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'... The computation of the first complete approximations of game-theoretic optimal strategies for fullscale poker is addressed. Several abstraction techniques are combined to represent the game of 2-player Texas Hold'em, having size O(10^18), using closely related models each having size ...'

The computation of the first complete approximations of game-theoretic optimal strategies for fullscale poker is addressed. Several abstraction techniques are combined to represent the game of 2-player Texas Hold'em, having size O(10^18), using closely related models each having size .

'... Poker is an interesting test-bed for arti cial intelligence research. It is a game of imperfect information, where multiple competing agents must deal with probabilistic knowledge, risk assessment, and possible deception, not unlike decisions made in the real world. Opponent modeling is another dicu ...'

Poker is an interesting test-bed for arti cial intelligence research. It is a game of imperfect information, where multiple competing agents must deal with probabilistic knowledge, risk assessment, and possible deception, not unlike decisions made in the real world. Opponent modeling is another dicult problem in decision-making applications, and it is essential to achieving high performance in poker. This paper describes the design considerations and architecture of the poker program Poki. In addition to methods for hand evaluation and betting strategy, Poki uses learning techniques to construct statistical models of each opponent, and dynamically adapts to exploit observed patterns and tendencies. The result is a program capable of playing reasonably strong poker, but there remains considerable research to be done to play at a world-class level. 1

'... We present new approximation methods for computing gametheoretic strategies for sequential games of imperfect information. At a high level, we contribute two new ideas. First, we introduce a new state-space abstraction algorithm. In each round of the game, there is a limit to the number of strategic ...'

We present new approximation methods for computing gametheoretic strategies for sequential games of imperfect information. At a high level, we contribute two new ideas. First, we introduce a new state-space abstraction algorithm. In each round of the game, there is a limit to the number of strategically different situations that an equilibrium-finding algorithm can handle. Given this constraint, we use clustering to discover similar positions, and we compute the abstraction via an integer program that minimizes the expected error at each stage of the game. Second, we present a method for computing the leaf payoffs for a truncated version of the game by simulating the actions in the remaining portion of the game. This allows the equilibrium-finding algorithm to take into account the entire game tree while having to explicitly solve only a truncated version. Experiments show that each of our two new techniques improves performance dramatically in Texas Hold’em poker. The techniques lead to a drastic improvement over prior approaches for automatically generating agents, and our agent plays competitively even against the best agents overall.

'... Artificial intelligence research has had great success in many clasic games such as chess, checkers, and othello. In these perfect-information domains, alpha-beta search is sufficient to achieve a high level of play. However Artificial intelligence research has long been criticized for focusing on d ...'

Artificial intelligence research has had great success in many clasic games such as chess, checkers, and othello. In these perfect-information domains, alpha-beta search is sufficient to achieve a high level of play. However Artificial intelligence research has long been criticized for focusing on deterministic domains of perfect information -- many problems in the real world exhibit properties of imperfect or incomplete information and non-determinism. Poker, the archetypal game studied by...

'... Game theory is the mathematical study of rational behavior in strategic environments. In many settings, most notably two-person zero-sum games, game theory provides particularly strong and appealing solution concepts. Furthermore, these solutions are efficiently computable in the complexity-theory s ...'

Game theory is the mathematical study of rational behavior in strategic environments. In many settings, most notably two-person zero-sum games, game theory provides particularly strong and appealing solution concepts. Furthermore, these solutions are efficiently computable in the complexity-theory sense. However, in most interesting potential applications in artificial intelligence, the solutions are difficult to compute using current techniques due primarily to the extremely large state-spaces of the environments. In this thesis, we propose new algorithms for tackling these computational difficulties. In one stream of research, we introduce automated abstraction algorithms for sequential games of imperfect information. These algorithms take as input a description of a game and produce a description of a strategically similar, but smaller, game as output. We present algorithms that are lossless (i.e., equilibrium-preserving), as well as algorithms that are lossy, but which can yield much smaller games while still retaining the most important features of the original game. In a second stream of research, we develop specialized optimization algorithms for finding ɛ-equilibria in sequential games of imperfect information. The algorithms are based on recent advances in nonsmooth convex optimization (namely the excessive gap technique) and provide significant improvements

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'... We present new approximation methods for computing game-theoretic strategies for sequential games of imperfect infor-mation. At a high level, we contribute two new ideas. First, we introduce a new state-space abstraction algorithm. In each round of the game, there is a limit to the number of strateg ...'

We present new approximation methods for computing game-theoretic strategies for sequential games of imperfect infor-mation. At a high level, we contribute two new ideas. First, we introduce a new state-space abstraction algorithm. In each round of the game, there is a limit to the number of strategically different situations that an equilibrium-finding algorithm can handle. Given this constraint, we use clus-tering to discover similar positions, and we compute the abstraction via an integer program that minimizes the ex-pected error at each stage of the game. Second, we present a method for computing the leaf payoffs for a truncated ver-sion of the game by simulating the actions in the remaining portion of the game. This allows the equilibrium-finding algorithm to take into account the entire game tree while

'... The field of Imperfect Information Games has interested researchers for many years, yet the field has failed to provide good competitive players to play some of the complex card games at the master level. The game of Poker is observed in this project, along with providing two Computer Poker Player s ...'

The field of Imperfect Information Games has interested researchers for many years, yet the field has failed to provide good competitive players to play some of the complex card games at the master level. The game of Poker is observed in this project, along with providing two Computer Poker Player solutions to the gaming problem, Anki – V1 and Anki – V2. These players, along with a few generic ones, were created in this project using methods ranging from Expert Systems to that of Simulation and Enumeration. Anki – V1 and Anki – V2 were tested against a range of hard-coded computer players, and a variety of human players to reach the conclusion that Anki – V2 displays behaviour at the intermediate level of human players. Finally, many interesting conclusions regarding poker strategies and human heuristics are observed and presented in this thesis. ii Acknowledgments I would like to thank Dr. Jessica Chen-Burger for her overwhelming support and help throughout the life-cycle of this project, and for the late nights she spent playing my Poker Players. I would also like to thank Mr. Richard Carter for his insight into the workings of some of the Poker players, and all the authors of the research quoted in my bibliography, especially the creators of Gala, Loki, Poki and PsOpti. I would also like to thank my parents, who have always been there to me, and inspire me every step of the way. And finally, I would like to acknowledge the calming contribution of my lab-fellows, without whom, completing this dissertation couldn't have been nearly this much fun. iii Declaration I declare that this thesis was composed by myself, that the work contained herein is my own except where explicitly stated otherwise in the text, and that this work has not been submitted for any other degree or professional qualification except as specified.

'... Games have always been a strong driving force in artificial intelligence. In the last ten years huge improvements have been made in perfect information games like chess and othello and the strongest computer agents can beat the strongest human players. This is not the case for imperfect information ...'

Games have always been a strong driving force in artificial intelligence. In the last ten years huge improvements have been made in perfect information games like chess and othello and the strongest computer agents can beat the strongest human players. This is not the case for imperfect information games such as poker and bridge where creating an expert computer player has shown to be much harder. Previous research in poker has either adressed fixed-limit poker or simplified variations of poker games. This paper tries to extend known techniqes successfully used in fixed-limit poker to no-limit. Nolimit poker increases the size of the game tree dramatically. To reduce the complexity an abstracted model of the game is created. The abstracted model is transformed to a matrix representation. Finding an optimal strategy for the abstracted model is now a minimization problem using linear programming techniques. The result is a set of pseudo-optimal strategies for no-limit Texas Hold’em that perform well as long as the