Fast algorithms for finding proper strategies in game trees

We show how to find a normal form proper equilibrium in behavior strategies of a given two-player zero-sum extensive form game with imperfect information but perfect recall. Our algorithm solves a finite sequence of linear programs and runs in polynomial time. For the case of a perfect information game, we show how to find a normal form proper equilibrium in linear time by a simple backwards induction procedure.

[1]  H. W. Kuhn,et al.  11. Extensive Games and the Problem of Information , 1953 .

[2]  M. Dresher.  The Mathematics of Games of Strategy: Theory and Applications , 1981 .

[3]  E. Damme.  Refinements of the Nash Equilibrium Concept , 1983 .

[4]  E. Damme.  A relation between perfect equilibria in extensive form games and proper equilibria in normal form games , 1984 .

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

[6]  E. Vandamme.  Stability and perfection of nash equilibria , 1987 .

[7]  Eddie Dekel,et al.  Lexicographic Probabilities and Equilibrium Refinements , 1991 .

[8]  Yoshitsugu Yamamoto,et al.  A path-following procedure to find a proper equilibrium of finite games , 1993 .

[9]  Bernhard von Stengel,et al.  Fast algorithms for finding randomized strategies in game trees , 1994, STOC '94.

[10]  D. Koller,et al.  Efficient Computation of Equilibria for Extensive Two-Person Games , 1996 .

[11]  B. Stengel,et al.  Efficient Computation of Behavior Strategies , 1996 .

[12]  Avi Pfeffer,et al.  Representations and Solutions for Game-Theoretic Problems , 1997, Artif. Intell..

[13]  Jonathan Schaeffer,et al.  Approximating Game-Theoretic Optimal Strategies for Full-scale Poker , 2003, IJCAI.

[14]  T. Sandholm,et al.  Finding equilibria in large sequential games of incomplete information ∗ , 2005 .

[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]  Peter Bro Miltersen,et al.  Computing sequential equilibria for two-player games , 2006, SODA '06.

[17]  Andrew McLennan,et al.  Gambit: Software Tools for Game Theory , 2006 .

[18]  Peter Bro Miltersen,et al.  Computing Proper Equilibria of Zero-Sum Games , 2006, Computers and Games.

[19]  Tuomas Sandholm,et al.  Lossless abstraction of imperfect information games , 2007, JACM.

[20]  Paul W. Goldberg,et al.  The Complexity of Computing a Nash Equilibrium , 2009, SIAM J. Comput..

[21]  Peter Bro Miltersen,et al.  Computing a quasi-perfect equilibrium of a two-player game , 2010 .