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In two-player zero-sum games, if both players minimize their average externalregret, then the average of the strategy profiles converges to a Nashequilibrium. For n-player general-sum games, however, theoretical guaranteesfor regret minimization are less understood. Nonetheless, Counterfactual RegretMinimization (CFR), a popular regret minimization algorithm for extensive-formgames, has generated winning three-player Texas Hold'em agents in the AnnualComputer Poker Competition (ACPC). In this paper, we provide the first set oftheoretical properties for regret minimization algorithms in non-zero-sum gamesby proving that solutions eliminate iterative strict domination. We formallydefine \emph{dominated actions} in extensive-form games, show that CFR avoidsiteratively strictly dominated actions and strategies, and demonstrate thatremoving iteratively dominated actions is enough to win a mock tournament in asmall poker game. In addition, for two-player non-zero-sum games, we bound theworst case performance and show that in practice, regret minimization can yieldstrategies very close to equilibrium. Our theoretical advancements lead us to anew modification of CFR for games with more than two players that is moreefficient and may be used to generate stronger strategies than previouslypossible. Furthermore, we present a new three-player Texas Hold'em poker agentthat was built using CFR and a novel game decomposition method. Our new agentwins the three-player events of the 2012 ACPC and defeats the winningthree-player programs from previous competitions while requiring less resourcesto generate than the 2011 winner. Finally, we show that our CFR modificationcomputes a strategy of equal quality to our new agent in a quarter of the timeof standard CFR using half the memory.

Regret Minimization in Non-Zero-Sum Games with Applications to Building Champion Multiplayer Computer Poker Agents