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A competitive Texas Hold’em poker player via automated abstraction and realtime equilibrium computation
 IN PROCEEDINGS OF THE NATIONAL CONFERENCE ON ARTIFICIAL INTELLIGENCE (AAAI
, 2006
"... We present a game theorybased headsup Texas Hold’em poker player, GS1. To overcome the computational obstacles stemming from Texas Hold’em’s gigantic game tree, the player employs our automated abstraction techniques to reduce the complexity of the strategy computations. Texas Hold’em consists of ..."
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Cited by 45 (15 self)
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We present a game theorybased headsup Texas Hold’em poker player, GS1. To overcome the computational obstacles stemming from Texas Hold’em’s gigantic game tree, the player employs our automated abstraction techniques to reduce the complexity of the strategy computations. Texas Hold’em consists of four betting rounds. Our player solves a large linear program (offline) to compute strategies for the abstracted first and second rounds. After the second betting round, our player updates the probability of each possible hand based on the observed betting actions in the first two rounds as well as the revealed cards. Using these updated probabilities, our player computes in realtime an equilibrium approximation for the last two abstracted rounds. We demonstrate that our player, which incorporates very little pokerspecific knowledge, is competitive with leading pokerplaying programs which incorporate extensive domain knowledge, as well as with advanced human players.
Nash Equilibrium and the History of Economic Theory
, 1996
"... John Nash's formulation of noncooperative game theory was one of the great breakthroughs in the history of social science. Nash's work in this area is reviewed in its historical context, to better understand how the fundamental ideas of noncooperative game theory were developed and how they change ..."
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Cited by 32 (3 self)
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John Nash's formulation of noncooperative game theory was one of the great breakthroughs in the history of social science. Nash's work in this area is reviewed in its historical context, to better understand how the fundamental ideas of noncooperative game theory were developed and how they changed the course of economic theory.
Better automated abstraction techniques for imperfect information games, with application to Texas Hold’em poker
 In International Conference on Autonomous Agents and MultiAgent Systems (AAMAS
, 2007
"... 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 statespace abstraction algorithm. In each round of the game, there is a limit to the number of strategic ..."
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Cited by 24 (8 self)
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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 statespace abstraction algorithm. In each round of the game, there is a limit to the number of strategically different situations that an equilibriumfinding 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 equilibriumfinding 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.
Lossless abstraction of imperfect information games
 Journal of the ACM
, 2007
"... Abstract. Finding an equilibrium of an extensive form game of imperfect information is a fundamental problem in computational game theory, but current techniques do not scale to large games. To address this, we introduce the ordered game isomorphism and the related ordered game isomorphic abstractio ..."
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Cited by 21 (9 self)
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Abstract. Finding an equilibrium of an extensive form game of imperfect information is a fundamental problem in computational game theory, but current techniques do not scale to large games. To address this, we introduce the ordered game isomorphism and the related ordered game isomorphic abstraction transformation. For a multiplayer sequential game of imperfect information with observable actions and an ordered signal space, we prove that any Nash equilibrium in an abstracted smaller game, obtained by one or more applications of the transformation, can be easily converted into a Nash equilibrium in the original game. We present an algorithm, GameShrink, for abstracting the game using our isomorphism exhaustively. Its complexity is Õ(n2), where n is the number of nodes in a structure we call the signal tree. It is no larger than the game tree, and on nontrivial games it is drastically smaller, so GameShrink has time and space complexity sublinear in the size of the game tree. Using GameShrink, we find an equilibrium to a poker game with 3.1 billion nodes—over four orders of magnitude more than in the largest poker game solved previously. To address even larger games, we introduce approximation methods that do not preserve equilibrium, but nevertheless yield (ex post) provably closetooptimal strategies.
Dealing with Imperfect Information in Poker
, 1998
"... Poker provides an excellent testbed for studying decisionmaking under conditions of uncertainty. There are many benefits to be gained from designing and experimenting with poker programs. It is a game of imperfect knowledge, where multiple competing agents must understand estimation, prediction, ri ..."
