The complexity of computing a Nash equilibrium

We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recently-established equivalence between polynomial time solvability of normal form games and graphical games, establishing that these kinds of games can simulate a PPAD-complete class of Brouwer functions.

[1]  Bronisław Knaster,et al.  Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe , 1929 .

[2]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[3]  J. Nash.  NON-COOPERATIVE GAMES , 1951, Classics in Game Theory.

[4]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

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

[6]  Robert Wilson,et al.  Computing Equilibria of N-Person Games , 1971 .

[7]  J. Rosenmüller.  On a Generalization of the Lemke–Howson Algorithm to Noncooperative N-Person Games , 1971 .

[8]  B. Curtis Eaves,et al.  Homotopies for computation of fixed points , 1972, Math. Program..

[9]  C. E. Lemke,et al.  Simplicial Approximation of an Equilibrium Point for Non-Cooperative N-Person Games , 1973 .

[10]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[11]  M. Garey.  Johnson: computers and intractability: a guide to the theory of np- completeness (freeman , 1979 .

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

[13]  Gerard van der Laan,et al.  A restart algorithm for computing fixed points without an extra dimension , 1979, Math. Program..

[14]  A. Rubinstein.  Equilibrium in supergames with the overtaking criterion , 1979 .

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

[16]  Gerard van der Laan,et al.  On the Computation of Fixed Points in the Product Space of Unit Simplices and an Application to Noncooperative N Person Games , 1982, Math. Oper. Res..

[17]  Mihalis Yannakakis,et al.  How easy is local search? , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[18]  Christos H. Papadimitriou,et al.  Exponential lower bounds for finding Brouwer fixed points , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

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

[20]  Christos H. Papadimitriou,et al.  On Total Functions, Existence Theorems and Computational Complexity , 1991, Theor. Comput. Sci..

[21]  Anne Condon,et al.  The Complexity of Stochastic Games , 1992, Inf. Comput..

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

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

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

[25]  Eric J. Friedman,et al.  Learning and Implementation on the Internet , 1997 .

[26]  Russell Impagliazzo,et al.  The Relative Complexity of NP Search Problems , 1998, J. Comput. Syst. Sci..

[27]  Daphne Der-Fen Liu,et al.  On L(d, 1)-labelings of graphs , 2000, Discret. Math..

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

[29]  Satinder Singh,et al.  An Efficient Exact Algorithm for Singly Connected Graphical Games , 2002, NIPS 2002.

[30]  J. Geanakoplos.  Nash and Walras equilibrium via Brouwer , 2003 .

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

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

[33]  Christos H. Papadimitriou,et al.  The complexity of pure Nash equilibria , 2004, STOC '04.

[34]  Richard J. Lipton,et al.  Nash Equilibria via Polynomial Equations , 2004, LATIN.

[35]  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.

[36]  Tim Roughgarden,et al.  Computing equilibria in multi-player games , 2005, SODA '05.

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

[38]  Daniel M. Kane,et al.  On the complexity of two-player win-lose games , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

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

[40]  Constantinos Daskalakis,et al.  Computing Pure Nash Equilibria via Markov Random Fields , 2005, ArXiv.

[41]  Christos H. Papadimitriou,et al.  The Game World Is Flat: The Complexity of Nash Equilibria in Succinct Games , 2006, ICALP.

[42]  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).

[43]  Grant Schoenebeck,et al.  The computational complexity of nash equilibria in concisely represented games , 2006, EC '06.

[44]  S. Teng,et al.  On the Approximation and Smoothed Complexity of , 2006 .

[45]  Paul W. Goldberg,et al.  Reducibility among equilibrium problems , 2006, STOC '06.

[46]  Edith Elkind,et al.  Nash equilibria in graphical games on trees revisited , 2006, EC '06.

[47]  Aranyak Mehta,et al.  A Note on Approximate Nash Equilibria , 2006, WINE.

[48]  Xi Chen,et al.  Computing Nash Equilibria: Approximation and Smoothed Complexity , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[49]  Amin Saberi,et al.  Leontief economies encode nonzero sum two-player games , 2006, SODA '06.

[50]  Kousha Etessami,et al.  On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract) , 2010, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[51]  Adam Tauman Kalai,et al.  The myth of the folk theorem , 2008, Games Econ. Behav..

[52]  Christos H. Papadimitriou,et al.  Computing Equilibria in Anonymous Games , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[53]  Evangelos Markakis,et al.  New algorithms for approximate Nash equilibria in bimatrix games , 2007, Theor. Comput. Sci..

[54]  Xi Chen,et al.  The approximation complexity of win-lose games , 2007, SODA '07.

[55]  Paul G. Spirakis,et al.  An Optimization Approach for Approximate Nash Equilibria , 2007, WINE.

[56]  Aranyak Mehta,et al.  Progress in approximate nash equilibria , 2007, EC '07.

[57]  Computing correlated equilibria in multi-player games , 2008, JACM.

[58]  Yoav Shoham,et al.  Computer science and game theory , 2008, CACM.

[59]  Christos H. Papadimitriou,et al.  Discretized Multinomial Distributions and Nash Equilibria in Anonymous Games , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.