Consistent collusion-proofness and correlation in exchange economies

We present a feasible strategic market mechanism with finitely many agents whose Nash, semi-strong Nash and coalition-proof Nash equilibria fully implement the Walrasian equilibria. We define a strategic equilibrium concept, called correlated semi-strong equilibrium, and show that the Walrasian equilibria can be implemented by these equilibria, and also by the coalition-proof correlated equilibria of our mechanism. We show that these two concepts, suitably modified with transfers, fully implement the Pareto optimal allocations.     

Double implementation of the ratio correspondence by a market mechanism

To overcome deficits of the Lindahl solution concept when the economy does not exhibit constant returns to scale, Kaneko (1977a) introduced the concept of aratio equilibrium. Theratio correspondence selects for each economy its set of ratio equilibrium allocations. In this paper we provide a simple market game thatdouble implements the ratio correspondence in Nash and strong equilibria.     

Virtual Nash implementation with admissible support

A social choice correspondence (SCC) is virtually implementable if it is ?$?$-close (in the probability simplex) to some (exactly) implementable correspondence [Abreu, D., Sen, A., 1991. Virtual Implementation in Nash Equilibrium. Econometrica 59, 997–1021] proved that, without restriction on the set of alternatives receiving strictly positive probability at equilibrium, every SCC is virtually implementable in Nash Equilibrium. We study virtual implementation when the supports of equilibrium lotteries are restricted. We provide a necessary and sufficient condition, imposing joint restrictions on SCCs and admissible supports. Next, we discuss how to construct supports, and we underline an important difficulty. Finally, we study virtual implementation when the support is restricted to the efficient or individually rational alternatives.     

Implementability via protective equilibria

We present a notion of non-cooperative strategic equilibrium for games generated by social choice functions, and fully characterize the class of those functions which are directly implementable under this equilibrium concept. Correct preference revelation turns out to always be such an equilibrium for the games generated by this class of implementable functions.     

Coalitional stability in the location problem with single-dipped preferences: An application of the minimax theorem

This study considers a situation in which agents choose the location of a public facility from a street according to a given mechanism. Agents have single-dipped preferences over a set of feasible locations. We analyze coalitional behavior for any given mechanism for this situation. We identify a necessary and sufficient condition for a mechanism to possess a strong Nash equilibrium by applying the minimax theorem of von Neumann and Morgenstern (1944). We introduce a class of core solutions and show that these solutions are characterized by strong Nash implementability. As a byproduct of these results, we propose a simple mechanism that implements any core solution in strong Nash equilibria.     

Strategic behavior in non-atomic games

In order to remedy the possible loss of strategic interaction in non-atomic games with a societal choice, this study proposes a refinement of Nash equilibrium, strategic equilibrium. Given a non-atomic game, its perturbed game is one in which every player believes that he alone has a small, but positive, impact on the societal choice; and a distribution is a strategic equilibrium if it is a limit point of a sequence of Nash equilibrium distributions of games in which each player’s belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that all of them must be Nash. We also show that all regular equilibria of smooth non-atomic games are strategic. Moreover, it is displayed that in many economic applications, the set of strategic equilibria coincides with that of Nash equilibria of large finite games.     

Nash implementing social choice rules with restricted ranges

We consider Nash implementation of social choice rules with restricted ranges, i.e., where some of the available alternatives are chosen at no preference profile. We show that the appropriate adaptation of Maskin monotonicity to this context depends on the range of the mechanisms: the wider is this range, the weaker is the monotonicity condition to be used. As a result, mechanisms employing outcome functions which allow for out-of-range alternatives at off-equilibrium messages can Nash implement social choice rules which fail to be Nash implementable by mechanisms whose ranges are restricted to the range of the social choice rule to be implemented. The Walrasian social choice correspondence is a particular instance of this. Moreover, social choice rules which are not Maskin monotonic can be “monotonized” by the addition of artificial out-of-range alternatives—a point we illustrate through Solomon’s Dilemma.     

Equilibrium with coordination

The obvious equilibrium concepts in the simplest institutions for transferring ownership of commodities—bilateral exchange—are neither Nash equilibria nor cooperative equilibria. To study such equilibria as special cases of equilibria of a social system it is necessary to introduce coordination. Two or more agents coordinate their actions, if, when they consider an alternative to a state, they take as given—for agents with whom they coordinate—the alternative state. If there is no coordination we obtain Nash equilibrium as a special case. If there is complete coordination we obtain optimality as a special case. The main result is an existence theorem for a social system with coordination. This theorem is then applied to prove existence of exchange equilibria in an economy with bilateral exchange.     

