Extensive games are often used to model the interactions of multiple agents within an environment. Much recent work has focused on increasing the size of an extensive game that can be feasibly solved. Despite these improvements, many interesting games are still too large for such techniques. A common approach for computing strategies in these large games is to first employ an abstraction technique to reduce the original game to an abstract game that is of a manageable size. This abstract game is then solved and the resulting strategy is used in the original game. Most top programs in recent AAAI Computer Poker Competitions use this approach. The trend in this competition has been that strategies found in larger abstract games tend to beat strategies found in smaller abstract games. These larger abstract games have more expressive strategy spaces and therefore contain better strategies. In this paper we present a new method for computing strategies in large games. This method allows us to compute more expressive strategies without increasing the size of abstract games that we are required to solve. We demonstrate the power of the approach experimentally in both small and large games, while also providing a theoretical justification for the resulting improvement.
Extensive-form games are a powerful model for representing interactions between agents. Nash equilibrium strategies are a common solution concept for extensive-form games and, in two-player zero-sum games, there are efficient algorithms for calculating such strategies. In large games, this computation may require too much memory and time to be tractable. A standard approach in such cases is to apply a lossy state-space abstraction technique to produce a smaller abstract game that can be tractably solved, while hoping that the resulting abstract game equilibrium is close to an equilibrium strategy in the unabstracted game. Recent work has shown that this assumption is unreliable, and an arbitrary Nash equilibrium in the abstract game is unlikely to be even near the least suboptimal strategy that can be represented in that space. In this work, we present for the first time an algorithm which efficiently finds optimal abstract strategies --- strategies with minimal exploitability in the unabstracted game. We use this technique to find the least exploitable strategy ever reported for two-player limit Texas hold'em.
Uncertainty in poker stems from two key sources, the shuffled deck and an adversary whose strategy is unknown. One approach is to find a pessimistic game theoretic solution (i.e. a Nash equilibrium), but human players have idiosyncratic weaknesses that can be exploited if a model of their strategy can be learned by observing their play. However, games against humans last for at most a few hundred hands so learning must be fast to be effective. We explore two approaches to opponent modelling in the context of Kuhn poker, a small game for which game theoretic solutions are known. Parameter estimation and expert algorithms are both studied. Experiments demonstrate that, even in this small game, convergence to maximally exploitive solutions in a small number of hands is impractical, but that good (i.e.
A fundamental challenge in imperfect-information games is that states do not have well-defined values. As a result, depth-limited search algorithms used in single-agent settings and perfect-information games do not apply. This paper introduces a principled way to conduct depth-limited solving in imperfect-information games by allowing the opponent to choose among a number of strategies for the remainder of the game at the depth limit. Each one of these strategies results in a different set of values for leaf nodes. This forces an agent to be robust to the different strategies an opponent may employ. We demonstrate the effectiveness of this approach by building a master-level heads-up no-limit Texas hold'em poker AI that defeats two prior top agents using only a 4-core CPU and 16 GB of memory. Developing such a powerful agent would have previously required a supercomputer.
Creating strong agents for games with more than two players is a major open problem in AI. Common approaches are based on approximating game-theoretic solution concepts such as Nash equilibrium, which have strong theoretical guarantees in two-player zero-sum games, but no guarantees in non-zero-sum games or in games with more than two players. We describe an agent that is able to defeat a variety of realistic opponents using an exact Nash equilibrium strategy in a 3-player imperfect-information game. This shows that, despite a lack of theoretical guarantees, agents based on Nash equilibrium strategies can be successful in multiplayer games after all.