Rank-1 bimatrix games: a homeomorphism and a polynomial time algorithm

Given a rank-1 bimatrix game (A,B), i.e., where rank(A+B)=1, we construct a suitable linear subspace of the rank-1 game space and show that this subspace is homeomorphic to its Nash equilibrium correspondence. Using this homeomorphism, we give the first polynomial time algorithm for computing an exact Nash equilibrium of a rank-1 bimatrix game. This settles an open question posed by Kannan and Theobald (SODA'07). In addition, we give a novel algorithm to enumerate all the Nash equilibria of a rank-1 game and show that a similar technique may also be applied for finding a Nash equilibrium of any bimatrix game. Our approach also provides new proofs of important classical results such as the existence and oddness of Nash equilibria, and the index theorem for bimatrix games. Further, we extend the rank-1 homeomorphism result to a fixed rank game space, and give a fixed point formulation on [0,1]k for solving a rank-k game. The homeomorphism and the fixed point formulation are piece-wise linear and considerably simpler than the classical constructions.

[1]  Thorsten Theobald,et al.  Games of fixed rank: a hierarchy of bimatrix games , 2005, SODA '07.

[2]  Bernhard von Stengel,et al.  Strategic Characterization of the Index of an Equilibrium , 2008, SAGT.

[3]  Nesa L'abbe Wu,et al.  Linear programming and extensions , 1981 .

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

[5]  Robert Wilson,et al.  A global Newton method to compute Nash equilibria , 2003, J. Econ. Theory.

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

[7]  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).

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

[9]  Christos H. Papadimitriou,et al.  Algorithms, Games, and the Internet , 2001, ICALP.

[10]  Arkadi Predtetchinski.  A general structure theorem for the Nash equilibrium correspondence , 2009, Games Econ. Behav..

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

[12]  Jeremy I. Bulow,et al.  Matching and Price Competition , 2003 .

[13]  L. Shapley.  A note on the Lemke-Howson algorithm , 1974 .

[14]  Paul G. Spirakis,et al.  Exploiting Concavity in Bimatrix Games: New Polynomially Tractable Subclasses , 2010, APPROX-RANDOM.

[15]  Srihari Govindan,et al.  Equivalence and Invariance of the Index and Degree of Nash Equilibria , 1997 .

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

[17]  Thorsten Theobald,et al.  Enumerating the Nash equilibria of rank 1-games , 2007, ArXiv.

[18]  B. Stengel.  Algorithmic Game Theory: Equilibrium Computation for Two-Player Games in Strategic and Extensive Form , 2007 .

[19]  Tim Roughgarden,et al.  Algorithmic Game Theory , 2007 .

[20]  P. Jean-Jacques Herings,et al.  Homotopy methods to compute equilibria in game theory , 2006, Equilibrium Computation.

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

[22]  Rahul Savani,et al.  Hard‐to‐Solve Bimatrix Games , 2006 .