A simple and numerically stable primal-dual algorithm for computing Nash-equilibria in sequential games with incomplete information

Abstract

We present a simple primal-dual algorithm for computing approximate Nash-equilibria in two-person zero-sum sequential games with incomplete information and perfect recall (like Texas Hold'em Poker). Our algorithm is numerically stable, performs only basic iterations (i.e matvec multiplications, clipping, etc., and no calls to external first-order oracles, no matrix inversions, etc.), and is applicable to a broad class of two-person zero-sum games including simultaneous games and sequential games with incomplete information and perfect recall. The applicability to the latter kind of games is thanks to the sequence-form representation which allows us to encode any such game as a matrix game with convex polytopial strategy profiles. We prove that the number of iterations needed to produce a Nash-equilibrium with a given precision is inversely proportional to the precision. As proof-of-concept, we present experimental results on matrix games on simplexes and Kuhn Poker.