On the complexity of approximating a Nash equilibrium

We show that computing a relative---that is, multiplicative as opposed to additive---approximate Nash equilibrium in two-player games is PPAD-complete, even for constant values of the approximation. Our result is the first constant inapproximability result for the problem, since the appearance of the original results on the complexity of the Nash equilibrium [8, 5, 7]. Moreover, it provides an apparent---assuming that PPAD ⊈ TIME(nO(log n))---dichotomy between the complexities of additive and relative notions of approximation, since for constant values of additive approximation a quasi-polynomial-time algorithm is known [22]. Such a dichotomy does not arise for values of the approximation that scale with the size of the game, as both relative and additive approximations are PPAD-complete [7]. As a byproduct, our proof shows that the Lipton-Markakis-Mehta sampling lemma is not applicable to relative notions of constant approximation, answering in the negative direction a question posed to us by Shang-Hua Teng [26].

[1]  Paul G. Spirakis,et al.  A Graph Spectral Approach for Computing Approximate Nash Equilibria , 2009, Electron. Colloquium Comput. Complex..

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

[3]  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).

[4]  Robert Krauthgamer,et al.  How hard is it to approximate the best Nash equilibrium? , 2009, SODA.

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

[6]  Panagiota N. Panagopoulou,et al.  Polynomial algorithms for approximating Nash equilibria of bimatrix games , 2006, Theor. Comput. Sci..

[7]  Amin Saberi,et al.  Approximating nash equilibria using small-support strategies , 2007, EC '07.

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

[9]  Christos H. Papadimitriou,et al.  On oblivious PTAS's for nash equilibrium , 2009, STOC '09.

[10]  Evangelos Markakis,et al.  New algorithms for approximate Nash equilibria in bimatrix games , 2010, Theor. Comput. Sci..

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

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

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

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

[15]  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).

[16]  Aranyak Mehta,et al.  A note on approximate Nash equilibria , 2006, Theor. Comput. Sci..

[17]  J. Nash.  Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[18]  Yang Cai,et al.  On minmax theorems for multiplayer games , 2011, SODA '11.

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

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

[21]  Paul G. Spirakis,et al.  Efficient Algorithms for Constant Well Supported Approximate Equilibria in Bimatrix Games , 2007, ICALP.

[22]  H PapadimitriouChristos,et al.  The complexity of computing a Nash equilibrium , 2009 .

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

[24]  D. Bernoulli.  Exposition of a New Theory on the Measurement of Risk , 1954 .

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

[26]  N. S. Barnett,et al.  Private communication , 1969 .

[27]  Paul G. Spirakis,et al.  Practical and Efficient Approximations of Nash Equilibria for Win-Lose Games Based on Graph Spectra , 2010, WINE.

[28]  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).

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

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

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

[32]  Christos H. Papadimitriou,et al.  On a Network Generalization of the Minmax Theorem , 2009, ICALP.