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

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 only performs 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 [10] which allows us to encode any such game as a matrix game with convex polytopial profiles. We prove that the number of iterations needed to produce a Nash equilibrium with a given precision is inversely proportional to the precision. We also present experimental results on simulated and real games (Kuhn poker). keywords: algorithmic-game theory; sequential game; incomplete information; perfect recall; approximate Nash equilibrium; primal-dual algorithm; convex-optimization