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.     

Learning by trial and error

A person learns by trial and error if he occasionally tries out new strategies, rejecting choices that are erroneous in the sense that they do not lead to higher payoffs. In a game, however, strategies can become erroneous due to a change of behavior by someone else. We introduce a learning rule in which behavior is conditional on whether a player experiences an error of the first or second type. This rule, called interactive trial and error learning, implements Nash equilibrium behavior in any game with generic payoffs and at least one pure Nash equilibrium.     

Testing subgame perfection apart from fairness in ultimatum games

We present an experiment designed to separate the two commonplace explanations for behavior in ultimatum games—subjects’ concern for fairness versus the failure of subgame perfection as an equilibrium refinement. We employ a tournament structure of the bargaining interaction to eliminate the potential for fairness to influence behavior. Comparing the results of the tournament game with two control treatments affords us a clean test of subgame perfection as well as a measure fairness-induced play. We find after 10 iterations of play that about half of all non-subgame-perfect demands are due to fairness, and the rest to imperfect learning. However, as suggested by models of learning, we also confirm that the ultimatum game presents an especially difficult environment for learning subgame perfection. Electronic Supplementary Material Supplementary material is available in the online version of this article at . JEL Classification C91, D64, J52     

On continuous-time Markov processes in bargaining

For bilateral stochastic bargaining procedures embedded in stable homogeneous continuous-time Markov processes, we show unusual limit results when time between rounds vanish. Standard convergence results require that some states are instantaneous.     

Conditions for equivalence between sequentiality and subgame perfection

We characterize the class of finite extensive forms for which the sets of Subgame Perfect and Sequential equilibrium strategy profiles coincide for any possible payoff function. In addition, we identify the class of finite extensive forms for which the outcomes induced by these two solution concepts coincide. We thank Luis Corchón, Francesco De Sinopoli, Herbert Gintis, Sjaak Hurkens, Francisco Marhuenda, Eric Maskin and four anonymous referees for helpful comments. Carlos thanks the financial support of the Spanish MEC through its FPI Grant BES-2003-0822. Cristian thanks the grant Plan de Formación de Personal Investigador de la Comunidad de Madrid and the financial support from project No. SEJ2004-00968/ECON from the Spanish Ministry of Education and Science. The usual disclaimer applies.     

Stability,sequential rationality,and subgame consistency

In this paper we analyze dynamic strategic behavior by means of the idea of "stability." We develop a solution concept of "sequentially stable equilibrium (SSE)" that satisfies subgame consistency. All SSEs are characterized by the largest stable set that yields exactly all the backward induction outcomes. We also provide a refinement of the SSE. We compare the SSE and its refinement with some existing solution concepts in the literature We are grateful to the editor and two anonymous referees for very useful comments and suggestions that led to this revision. We thank Chih Chang, Yi-Chun Chen, Chenying Huang, Huiwen Koo, Man-Chung Ng, Shyh-fang Ueng, and Chun-Hsien Yeh for helpful comments and discussions. We also thank participants in seminars at Academia Sinica, National Dong Hwa University, and the Third Pan-Pacific Game Theory Conference. Financial support from the Social Sciences and Humanities Research Council of Canada (SSHRC), the National Science Council of Taiwan, and the Economic and Social Research Council of the UK is gratefully acknowledged. The usual disclaimer applies.     

A multi-agent bilateral bargaining model with endogenous protocol

People do bargain over how to bargain. We examine the role of individuals’ ability to pursue certain bargaining protocols in a multi-agent bilateral bargaining model. Bargaining protocols are not completely settled, but will emerge endogenously in equilibrium. We show that players’ ability to partially influence bargaining protocols plays a crucial role in determining equilibrium outcomes. When discounting is not too high, there are multiple subgame perfect equilibria, including inefficient ones. As the number of players increases, both the set of discount factors that support multiple equilibrium outcomes and the set of the first proposing player’s equilibrium payoffs expand. The maximum loss of efficiency increases with respect to the discount factor. We would like to thank Hongbin Cai, John Conlon, Andrew Daughety, Taiji Furusawa, Byoung Heon Jun, Akira Okada, Ping Wang, and two anonymous referees for their comments and suggestions. Both authors gratefully acknowledge financial support from the Social Sciences and Humanities Research Council of Canada.     

Self-Fulfilling Mechanisms in Bayesian Games

Amechanismfor a Bayesian gameGis a mapping μ from the set of states of nature to the set of players' actions. μ isself-fulfillingif players are truthful at the communication stage and, given the information revealed by μ, no player can gain in unilaterally deviating from the action prescribed by the mechanism. We investigate the properties of self-fulfilling mechanisms and we show in particular that they correspond to inert solutions of the infinitely repeated game generated byG. We also discuss applications to market games, regulation, and R&D games.Journal of Economic LiteratureClassification Numbers: D82, C72.     

A model of noisy introspection

We present a theoretical model of noisy introspection designed to explain behavior in games played only once. The model determines layers of beliefs about others' beliefs about …, etc., but allows for surprises by relaxing the equilibrium requirement that belief distributions coincide with decision distributions. Noise is injected into iterated conjectures about others' decisions and beliefs, which causes the predictions to differ from those of deterministic models of iterated thinking, e.g., rationalizability. The paper contains a convergence proof that implies existence and uniqueness of the outcome of the iterated thought process. In addition, estimated introspection and noise parameters for data from 37 one-shot matrix games are reported. The accuracy of the model is compared with that of several alternatives.     

Population growth and technological change in a global warming model

Global warming (GW) is now recognized as a significant threat to sustainable development on an international scale. After providing some introductory background material, we introduce a benchmark dynamic game within which to study the GW problem. The model allows for population growth and is subsequently generalized to allow for changes in technology. In each case, a benchmark “Business as Usual” (BAU) equilibrium is analyzed and contrasted with the efficient solution. Furthermore, a complete characterization is provided in the benchmark model of the entire subgame perfect equilibrium value correspondence.JEL Classification Q54, D99, O12This paper builds on the Woytinsky Lecture delivered by Roy Radner at the University of Michigan, March 11, 1998. We would like to thank Graciela Chichilnisky, Arnulf Grubler, Geoffrey Heal, Leonid Hurwicz, Jill Jaeger, Peter de Janosi, Paul Kleindorfer, Giuseppe Lopomo, Thomas Schelling, Tapan Mitra, and Michael Toman for helpful discussions and references, and Andrew King, Charlotte V. Kuh, and Frank Sinden for comments on an earlier draft. In the case of the present enterprise, much of whose scope lies outside of our previous expertise, it is even more important than usual to emphasize that we are responsible for all errors.     

废旧家电回收再利用体系构建中的博弈均衡分析

近些年来,家电产品的更新换代越来越快,淘汰下来的家电废弃物数量惊人,对这些家电废弃物如何处理,已经成为全世界关注的焦点。随着我国废旧家电高峰期的到来,构建废旧家电回收再利用体系,对于提高我国家电产业的国际竞争力,促进经济增长方式的转变,具有非常重要的意义。本文通过利益相关方博弈均衡分析,提出了废旧家电回收再利用体系构建过程中的政府政策设计要点。具体包括：建立资源回收再利用信息支持服务系统,弥补市场缺陷;加强环保立法,合理确定垃圾收费水平;资助废旧物料再生技术的研究项目,建立环境无害化技术支撑体系;以资源高效利用为核心。鼓励再生企业和生产企业使用再生物料．建立配套政策支持体系。     

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.