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Cited by 15 (0 self)
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Poker provides an excellent testbed for studying decisionmaking under conditions of uncertainty. There are many benefits to be gained from designing and experimenting with poker programs. It is a game of imperfect knowledge, where multiple competing agents must understand estimation, prediction, risk management, deception, counterdeception, and agent modeling. New evaluation techniques for estimating the strength and potential of a poker hand are presented. This thesis describes the implementation of a program that successfully handles all aspects of the game, and uses adaptive opponent modeling to improve performance. Acknowledgements The author would like to acknowledge the following: ffl A big thanks to Jonathan Schaeffer, for being an excellent supervisor. ffl Darse Billings, for all his time, ideas and insightful input. ffl Duane Szafron, for contributing to the poker research group. ffl The National Sciences and Engineering Research Council, for providing financial support...
DNA starts to learn poker
 Proc. DNA7, Lecture Notes in Computer Science
, 2003
"... Abstract. DNA is used to implement a simplified version of poker. Strategies are evolved that mix bluffing with telling the truth. The essential features are (1) to wait your turn, (2) to default to the most conservative course, (3) to probabilistically override the default in some cases, and (4) to ..."
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Cited by 5 (1 self)
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Abstract. DNA is used to implement a simplified version of poker. Strategies are evolved that mix bluffing with telling the truth. The essential features are (1) to wait your turn, (2) to default to the most conservative course, (3) to probabilistically override the default in some cases, and (4) to learn from payoffs. Two players each use an independent population of strategies that adapt and learn from their experiences in competition. 1
Three player, two strategy maximally entangled quantum games, to appear
 in Proceedings, 9th International Conference in Pure Mathematics, Pakistan Mathematical Society, Islamabad
"... We develop an octonionic representation of the payoff function for a three player, two strategy, maximally entangled quantum game. 1 A Formalism for Quantization Up until now, to quantize games most authors have, like Meyer, focused their efforts on the quantization of the players ’ strategy spaces, ..."
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Cited by 2 (0 self)
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We develop an octonionic representation of the payoff function for a three player, two strategy, maximally entangled quantum game. 1 A Formalism for Quantization Up until now, to quantize games most authors have, like Meyer, focused their efforts on the quantization of the players ’ strategy spaces, essentially the domain of the payoff function that defines the game to be quantized (see for example [7, 8, 11, 12]). The principal technique is to identify these strategy spaces with an orthogonal basis of some quantum system, in order that players may now take superpositions or even mixed superpositions of strategic choices by acting on the system via quantum operations. In addition, players may now correlate their strategic choices via the entanglement of the joint states of the system. Frequently, mere access to the higher randomization of superposition (as opposed to real probabilistic combination) or the correlation of strategic choices via entanglement allows payoffs to the players superior to those available in the game and its classical extensions. In [3] a mathematical formalism for game quantization that focuses on the quantization of the payouts of the original game G to be quantized, and expresses the quantized version as a (proper) extension of the original payout
Algorithms for abstracting and solving imperfect information games
, 2007
"... Game theory is the mathematical study of rational behavior in strategic environments. In many settings, most notably twoperson zerosum games, game theory provides particularly strong and appealing solution concepts. Furthermore, these solutions are efficiently computable in the complexitytheory s ..."
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Cited by 2 (1 self)
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Game theory is the mathematical study of rational behavior in strategic environments. In many settings, most notably twoperson zerosum games, game theory provides particularly strong and appealing solution concepts. Furthermore, these solutions are efficiently computable in the complexitytheory 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 statespaces 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., equilibriumpreserving), 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 ɛequilibria 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
A competitive Texas Hold'em . . .
, 2006
"... We present our game theorybased headsup Texas Hold’em poker player. To overcome the computational obstacles stemming from Texas Hold’em’s gigantic game tree, our player employs automated abstraction techniques to reduce the complexity of the strategy computations. In addition to this statespace a ..."
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We present our game theorybased headsup Texas Hold’em poker player. To overcome the computational obstacles stemming from Texas Hold’em’s gigantic game tree, our player employs automated abstraction techniques to reduce the complexity of the strategy computations. In addition to this statespace abstraction, our player uses roundbased abstraction in conjunction with both offline and realtime equilibrium approximation. Texas Hold’em consists of four betting rounds. Our player solves a large linear program (offline) to compute strategies for the abstracted first and second rounds. After the second betting round, our player updates the probability of each possible hand based on the observed betting actions in the first two rounds as well as the revealed cards. Using these updated probabilities, our player computes in realtime an equilibrium approximation for the last two abstracted rounds. We demonstrate that our player, which does not directly incorporate any pokerspecific expert knowledge, is competitive with leading pokerplaying programs which do incorporate such domainspecific knowledge, as well as with advanced human players.