# Hard-to-solve Bimatrix Games

Bernhard von Stengel Department of Mathematics London School of Economics 3pm Tuesday 22nd February 2005 Room 2511, JCMB, King's Buildings

A bimatrix game is a two-player game in strategic form, a basic model in game theory. A Nash equilibrium is a pair of (possibly randomized) strategies, one for each player, so that no player can do better by unilaterally changing his or her strategy. The problem of finding one Nash equilibrium of a bimatrix game is considered as "one of the most important concrete open questions on the boundary of P [polynomial-time computability] today" (Papadimitriou, 2001).

In this talk, which will introduce the main concepts and geometric tools, we show that the commonly used Lemke-Howson algorithm for finding one equilibrium of a bimatrix game is NOT polynomial. This question had been open for some time. The algorithm is a pivoting method similar to the simplex algorithm for linear programming. We present a class of square bimatrix games for which the shortest Lemke-Howson path grows exponentially in the dimension of the game d. We construct the games using pairs of dual cyclic polytopes with 2d facets in d-space.

Joint work with Rahul Savani (LSE).