Iterated Strict Dominance in General Games

We offer a definition of iterated elimination of strictly dominated strategies (IESDS) for games with (in)finite players, (non)compact strategy sets, and (dis)continuous payoff functions. IESDS is always a well-defined order independent procedure that can be used to solve Nash equilibrium in dominance-solvable games. We characterize IESDS by means of a "stability"" criterion, and offer a sufficient and necessary epistemic condition for IESDS. We show by an example that IESDS may generate spurious Nash equilibria in the class of Reny's better-reply secure games. We provide sufficient/necessary conditions under which IESDS preserves the set of Nash equilibria." Nous donnons une definition de l'elimination iterative des strategies qui sont strictement donimees (EISSD) pour les jeux avec un nombre fini (ou infini) de joueurs , des ensembles de strategies compactes (ou non-compactes), et des fonctions de gains continues (ou non-continues). Le processus EISSD est bien defini et independant de l'ordre d'elimination. Nous donnons une caracterisation du processus EISSD en utilisant un critere de stabilite et offrons une condition epistemologique. Nous demontrons que le processus EISSD peut produire des equilibres faux dans la classe des jeux de meilleures reponses securitaires de Reny. Nous donnons des conditions necessaires et suffisantes pour que le processus EISSD conserve l'ensemble des equilibre de Nash.

[1]  Paul R. Milgrom,et al.  Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities , 1990 .

[2]  R. Aumann.  Correlated Equilibrium as an Expression of Bayesian Rationality Author ( s ) , 1987 .

[3]  H. Carlsson,et al.  Global Games and Equilibrium Selection , 1993 .

[4]  M. Jackson.  Implementation in Undominated Strategies: A Look at Bounded Mechanisms , 1992 .

[5]  Polly S Nichols,et al.  Agreeing to disagree. , 2005, General dentistry.

[6]  K. Fan.  Applications of a theorem concerning sets with convex sections , 1966 .

[7]  Larry G. Epstein.  Preference, Rationalizability and Equilibrium , 1997 .

[8]  Krzysztof R. Apt,et al.  Order independence and rationalizability , 2005, TARK.

[9]  R. Aumann.  Backward induction and common knowledge of rationality , 1995 .

[10]  Barton L. Lipman,et al.  A Note on the Implications of Common Knowledge of Rationality , 1994 .

[11]  X. Vives.  Nash equilibrium with strategic complementarities , 1990 .

[12]  S. Vajda,et al.  GAMES AND DECISIONS; INTRODUCTION AND CRITICAL SURVEY. , 1958 .

[13]  Robert J. Aumann,et al.  Interactive epistemology I: Knowledge , 1999, Int. J. Game Theory.

[14]  Larry Samuelson,et al.  “Cautious” utility maximization and iterated weak dominance , 1992 .

[15]  Eric van Damme,et al.  Non-Cooperative Games , 2000 .

[16]  Adam Brandenburger,et al.  Rationalizability and Correlated Equilibria , 1987 .

[17]  Paul R. Milgrom,et al.  Coalition-Proofness and Correlation with Arbitrary Communication Possibilities , 1996 .

[18]  Ariel Rubinstein,et al.  A Course in Game Theory , 1995 .

[19]  Bayesian Rationality,et al.  CORRELATED EQUILIBRIUM AS AN EXPRESSION OF , 1987 .

[20]  Tilman B oumi rgers.  PURE STRATEGY DOMINANCE , 1993 .

[21]  S. Morris,et al.  Global Games: Theory and Applications , 2001 .

[22]  H. Moulin.  Dominance solvability and cournot stability , 1984 .

[23]  E. Maskin,et al.  The Existence of Equilibrium in Discontinuous Economic Games, I: Theory , 1986 .

[24]  Martin Meier,et al.  Hierarchies of beliefs for compact possibility models , 2005 .

[25]  M. Dufwenberg,et al.  Existence and Uniqueness of Maximal Reductions Under Iterated Strict Dominance , 2002 .

[26]  Gerard Debreu,et al.  A Social Equilibrium Existence Theorem* , 1952, Proceedings of the National Academy of Sciences.

[27]  B. Bernheim.  Rationalizable Strategic Behavior , 1984 .

[28]  D. M. Topkis.  Equilibrium Points in Nonzero-Sum n-Person Submodular Games , 1979 .

[29]  Stephen Morris,et al.  Robust Implementation: The Role of Large Type Spaces , 2005 .

[30]  T. Tan,et al.  The Bayesian foundations of solution concepts of games , 1988 .

[31]  I. Glicksberg.  A FURTHER GENERALIZATION OF THE KAKUTANI FIXED POINT THEOREM, WITH APPLICATION TO NASH EQUILIBRIUM POINTS , 1952 .

[32]  R. Aumann,et al.  Epistemic Conditions for Nash Equilibrium , 1995 .

[33]  Xiao Luo,et al.  A Unified Approach to Information, Knowledge, and Stability , 2004 .

[34]  L. Samuelson.  Modeling knowledge in economic analysis , 2004 .

[35]  Larry Samuelson,et al.  Dominated strategies and common knowledge , 1992 .

[36]  P. Reny.  On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games , 1999 .

[37]  H. Keisler,et al.  ADMISSIBILITY IN GAMES , 2008 .

[38]  P. Malliavin.  Infinite dimensional analysis , 1993 .

[39]  Eitan Zemel,et al.  On the order of eliminating dominated strategies , 1990 .

[40]  Michael Meyling,et al.  Axiomatic Set Theory , 2008 .

[41]  Thomas Mariotti.  Hierarchies of compact beliefs and rationalizable behavior , 2003 .

[42]  David Pearce.  Rationalizable Strategic Behavior and the Problem of Perfection , 1984 .

[43]  Xiao Luo.  General systems and ϕ-stable sets — a formal analysis of socioeconomic environments , 2001 .

[44]  Jeroen M. Swinkels,et al.  Order Independence for Iterated Weak Dominance , 1997, Games Econ. Behav..