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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 62 (21 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.
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 36 (10 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 32 (13 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.
Bilateral bargaining with onesided twotype uncertainty
 Proceedings of the International Joint Conference on Web Intelligence and Intelligent Agent TechnologyVolume 02
, 2009
"... It is a challenging problem to find agents ’ rational strategies in bargaining with incomplete information. In this paper we perform a game theoretic analysis of agents ’ rational strategies in finite horizon bilateral bargaining with onesided uncertainty regarding agents ’ reserve prices. The neg ..."
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Cited by 5 (4 self)
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It is a challenging problem to find agents ’ rational strategies in bargaining with incomplete information. In this paper we perform a game theoretic analysis of agents ’ rational strategies in finite horizon bilateral bargaining with onesided uncertainty regarding agents ’ reserve prices. The negotiation setting considered in this paper has four features: alternatingoffers bargaining protocol, finite horizon, twotype uncertainty about agents ’ reserve prices, and discount factors. The main contribution of this paper is the development of a novel algorithm to find a pure strategy sequential equilibrium in the setting we study. Our algorithm is based on the combination of game theoretic analysis and search techniques which finds agents ’ equilibrium in pure strategies when they exist. 1.
Computing a Proper Equilibrium of a Bimatrix Game
"... We provide the first pivotingtype algorithm that computes an exact proper equilibrium of a bimatrix game. This is achieved by using Lemke’s algorithm to solve a linear complementarity problem (LCP) of polynomial size. This also proves that computing a simple refinement of proper equilibria for bima ..."
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Cited by 4 (1 self)
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We provide the first pivotingtype algorithm that computes an exact proper equilibrium of a bimatrix game. This is achieved by using Lemke’s algorithm to solve a linear complementarity problem (LCP) of polynomial size. This also proves that computing a simple refinement of proper equilibria for bimatrix game is PPADcomplete. The algorithm also computes a witness in the form of a parameterized strategy that is an εproper equilibrium for any given sufficiently small ε, allowing polynomialtime verification of the properties of the refined equilibrium. The same technique can be applied to matrix games (twoplayer zerosum), thereby computing a parameterized εproper strategy in polynomial time using linear programming.
On the Hardness and Existence of QuasiStrict Equilibria
"... This paper investigates the computational properties of quasistrict equilibrium, an attractive equilibrium refinement proposed by Harsanyi, which was recently shown to always exist in bimatrix games. We prove that deciding the existence of a quasistrict equilibrium in games with more than two pla ..."
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Cited by 3 (2 self)
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This paper investigates the computational properties of quasistrict equilibrium, an attractive equilibrium refinement proposed by Harsanyi, which was recently shown to always exist in bimatrix games. We prove that deciding the existence of a quasistrict equilibrium in games with more than two players is NPcomplete. We further show that, in contrast to Nash equilibrium, the support of quasistrict equilibrium in zerosum games is unique and propose a linear program to compute quasistrict equilibria in these games. Finally, we prove that every symmetric multiplayer game where each player has two actions at his disposal contains an efficiently computable quasistrict equilibrium which may itself be asymmetric.
Searching for pure strategy equilibria in bilateral bargaining with onesided uncertainty
 Proc. of the Ninth Int. Joint Conf. on Autonomous Agents and Multiagent Systems
, 2010
"... The problem of finding agents ’ rational strategies in bargaining with incomplete information is well known to be challenging. The literature provides a collection of results for very narrow uncertainty settings, but no generally applicable algorithm. In this paper, we focus on the alternatingoffer ..."
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Cited by 3 (1 self)
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The problem of finding agents ’ rational strategies in bargaining with incomplete information is well known to be challenging. The literature provides a collection of results for very narrow uncertainty settings, but no generally applicable algorithm. In this paper, we focus on the alternatingoffers finite horizon bargaining protocol with onesided uncertainty regarding agents ’ reserve prices. We provide an algorithm based on the combination of game theoretic analysis and search techniques which finds agents ’ equilibrium in pure strategies when they exist. Our approach is sound, complete and, in principle, can be applied to other uncertainty settings. Categories and Subject Descriptors
Computing a self–confirming equilibrium in two–player extensive–form games
, 2010
"... The Nash equilibrium is the most commonly adopted solution concept for non–cooperative interaction situations. However, it underlays on the assumption of common information that is hardly verified in many practical situations. When information is not common, the appropriate game theoretic solution c ..."
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Cited by 2 (2 self)
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The Nash equilibrium is the most commonly adopted solution concept for non–cooperative interaction situations. However, it underlays on the assumption of common information that is hardly verified in many practical situations. When information is not common, the appropriate game theoretic solution concept is the self–confirming equilibrium. It requires that every agent plays the best response to her beliefs and that the beliefs are correct on the equilibrium path. We present, to the best of our knowledge, the first study on the computation of a self–confirming equilibrium for two–player extensive–form games. We provide algorithms, we analyze the computational complexity, and we experimentally evaluate the performance of our algorithms in terms of computational time.