Nash implementation with a private good

Summary. I construct a general model of social planning problems, including mixed production economies and regulatory problems with negative externalities as special cases, and I give simple mechanisms for Nash implementation under three increasingly general sets of assumptions. I first construct a continuous mechanism to implement the (constrained) Lindahl allocations of an economy, and I then extend this to arbitrary social choice rules based on prices. I end with a mechani sm to implement any monotonic social choice rule, assuming only the existence of a private (not necessarily transferable) good. In that general case, each agent simply reports an upper contour set, an outcome, and I need two agents to make binary numerical announcements. I do not require the usual no-veto-power condition. Received: February 19, 1998; revised version: January 30, 2002     

Double moral hazard and renegotiation

We examine renegotiation in a double moral hazard model with an ex ante budget balancing constraint when both the principal and the agent are allowed to make a renegotiation offer even though the principal proposes an initial contract. Under a belief restriction, any perfect-Bayesian equilibrium leads to an allocation that is superior to the second-best allocation of the standard double moral hazard model without renegotiation. The result of this paper gives some reasons for the existence of intermediary organizations such as holding companies, law houses, consulting firms, investment banks or venture capital. The result can also provide the rationalization for a fund set up by a group of firms of the industry in which their product is legally required to be recyclable.     

Universality of Nash components

We show that Nash equilibrium components are universal for the collection of connected polyhedral sets. More precisely for every polyhedral set we construct a so-called binary game—a game where all players have two pure strategies and a common utility function with values either zero or one—whose success set (the set of strategy profiles where the maximal payoff of one is indeed achieved) is homeomorphic to the given polyhedral set. Since compact semi-algebraic sets can be triangulated, a similar result follows for the collection of connected compact semi-algebraic sets.We discuss implications of our results for the strategic stability of success sets, and use the results to construct a Nash component with index k for any fixed integer k.     

Non-discriminating renegotiation in a competitive insurance market

A variant of the Rothschild-Stiglitz model of a competitive insurance market is considered, where each uninformed firm is allowed to renegotiate the contracts that its customers initially sign, subject to the restriction that renegotiated contracts be offered to all the firm's customers. Such non-discriminating renegotiation is shown to weaken the profitability of cream skimming to the extent that there exists a unique equilibrium outcome. This outcome is that of Miyazaki and Spence i.e., the incentive-compatible pair of zero-profit contracts, if efficient; and the incentive-compatible, zero-profit pair of contracts maximizing low-risk utility, otherwise.     

The structure of the Nash equilibrium sets of standard 2-player games

Summary. In this paper I study a class of two-player games, in which both players action sets are [0,1] and their payoff functions are continuous in joint actions and quasi-concave in own actions. I show that a no-improper-crossing condition is both necessary and sufficient for a finite subset A of to be the set of Nash equilibria of such a game.Received: 21 November 2002, Revised: 9 September 2004, JEL Classification Numbers: C65, C72.I am grateful to an editor of the journal and an anonymous referee for their very helpful comments. I also would like to thank the seminar participants at City University of Hong Kong, Georgia State University, Northwestern University, and Rice University.     

Nash equilibrium when players account for the complexity of their forecasts

Nash equilibrium is often interpreted as a steady state in which each player holds the correct expectations about the other players' behavior and acts rationally. This paper investigates the robustness of this interpretation when there are small costs associated with complicated forecasts. The model consists of a two-person strategic game in which each player chooses a finite machine to implement a strategy in an infinitely repeated 2×2 game with discounting. I analyze the model using a solution concept called Nash Equilibrium with Stable Forecasts (ESF). My main results concern the structure of equilibrium machine pairs. They provide necessary and sufficient conditions on the form of equilibrium strategies and plays. In contrast to the “folk theorem,” these structural properties place severe restrictions on the set of equilibrium paths and payoffs. For example, only sequences of the one-shot Nash equilibrium can be generated by any ESF of the repeated game of chicken.     

Existence of Nash equilibria in finite extensive form games with imperfect recall: A counterexample

This paper provides an example showing that for finite extensive form games without perfect recall existence of Nash equilibria in behaviour strategies is not guaranteed in general.     

Nash equilibrium in games with incomplete preferences

Summary. This paper investigates Nash equilibrium under the possibility that preferences may be incomplete. I characterize the Nash-equilibrium-set of such a game as the union of the Nash-equilibrium-sets of certain derived games with complete preferences. These games with complete preferences can be derived from the original game by a simple linear procedure, provided that preferences admit a concave vector-representation. These theorems extend some results on finite games by Shapley and Aumann. The applicability of the theoretical results is illustrated with examples from oligopolistic theory, where firms are modelled to aim at maximizing both profits and sales (and thus have multiple objectives). Mixed strategy and trembling hand perfect equilibria are also discussed.Received: 22 September 2003, Revised: 24 June 2004, JEL Classification Numbers: D11, C72, D43.I would like to thank Jean-Pierre Benôit, Juan Dubra, Alejandrio Jofre, Debraj Ray, Kim-Sau Chung and the seminar participants at NYU and at the Universidad de Chile for their comments. I am most grateful to Efe Ok, for his comments, criticism, suggestions and questions.     

Uniqueness and comparative statics of Nash equilibrium in a tariff retaliation model

This paper considers uniqueness and comparative statics of Nash equilibrium of a tariff retaliation model. The approach to the problem is geometrical and reminiscent of the analysis for the free trade competitive equilibrium. If the countries have constant elasticity of substitution utility functions, some simple conditions can be used to prove uniqueness of the Nash equilibrium of the tariff retaliation game. The welfare effects of endowment changes are analyzed in terms of the standard terms of trade and volume of trade effects. If the elasticity of substitution of one of the countries is sufficiently high, immiserizing growth will not occur.      

