On the Complexity of Iterated Weak Dominance in Constant-Sum Games

In game theory, a player's action is said to be weakly dominated if there exists another action that, with respect to what the other players do, is never worse and sometimes strictly better. We investigate the computational complexity of the process of iteratively eliminating weakly dominated actions (IWD) in two-player constant-sum games, i.e., games in which the interests of both players are diametrically opposed. It turns out that deciding whether an action is eliminable via IWD is feasible in polynomial time whereas deciding whether a given subgame is reachable via IWD is NP-complete. The latter result is quite surprising as we are not aware of other natural computational problems that are intractable in constant-sum games. Furthermore, we slightly improve a result by Conitzer and Sandholm [6] by showing that typical problems associated with IWD in win-lose games with at most one winner are NP-complete.

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

[2]  D. Koller,et al.  The complexity of two-person zero-sum games in extensive form , 1992 .

[3]  Yoav Shoham,et al.  Multiagent Systems - Algorithmic, Game-Theoretic, and Logical Foundations , 2009 .

[4]  Felix A. Fischer,et al.  Computing Shapley's saddles , 2009, SECO.

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

[6]  Martin J. Osborne,et al.  An Introduction to Game Theory , 2003 .

[7]  J. Mertens,et al.  ON THE STRATEGIC STABILITY OF EQUILIBRIA , 1986 .

[8]  Roger B. Myerson,et al.  Game theory - Analysis of Conflict , 1991 .

[9]  Makoto Shimoji,et al.  On the equivalence of weak dominance and sequential best response , 2004, Games Econ. Behav..

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

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

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

[13]  Christian Ewerhart,et al.  Iterated Weak Dominance in Strictly Competitive Games of Perfect Information , 2002, J. Econ. Theory.

[14]  Yoav Shoham,et al.  Ranking games , 2009, Artif. Intell..

[15]  Howard Raiffa,et al.  Games and Decisions: Introduction and Critical Survey. , 1958 .

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

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

[18]  H. Jerome Keisler,et al.  Admissibility in Games1 , 2008 .

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

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

[21]  Felix A. Fischer,et al.  Computational aspects of Shapley's saddles , 2009, AAMAS.