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22
Evolutionary Game Theory
, 1995
"... Abstract. Experimentalists frequently claim that human subjects in the laboratory violate gametheoretic predictions. It is here argued that this claim is usually premature. The paper elaborates on this theme by way of raising some conceptual and methodological issues in connection with the very def ..."
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Cited by 642 (9 self)
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Abstract. Experimentalists frequently claim that human subjects in the laboratory violate gametheoretic predictions. It is here argued that this claim is usually premature. The paper elaborates on this theme by way of raising some conceptual and methodological issues in connection with the very definition of a game and of players ’ preferences, in particular with respect to potential context dependence, interpersonal preference dependence, backward induction and incomplete information.
Efficient Computation of Behavior Strategies
 GAMES AND ECONOMIC BEHAVIOR
, 1996
"... We propose the sequence form as a new strategic description for an extensive game with perfect recall. It is similar to the normal form but has linear instead of exponential complexity, and allows a direct representation and efficient computation of behavior strategies. Pure strategies and their mix ..."
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Cited by 46 (8 self)
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We propose the sequence form as a new strategic description for an extensive game with perfect recall. It is similar to the normal form but has linear instead of exponential complexity, and allows a direct representation and efficient computation of behavior strategies. Pure strategies and their mixed strategy probabilities are replaced by sequences of consecutive choices and their realization probabilities. A zerosum game is solved by a corresponding linear program that has linear size in the size of the game tree. General twoperson games are studied in the paper by Koller, Megiddo, and von Stengel in this journal issue.
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.
Finding equilibria in large sequential games of imperfect information
 In ACM Conference on Electronic Commerce
, 2006
"... Information ∗ ..."
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.
Finding Mixed Strategies with Small Supports in Extensive Form Games
 International Journal of Game Theory
, 1995
"... The complexity of algorithms that compute strategies or operate on them typically depends on the representation length of the strategies involved. One measure for the size of a mixed strategy is the number of strategies in its supportthe set of pure strategies to which it gives positive probabili ..."
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Cited by 23 (1 self)
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The complexity of algorithms that compute strategies or operate on them typically depends on the representation length of the strategies involved. One measure for the size of a mixed strategy is the number of strategies in its supportthe set of pure strategies to which it gives positive probability. This paper investigates the existence of "small" mixed strategies in extensive form games, and how such strategies can be used to create more efficient algorithms. The basic idea is that, in an extensive form game, a mixed strategy induces a small set of realization weights that completely describe its observable behavior. This fact can be used to show that for any mixed strategy ¯, there exists a realizationequivalent mixed strategy ¯ 0 whose size is at most the size of the game tree. For a player with imperfect recall, the problem of finding such a strategy ¯ 0 (given the realization weights) is NPhard. On the other hand, if ¯ is a behavior strategy, ¯ 0 can be constructed from...
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.
SelfAdmissible Sets
, 2004
"... We study a weakdominance analog to Pearce’s [28, 1984] fundamental solution concept of a bestresponse set. The concept, called a selfadmissible set (SAS), arises from an epistemic analysis of weak dominance in BrandenburgerFriedenbergKeisler [12, 2007]. Here, we ‘test’ the SAS concept by: (i) e ..."
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Cited by 5 (2 self)
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We study a weakdominance analog to Pearce’s [28, 1984] fundamental solution concept of a bestresponse set. The concept, called a selfadmissible set (SAS), arises from an epistemic analysis of weak dominance in BrandenburgerFriedenbergKeisler [12, 2007]. Here, we ‘test’ the SAS concept by: (i) examining which of the KohlbergMertens [22, 1986] axioms it satisfies; (ii) analyzing its behavior in the Finitely Repeated Prisoner’s Dilemma, Centipede, and the Chain Store Game; and (iii) characterizing it in perfectinformation games.
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