The complexity of computing a (perfect) equilibrium for an n-player extensive form game of perfect recall

We study the complexity of computing or approximating refinements of Nash equilibrium for a given finite n-player extensive form game of perfect recall (EFGPR), where n >= 3. Our results apply to a number of well-studied refinements, including sequential (SE), extensive-form perfect (PE), and quasi-perfect equilibrium (QPE). These refine Nash and subgame-perfect equilibrium. Of these, the most refined notions are PE and QPE. By classic results, all these equilibria exist in any EFGPR. We show that, for all these notions of equilibrium, approximating an equilibrium for a given EFGPR, to within a given desired precision, is FIXP_a-complete. We also consider the complexity of corresponding "almost" equilibrium notions, and show that they are PPAD-complete. In particular, we define "delta-almost epsilon-(quasi-)perfect" equilibrium, and show computing one is PPAD-complete. We show these notions refine "delta-almost subgame-perfect equilibrium" for EFGPRs, which is PPAD-complete. Thus, approximating one such (delta-almost) equilibrium for n-player EFGPRs, n >= 3, is P-time equivalent to approximating a (delta-almost) NE for a normal form game (NFG) with 3 or more players. NFGs are trivially encodable as EFGPRs without blowup in size. Thus our results extend the celebrated complexity results for NFGs to refinements of equilibrium in the more general setting of EFGPRs. For 2-player EFGPRs, analogous complexity results follow from the algorithms of Koller, Megiddo, and von Stengel (1996), von Stengel, van den Elzen, and Talman (2002), and Miltersen and Soerensen (2010). For n-player EFGPRs, an analogous result for Nash and subgame-perfect equilibrium was given by Daskalakis, Fabrikant, and Papadimitriou (2006). However, no analogous results were known for the more refined notions of equilibrium for EFGPRs with 3 or more players.

[1]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[2]  Kristoffer Arnsfelt Hansen,et al.  The Computational Complexity of Trembling Hand Perfection and Other Equilibrium Refinements , 2010, SAGT.

[3]  M. Dufwenberg.  Game theory. , 2011, Wiley interdisciplinary reviews. Cognitive science.

[4]  Kristoffer Arnsfelt Hansen,et al.  The Complexity of Approximating a Trembling Hand Perfect Equilibrium of a Multi-player Game in Strategic Form , 2014, SAGT.

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

[6]  James Renegar.  On the computational complexity and geome-try of the first-order theory of the reals , 1992 .

[7]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

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

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

[10]  Bernhard von Stengel,et al.  Computing Normal Form Perfect Equilibria for Extensive Two-Person Games , 2002 .

[11]  E. Damme.  Stability and perfection of Nash equilibria , 1987 .

[12]  Robert Wilson.  Computing Equilibria of Two-Person Games from the Extensive Form , 1972 .

[13]  J. Blair,et al.  Games with Imperfect Information , 1993 .

[14]  R. McKelvey,et al.  Computation of equilibria in finite games , 1996 .

[15]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part III: Quantifier Elimination , 1992, J. Symb. Comput..

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

[17]  Herbert E. Scarf,et al.  The Approximation of Fixed Points of a Continuous Mapping , 1967 .

[18]  R. Anderson.  “Almost” implies “near” , 1986 .

[19]  Francesco Mallegni,et al.  The Computation of Economic Equilibria , 1973 .

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

[21]  Christos H. Papadimitriou,et al.  On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence , 1994, J. Comput. Syst. Sci..

[22]  R. Selten.  Reexamination of the perfectness concept for equilibrium points in extensive games , 1975, Classics in Game Theory.

[23]  H. Kuhn.  Simplicial approximation of fixed points. , 1968, Proceedings of the National Academy of Sciences of the United States of America.

[24]  Lawrence E. Blume,et al.  The Algebraic Geometry of Perfect and Sequential Equilibrium , 1994 .

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

[26]  Troels Bjerre Lund.  Computing a proper equilibrium of a bimatrix game , 2012, EC.

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

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

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

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

[31]  Donald E. Knuth,et al.  The Art of Computer Programming: Volume 3: Sorting and Searching , 1998 .

[32]  Reinhard Selten.  Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit , 2016 .

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

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

[35]  Jianfei Shen,et al.  On the equivalence between (quasi-)perfect and sequential equilibria , 2014, Int. J. Game Theory.

[36]  D. S. Arnon,et al.  Algorithms in real algebraic geometry , 1988 .

[37]  Herbert E. Scarf,et al.  The Computation of Economic Equilibria , 1974 .

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

[39]  Kousha Etessami,et al.  On the Complexity of Nash Equilibria and Other Fixed Points , 2010, SIAM J. Comput..

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

[41]  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).

[42]  John Hillas.  ON THE RELATION BETWEEN PERFECT EQUILIBRIA IN EXTENSIVE FORM GAMES AND PROPER EQUILIBRIA IN NORMAL FORM GAMES , 1996 .

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

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

[45]  Jean-François Mertens.  Two examples of strategic equilibrium , 1995 .

[46]  J. Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I , 1989 .

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