strong Nash allocations coincide with Lindahl allocations for public goods economies when coalition patterns, preferences, and endowments are unknown to the designer. It will be noted that the mechanism presented here is feasible and continuous, and the implementation result is obtained without defining an artificial preference profile on prices announced by individuals. In addition, unlike most existing Nash-implementing mechanisms which need to distinguish the case of two agents from that of three or more agents, this paper provides a unified mechanism which is irrespective of the number of agents. Received: 19 August 1997/Accepted: 16 November 1998     

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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Reconsidering two-agent Nash implementation

In this paper, we reconsider the full characterization of two-agent Nash implementation provided in the celebrated papers by Moore and Repullo (Econometrica 58:1083–1099, 1990) and Dutta and Sen (Rev Econ Stud 58:121–128, 1991), since we are able to show that the characterizing conditions are not logically independent. We prove that an amended version of the conditions proposed in these papers is still necessary and sufficient for Nash implementability. Then, by using our necessary and sufficient condition, we show that Maskin’s impossibility result can be avoided under restrictions on the outcomes and the domain of preferences much weaker than those previously imposed by Moore and Repullo (Econometrica 58:1083–1099, 1990) and Dutta and Sen (Rev Econ Stud 58:121–128, 1991).     

Double implementation in Nash and strong Nash equilibria

 We deal with the implementation problem where agents in some groups can cooperate (or form coalitions) and in some others cannot. We consider a situation where the planner does not know whether agents in a group can cooperate or not. We provide a necessary and sufficient condition for a correspondence to be implementable in such a situation. Received: 28 February 1995/Accepted: 13 March 1996     

Anonymous implementation of the Lindahl correspondence: possibility and impossibility results

This paper deals with the existence of anonymous mechanisms to realize the Lindahl correspondence. We consider, in turn, constant and decreasing returns to scale technologies for producing public goods. In each case, we provide a continuous (but not smooth) and weakly balanced mechanism, which meets the two conditions. We then remark that they satisfy a property (see property NC), which is related to, but in fact stronger than anonymity. Finally, we prove that if a mechanism has this property, if it is weakly balanced and implements the Lindahl correspondence, then it cannot be differentiable around Nash equilibria.     

Nash implementation and the bargaining problem

We study Nash implementation of the bargaining solutions in a cake sharing set up. We argue that the minimal Pareto optimal, symmetric and Nash implementable SBS is the one inducing all Pareto optimal and midpoint-dominating utility vectors in each state.     

A universal meta bargaining implementation of the Nash solution

This paper follows van Damme (1986) in presenting a meta bargaining approach that justifies the Nash bargaining solution. But in contrast to van Damme's procedure our meta bargaining game is universal in the sense that all bargaining solutions are allowed as strategic choices in the meta bargaining game. Also our result holds true for any number n of players. Received: 31 July 2000/Accepted: 19 March 2001     

A simple proof of Maskin's theorem on Nash implementation

This paper gives a simple proof of the following result, which is due to Maskin [1]: If an n-person social choice rule f, with n3, satisfied monotonicity and no veto power, then f can be implemented in Nash equilibria.     

On the necessary and sufficient conditions for Nash implementation

The purpose of this paper is to provide a constructive way of checking whether or not a social choice correspondence can be implemented in Nash equilibria. The results apply when there are two or more players. The usefulness of this constructive approach is illustrated for the case of single-peaked preferences over , a two-person public good economy with monotonic preferences, and a two-person exchange economy with monotonic preferences.I am grateful to Rajeev Bhattacharya, William Thomson, Takehiko Yamato, and an anonymous referee, for helpful comments. This research was supported by grants from the Bank of Sweden Tercentenary Foundation and the Royal Swedish Academy of Sciences.     

Game forms for Nash implementation of general social choice correspondences

Several game forms are given for Nash implementing general social choice correspondences (SCC's) which satisfy Maskin's conditions of monotonicity and No Veto Power. The game forms have smaller strategy spaces than those used in previously studied mechanisms: the strategy for an individual consists of an alternative, two subsets (of alternatives), and a player number. For certain types of economic and political SCC's, including -majority rule, the Walrasian, and Lindahl correspondence, the strategy space reduces to an alternative and a vector, where the number of components of the vector is at most twice the dimension of the alternative space.This paper was inspired by Tatsuyoshi Saijo's dissertation. I am greatly indebted to Kim Border, Leo Hurwicz, and Tatsuyoshi Saijo for some very helpful discussions. I acknowledge support of NSF grant No. SES 8208184.     

Existence and Nash implementation of efficient sharing rules for a commonly owned technology

Suppose that a group of individuals owns collectively a technology which produces a consumption good by means of a (possibly heterogeneous) input. A sharing rule associates input contributions with a vector of consumptions that are technologically feasible. We show that the set of allocations obtained by any continuous sharing rule contains Pareto efficient allocations. We also present a mechanism that implements in Nash equilibrium the Pareto efficient allocations contained in an arbitrary sharing rule. Received: 29 June 1998/Accepted: 15 November 2000     

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.     

