A Generalized Strategy Eliminability Criterion and Computational Methods for Applying It

We define a generalized strategy eliminability criterion for bimatrix games that considers whether a given strategy is eliminable relative to given dominator & eliminee subsets of the players' strategies. We show that this definition spans a spectrum of eliminability criteria from strict dominance (when the sets are as small as possible) to Nash equilibrium (when the sets are as large as possible). We show that checking whether a strategy is eliminable according to this criterion is coNP-complete (both when all the sets are as large as possible and when the dominator sets each have size 1). We then give an alternative definition of the eliminability criterion and show that it is equivalent using the Minimax Theorem. We show how this alternative definition can be translated into a mixed integer program of polynomial size with a number of (binary) integer variables equal to the sum of the sizes of the eliminee sets, implying that checking whether a strategy is eliminable according to the criterion can be done in polynomial time, given that the eliminee sets are small. Finally, we study using the criterion for iterated elimination of strategies.

[1]  Bernhard von Stengel,et al.  Exponentially many steps for finding a Nash equilibrium in a bimatrix game , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[2]  Christos H. Papadimitriou,et al.  Algorithms, games, and the internet , 2001, STOC '01.

[3]  Eitan Zemel,et al.  Nash and correlated equilibria: Some complexity considerations , 1989 .

[4]  Peter Stone,et al.  A polynomial-time nash equilibrium algorithm for repeated games , 2003, EC '03.

[5]  Vincent Conitzer,et al.  Complexity Results about Nash Equilibria , 2002, IJCAI.

[6]  Xi Chen,et al.  3-NASH is PPAD-Complete , 2005, Electron. Colloquium Comput. Complex..

[7]  J. Neumann Zur Theorie der Gesellschaftsspiele , 1928 .

[8]  Yoav Shoham,et al.  Run the GAMUT: a comprehensive approach to evaluating game-theoretic algorithms , 2004, Proceedings of the Third International Joint Conference on Autonomous Agents and Multiagent Systems, 2004. AAMAS 2004..

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

[10]  Christos H. Papadimitriou,et al.  Three-Player Games Are Hard , 2005, Electron. Colloquium Comput. Complex..

[11]  Paul W. Goldberg,et al.  The complexity of computing a Nash equilibrium , 2006, STOC '06.

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

[13]  C. E. Lemke,et al.  Equilibrium Points of Bimatrix Games , 1964 .

[14]  Vincent Conitzer,et al.  Complexity of (iterated) dominance , 2005, EC '05.

[15]  Xiaotie Deng,et al.  Settling the Complexity of Two-Player Nash Equilibrium , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

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

[17]  Jean-Pierre Bourguignon,et al.  Mathematische Annalen , 1893 .

[18]  Tilman Börgers,et al.  Pure Strategy Dominance , 1993 .

[19]  Daphne Koller,et al.  A Continuation Method for Nash Equilibria in Structured Games , 2003, IJCAI.

[20]  L. G. H. Cijan A polynomial algorithm in linear programming , 1979 .

[21]  Krzysztof R. Apt,et al.  Uniform Proofs of Order Independence for Various Strategy Elimination Procedures , 2004, ArXiv.

[22]  L. Khachiyan Polynomial algorithms in linear programming , 1980 .

[23]  Leslie M. Marx,et al.  Order Independence for Iterated Weak Dominance , 2000, Games Econ. Behav..

[24]  Moshe Tennenholtz,et al.  Local-Effect Games , 2003, IJCAI.

[25]  Michael L. Littman,et al.  Graphical Models for Game Theory , 2001, UAI.

[26]  Eitan Zemel,et al.  The Complexity of Eliminating Dominated Strategies , 1993, Math. Oper. Res..

[27]  D. Knuth,et al.  A note on strategy elimination in bimatrix games , 1988 .

[28]  H. Kuk On equilibrium points in bimatrix games , 1996 .

[29]  Aranyak Mehta,et al.  Playing large games using simple strategies , 2003, EC '03.

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