Implementation with coalition formation: A complete characterization

The basic difference between Nash and strong (Nash) implementations originates from the possibility of coalition formation. The results of Nash implementation are applicable when agents in no group can cooperate (or form a coalition) and those of strong implementation are applicable when agents in every group can cooperate. In the present paper we consider an environment where agents in some groups can cooperate and agents in other groups cannot. We completely characterize the set of correspondences that are implementable in such a general environment, and we provide an algorithm which enables us to verify whether a correspondence satisfies the condition that appears in the characterization results. By applying the algorithm in an exchange economy, we show that the core, the definition of which depends on the possibility of cooperation, is implementable in the equilibrium corresponding to the cooperation possibility.     

政府随机审计的动态博弈模型及政策启示

本文在一个三阶段动态博弈框架下,对政府随机审计过程中的政府审计机关与政府审计对象以及国家与政府审计对象之间的动态博弈特征进行了分析。分析结果显示：政府审计机关与政府审计对象之间存在一个纯策略纳什均衡,而国家和政府审计对象之间存在一个混合策略纳什均衡。政府审计对象接受国家实质性审计的概率是政府审计对象被政府机关审计发现违规或不作为后的惩罚倍数的单调递减函数,而政府审计对象的违规或不作为的概率分别是政府审计对象被政府机关审计发现违规或不作为后的惩罚倍数和政府审计对象的应尽义务或责任的单调递减函数,却是政府审计机关审计行为成本的单调递增函数。同时,结合研究结论和中国政府审计全覆盖的现实,给出了一些政策建议。     

Implementation of Pareto efficient allocations

This paper considers Nash implementation and double implementation of Pareto efficient allocations for production economies. We allow production sets and preferences are unknown to the planner. We present a well-behaved mechanism that fully implements Pareto efficient allocations in Nash equilibrium. The mechanism then is modified to fully doubly implement Pareto efficient allocations in Nash and strong Nash equilibria. The mechanisms constructed in the paper have many nice properties such as feasibility and continuity. In addition, they use finite-dimensional message spaces. Furthermore, the mechanism works not only for three or more agents, but also for two-agent economies.     

Nash equilibrium reconsidered and an option for maximin

In this paper we will discuss some peculiarities of the Nash equilibrium which are at odds with its standard applications: (a) the underlying dynamic interpretation, (b) the incentive independency if equilibria are mixed, and (c) the unprofitability if equilibrium and maximin are mixed and the game is 2-by-2. Maximin is proposed as an alternative solution concept in “relevant situations”.     

Feasible implementation of taxation methods

This paper studies implementation of taxation methods in one-commodity environments in which the incomes of the agents are unknown to the planner. Feasibility out of equilibrium imposes that the mechanism depend on the environment. We present two mechanisms. The first one, which requires complete information, implements every taxation method in Nash, strong and coalition-proof equilibrium. The second, where informational requirements are relaxed, implements a large class of consistent and monotone methods in subgame perfect equilibrium. Neither mechanism employs the off-equilibrium devices used by the general theory. Under fully private information no method is implementable. Received: 12 March 1997 / Accepted: 21 July 1998     

Convergence in games with continua of equilibria

In game theory, the question of convergence of dynamical systems to the set of Nash equilibria has often been tackled. When the game admits a continuum of Nash equilibria, however, a natural and challenging question is whether convergence to the set of Nash equilibria implies convergence to a Nash equilibrium. In this paper we introduce a technique developed in Bhat and Bernstein (2003) as a useful way to answer this question. We illustrate it with the best-response dynamics in the local public good game played on a network, where continua of Nash equilibria often appear.     

Equilibrium selection in games: the mollifier method

This paper introduces an embedding of a Nash equilibrium into a sequence of perturbed games, which achieves continuous differentiability of best responses by mollifying them over a continuously differentiable density with compact support (window size). Along any sequence with shrinking window size, the equilibria are single-valued whenever the function has a regular Jacobian and the set of equilibria where it is singular has measure zero. We achieve a further reduction of the equilibrium set by inserting within the embedding a procedure that eliminates the strict interior of equilibrium sets. The mollifier can be approximated consistently using kernel density regression, and we sketch a non-stationary stochastic optimization algorithm that uses this approximation and converges with probability one to an equilibrium of the original game.     