浅析知识产权保护中混合策略纳什均衡模型的应用

知识产权保护不但受制于立法的完善和执法的力度,而且取决于自觉维权的意识和守法的环境。我国企业和科研机构知识产权保护意识薄弱,缺乏维护自身合法权益的能力,尊重他人知识产权的观念尚未形成社会主流。本文运用博弈论方法,分析了一个知识产权保护博弈案例中的混合策略纳什均衡解,得出从增加知识产权主体自觉维权意识和加大侵权者成本两方面实现知识产权有效保护的结论。     

Teaching Nash Equilibrium and Dominance: A Classroom Experiment on the Beauty Contest

Abstract: The authors' aim in this article was to show how the use of classroom experiments may be a good pedagogical tool to teach the Nash equilibrium (NE) concept. The basic game is a version of the beauty contest game (BCG), a simple guessing game in which repetition lets students react to other players' choices and converge iteratively to the equilibrium solution. The authors perform this experiment with undergraduate students with no previous training in game theory. After four rounds, they observe a clear decreasing tendency in the average submitted number in all groups. Thus, the findings show that by playing a repeated BCG, students quickly learn how to reach the NE solution.     

Stochastic uncoupled dynamics and Nash equilibrium

In this paper we consider dynamic processes, in repeated games, that are subject to the natural informational restriction of uncoupledness. We study the almost sure convergence of play (the period-by-period behavior as well as the long-run frequency) to Nash equilibria of the one-shot stage game, and present a number of possibility and impossibility results. Basically, we show that if in addition to random experimentation some recall, or memory, is introduced, then successful search procedures that are uncoupled can be devised. In particular, to get almost sure convergence to pure Nash equilibria when these exist, it suffices to recall the last two periods of play.     

团队生产的纳什均衡及合作博弈空间

本文在霍姆斯特姆所做贡献的基础上认为团队生产问题在本质上等同于囚徒困境问题,并指出霍姆斯特姆的实施强制契约的解决方法存在逻辑漏洞,在现实社会中一般是无效的.本文依据社会经济人个人效用最大化的条件定义了团队生产的合作战略博弈空间,合作战略博弈空间的均衡极点(即个人效用最大化点)就是团队生产收益分配博弈的纳什均衡解.如果个人收益的总和等于集体收益的最优值,那么个人理性与集体理性之间就达到统一,处于帕累托最优状态,即不存在道德风险;反之,则说明存在道德风险.并给出了在团队生产中通过改变合作战略博弈空间达到帕累托最优的一个特例,指出了科学合理地改变合作战略博弈空间在社会经济发展中的重要作用     

Coalitional Bayesian Nash implementation in differential information economies

Summary. A mechanism coalitionally implements a social choice set if any outcome of the social choice set can be achieved as a coalitional Bayesian Nash equilibrium of a mechanism and vice versa. We say that a social choice set is coalitionally implementable if there is a mechanism which coalitionally implements it. Our main theorem proves that a social choice set is coalitionally implementable if and only if it is interim individually rational, interim efficient, coalitional B ayesian incentive compatible, and satisfies a coalitional Bayesian monotonicity condition as well as a closure condition. As an application of our main result, we show that the private core and the private Shapley value of an economy with differential information are coalitionally implementable. Received: January 12, 1998; revised version: March 30, 2000     

On the convergence to the Nash bargaining solution for action-dependent bargaining protocols

We consider a non-cooperative multilateral bargaining game and study an action-dependent bargaining protocol, that is, the probability with which a player becomes the proposer in a round of bargaining depends on the identity of the player who previously rejected. An important example is the frequently studied rejector-becomes-proposer protocol. We focus on subgame perfect equilibria in stationary strategies which are shown to exist and to be efficient. Equilibrium proposals do not depend on the probability to propose conditional on the rejection by another player. We consider the limit, as the bargaining friction vanishes. In case no player has a positive probability to propose conditional on his rejection, each player receives his utopia payoff conditional on being recognized. Otherwise, equilibrium proposals of all players converge to a weighted Nash bargaining solution, where the weights are determined by the probability to propose conditional on one's own rejection.     

投资决策中个人理性与集体理性博弈分析及对策研究

针对近期多起曝光的企业对外投资、担保大案,本文应用博弈论分析方法,在强调个人理性与集体理性矛盾的基础上研究投资决策中两者的完全信息博弈及无限阶段重复博弈,并建立相应的定量分析模型,通过分析博弈参与人均衡行为,指出投资决策中寻租的存在的必然性及巨大危害,并针对性的提出了适合实际操作的对策建议.     

Nash consistent representation of effectivity functions through lottery models

Effectivity functions for finitely many players and alternatives are considered. It is shown that every monotonic and superadditive effectivity function can be augmented with equal chance lotteries to a finite lottery model—i.e., an effectivity function that preserves the original effectivity in terms of supports of lotteries—which has a Nash consistent representation. The latter means that there exists a finite game form which represents the lottery model and which has a Nash equilibrium for any profile of utility functions satisfying the minimal requirement of respecting first order stochastic dominance among lotteries. No additional condition on the original effectivity function is needed.     

Continuity of the first price auction Nash equilibrium correspondence

Summary. Despite the complexity of the first price auction in the general asymmetric case, analytical results have started to emerge in the literature. Authors have also searched to gain insights by computing numerical estimates of the equilibria for some probability distributions of the valuations. This paper proves that the Nash equilibrium of the first price auction depends continuously, for the weak topology, on the valuation distributions and thus brings robustness to the numerical results as well as some theoretical results. As an example of application, we disprove a conjecture of comparative statics. Received: February 1, 1999; revised version: July 27, 2001