Strategy-proof,efficient, and nonbossy quota allocations

We consider the problem of designing a mechanism to allocate objects to agents when each agent has a quota that must be filled exactly. Agents are assumed to have responsive preferences over items. We show that the only strategy-proof, Pareto optimal, and nonbossy mechanisms are sequential dictatorships. We also show that the only strategy-proof, Pareto optimal, nonbossy, and neutral mechanisms are serial dictatorships. Since these negative results hold for responsive preferences, they hold for more general preferences as well. We are grateful to Atila Abdulkadiroğlu, Syed Nageeb Mustafa Ali, Douglas Bernheim, Navin Kartik, Jon Levin, Paul Milgrom, and Azeem Shaikh and two anonymous referees and an editor for helpful discussions. This research has been supported by the Burt and Deedee McMurtry Stanford Graduate Fellowship.     

A simple procedure for finding equitable allocations of indivisible goods

The paper investigates how far a particular procedure, called the “descending demand procedure,” can take us in finding equitable allocations of indivisible goods. Both interpersonal and intrapersonal criteria of equitability are considered. It is shown that the procedure generally fares well on an interpersonal criterion of “balancedness”; specifically, the resulting allocations are Pareto-optimal and maximize the well-being of the worst-off individual. As a criterion of intrapersonal equitability, the property of envy-freeness is considered. To accommodate envy-freeness, a modification of the basic procedure is suggested. With two individuals, the modified procedure is shown to select the envy-free allocations that are balanced, i.e. the allocations that maximize the well-being of the worse-off individual among all envy-free allocations. Received: 3 March 2000/Accepted: 27 November 2000     

Money-egalitarian-equivalent and gain-maximin allocations of indivisible items with monetary compensation

Suppose that a certain quantity M of money and a finite number of indivisible items are to be distributed among n people, all of whom have equal claims on the whole. Different allocations are presented using various criteria of fairness in the special case where each player's utility function is additively separable. An allocation is “money-egalitarian-equivalent” (MEE) if each player's monetary valuation of his or her bundle is a fixed constant. We show that there is an essentially unique allocation that is MEE and Pareto-optimal; it is also envy-free. Alternatively, the “gain” of a player may be defined as the difference between how the player evaluates his bundle and an exact nth part of the whole according to his numerical evaluation of the whole. A “gain-maximin” criterion would maximize the minimum gain obtained by any player. We show that Knaster's procedure finds an allocation which is optimal under the gain-maximin criterion. That allocation is not necessarily envy-free, so we also find the envy-free allocation that is optimal under the gain-maximin criterion among all envy-free allocations. It turns out that, even though there exist allocations that are simultaneously envy-free and Pareto-optimal, this optimal allocation may fail to be Pareto-optimal, and it may also violate monotonicity criteria. Received: 30 September 1996/Accepted: 6 March 2002 The author would like to thank Professor William Thomson for a discussion on this subject; and he would like to thank the anonymous referees, who made many substantive suggestions for improving this paper – shortening it, streamlining the arguments, improving the terminology, making further ties with the literature, and improving the exposition.     

Maximal symmetry and the Nash solution

The Nash Bargaining Solution is characterised by using the new axiom of Maximal Symmetry in place of Nash's Independence of Irrelevant Alternatives and Symmetry. This axiom expresses the idea that a fair arbitrator should treat symmetric alternatives in the same way, subject to the feasibility constraint. An advantage of the proposed characterisations is that they are valid on a wide set of domains, in particular domains including, or consisting of, non-convex problems. Received: 16 March 1998/Accepted: 13 October 1998     

An axiomatization of the Nash bargaining solution

I prove that ‘Disagreement Point Convexity’ and ‘Midpoint Domination’ characterize the Nash bargaining solution on the class of two-player bargaining problems and on the class of smooth bargaining problems. I propose an example to show that these two axioms do not characterize the Nash bargaining solution on the class of bargaining problems with more than two players. I prove that the other solutions that satisfy these two properties are not lower hemi-continuous. These different results refine the analysis of Chun (Econ Lett 34:311–316, 1990). I also highlight a rather unexpected link with the result of Dagan et al. (Soc Choice Welfare 19:811–823, 2002).     

A characterization of the Nash bargaining solution

We characterize the Nash bargaining solution replacing the axiom of Independence of Irrelevant Alternatives with three independent axioms: Independence of Non-Individually Rational Alternatives, Twisting, and Disagreement Point Convexity. We give a non-cooperative bargaining interpretation to this last axiom. Received: 4 September 2000/Accepted: 6 September 2001     

On two-person Nash implementable choice functions

An elementary proof of a theorem on two-person Nash implementable choice functions is provided.