Stable matchings and the small core in Nash equilibrium in the college admissions problem

Both rematching proof and strong equilibrium outcomes are stable with respect to the true preferences in the marriage problem. We show that not all rematching proof or strong equilibrium outcomes are stable in the college admissions problem. But we show that both rematching proof and strong equilibrium outcomes in truncations at the match point are all stable in the college admissions problem. Further, all true stable matchings can be achieved in both rematching proof and strong equilibrium in truncations at the match point. We show that any Nash equilibrium in truncations admits one and only one matching, stable or not. Therefore, the core at a Nash equilibrium in truncations must be small. But examples exist such that the set of stable matchings with respect to a Nash equilibrium may contain more than one matching. Nevertheless, each Nash equilibrium can only admit at most one true stable matching. If, indeed, there is a true stable matching at a Nash equilibrium, then the only possible equilibrium outcome will be the true stable matching, no matter how different are players' equilibrium strategies from the true preferences and how many other unstable matchings are there at that Nash equilibrium. Thus, we show that a necessary and sufficient condition for the stable matching rule to be implemented in a subset of Nash equilibria by the direct revelation game induced by a stable mechanism is that every Nash equilibrium profile in that subset admits one and only one true stable matching. Received: 30 December 1998 / Accepted: 12 October 2001 This paper is a revision of the paper “Manipulation and Stability in a College Admissions Problem” circulated since 1994. I thank Rich McLean, Abraham Neyman, Mark Satterthwaite, Sang-Chul Suh, and Tetsuji Yamada for helpful discussions. I thank the associate editor and the two anonymous referees for their helpful comments that have greatly improved the paper. I am grateful to the Kellogg G.S.M. at the Northwestern University for the hospitality for my visit. Any errors are mine.     

Taxation in an economy with private provision of public goods

This paper analyses the effects of taxation and subsidies in an economy with private provision of a public good. It is shown that in a situation where all individuals contribute, taxation affects the equilibrium allocation if and only if at least one individual's voluntary contribution to the public good has an impact on the aggregate tax payments of the others. We then consider linear nonneutral tax-subsidy schemes and analyse efficiency and uniqueness of the resulting Nash equilibria. We show that an efficient Nash equilibrium, where all individuals contribute, will in general not be unique, and establish a non-uniformity property which a tax-subsidy scheme must fulfil in order to induce a unique interior equilibrium that is efficient. Throughout the paper it is assumed that individuals fully understand and take into account the government's budget constraint. Received: 3 November 1997 / Accepted: 23 March 1999     

Constitutional implementation

We consider the problem of implementing a social choice correspondence H in Nash equilibrium when the constitution of the society is given by an effectivity function E. It is assumed that the effectivity function of , is a sub-correspondence of E. We found necessary and efficient conditions for a game form to implement H (in Nash equilibria), and to satisfy, at the same time, that , the effectivity function of , is a sub-correspondence of (which guarantees that is compatible with E). We also find sufficient conditions for the coincidence of the set of winning coalitions of and , and for . All our results are sharp as is shown by suitable examples. Received: 15 December 2000 / Accepted: 3 September 2001     

On pure-strategy Nash equilibria in price–quantity games

This paper examines the existence and characteristics of pure-strategy Nash equilibria in oligopoly models in which firms simultaneously set prices and quantities. Existence of a pure-strategy equilibrium is proved for a class of price–quantity games. If the demand function is continuous, then the equilibrium outcome is similar to that of a price-only model. With discontinuous demand and limited spillover, there are rationing equilibria in which combined production falls short of market demand. Moreover, there might again be an equilibrium reflecting the outcome of a price game. Competition in price and quantity thus yields Bertrand outcomes under a variety of market conditions.     

Large games with countable characteristics

We consider anonymous games with an atomless probability space of players in which players’ characteristics are countable. Our main result shows that the set of equilibrium distributions coincides with the set of distributions induced by equilibrium strategies together with the function assigning characteristics to players. This result implies the existence of Nash equilibria in continuous large games with countable characteristics.