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Efficient Nash equilibrium approximation through Monte Carlo counterfactual regret minimization

Recently, there has been considerable progress towards algorithms for approximating Nash equilibrium strategies in extensive games. One such algorithm, Counterfactual Regret Minimization (CFR), has proven to be effective in two-player zero-sum poker domains. While the basic algorithm is iterative and performs a full game traversal on each iteration, sampling based approaches are possible. For instance, chance-sampled CFR considers just a single chance outcome per traversal, resulting in faster but less precise iterations. While more iterations are required, chance-sampled CFR requires less time overall to converge. In this work, we present new sampling techniques that consider sets of chance outcomes during each traversal to produce slower, more accurate iterations. By sampling only the public chance outcomes seen by all players, we take advantage of the imperfect information structure of the game to (i) avoid recomputation of strategy probabilities, and (ii) achieve an algorithmic speed improvement, performing O(n2) work at terminal nodes in O(n) time. We demonstrate that this new CFR update converges more quickly than chance-sampled CFR in the large domains of poker and Bluff.