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Solving Systems of Polynomial Equations
 American Mathematical Society, CBMS Regional Conferences Series, No 97
, 2002
"... Abstract. One of the most classical problems of mathematics is to solve systems of polynomial equations in several unknowns. Today, polynomial models are ubiquitous and widely applied across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, gam ..."
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Cited by 145 (10 self)
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Abstract. One of the most classical problems of mathematics is to solve systems of polynomial equations in several unknowns. Today, polynomial models are ubiquitous and widely applied across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, game theory, statistics, machine learning, control theory, and numerous other areas. The set of solutions to a system of polynomial equations is an algebraic variety, the basic object of algebraic geometry. The algorithmic study of algebraic varieties is the central theme of computational algebraic geometry. Exciting recent developments in symbolic algebra and numerical software for geometric calculations have revolutionized the field, making formerly inaccessible problems tractable, and providing fertile ground for experimentation and conjecture. The first half of this book furnishes an introduction and represents a snapshot of the state of the art regarding systems of polynomial equations. Afficionados of the wellknown text books by Cox, Little, and O’Shea will find familiar themes in the first five chapters: polynomials in one variable, Gröbner
The Challenge of Poker
 Artificial Intelligence
, 2001
"... Poker is an interesting testbed 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 ..."
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Cited by 110 (9 self)
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Poker is an interesting testbed 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 decisionmaking 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 worldclass level. 1
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.
Famous trails to Paul Erdős
 MATHEMATICAL INTELLIGENCER
, 1999
"... The notion of Erdős number has floated around the mathematical research community for more than thirty years, as a way to quantify the common knowledge that mathematical and scientific research has become a very collaborative process in the twentieth century, not an activity engaged in solely by ..."
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Cited by 18 (0 self)
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The notion of Erdős number has floated around the mathematical research community for more than thirty years, as a way to quantify the common knowledge that mathematical and scientific research has become a very collaborative process in the twentieth century, not an activity engaged in solely by isolated individuals. In this paper we explore some (fairly short) collaboration paths that one can follow from Paul Erdős to researchers inside and outside of mathematics. In particular, we find that all the Fields Medalists up through 1998 have Erdős numbers less than 6, and that over 60 Nobel Prize winners in physics, chemistry, economics, and medicine have Erdős numbers less than 9.
Opponent Modeling in Poker: Learning and Acting in a Hostile and Uncertain Environment
, 2002
"... Artificial intelligence research has had great success in many clasic games such as chess, checkers, and othello. In these perfectinformation domains, alphabeta search is sufficient to achieve a high level of play. However Artificial intelligence research has long been criticized for focusing on d ..."
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Cited by 16 (0 self)
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Artificial intelligence research has had great success in many clasic games such as chess, checkers, and othello. In these perfectinformation domains, alphabeta 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 nondeterminism. Poker, the archetypal game studied by...
The Effectiveness of Opponent Modelling in a Small Imperfect Information Game
, 2006
"... Opponent modelling is an important issue in games programming today. Programs which
do not perform opponent modelling are unlikely to take full advantage of the mistakes
made by an opponent. Additionally, programs which do not adapt over time become less
of a challenge to players, causing these pla ..."
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Cited by 4 (1 self)
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Opponent modelling is an important issue in games programming today. Programs which
do not perform opponent modelling are unlikely to take full advantage of the mistakes
made by an opponent. Additionally, programs which do not adapt over time become less
of a challenge to players, causing these players to lose interest. While opponent modelling
can be a difficult challenge in perfect information games, where the full state of the game
is known to all players at all times, it becomes an even more difficult task in games of
imperfect information, where players are not always able to observe the actual state of
the game. This thesis studies the problem of opponent modelling in Kuhn Poker, a small
imperfect information game that contains several properties that make realworld poker
games interesting. Two basic types of opponent modelling are studied, explicit modelling
and implicit modelling, and their effectiveness is compared.
Strippeddown poker: A classroom game with signaling and bluffing, February 2005. Working paper. Available at http://economics.eller.arizona.edu/ downloads/working_papers/EconWP05%11.pdf
"... This paper proposes a simplified version of poker as an instructional classroom game. In spite of the game’s simplicity, it provides an excellent illustration of a number of topics: signaling, bluffing, mixed strategies, the value of information, and Bayes ’ Rule. We first briefly cover the history ..."
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Cited by 3 (0 self)
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This paper proposes a simplified version of poker as an instructional classroom game. In spite of the game’s simplicity, it provides an excellent illustration of a number of topics: signaling, bluffing, mixed strategies, the value of information, and Bayes ’ Rule. We first briefly cover the history of poker in gametheoretic contexts. Next we characterize Strippeddown Poker: how to play it, what makes it an interesting classroom game, and how to teach its solution to students. We discuss possible applications of this model to realworld interactions, such as litigation, tax evasion, and domestic or international diplomacy. Finally, we suggest modifications of the game either for use in class or as homework problems.
Artificial intelligence and data mining applied to nolimit Texas Hold’em
, 2007
"... This project has two goals. First, to create a poker playing computer softbot for Sit and Go tournaments. This bot will be based on statistics and known strategies used by real players and should at least play better than a player new to the game. Second, to data mine records of poker hands played i ..."
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This project has two goals. First, to create a poker playing computer softbot for Sit and Go tournaments. This bot will be based on statistics and known strategies used by real players and should at least play better than a player new to the game. Second, to data mine records of poker hands played in the past and look for potentially interesting patterns that might be useful in creating a poker bot. The goal here is to at least find some interesting statistics about the way people play poker, and perhaps to even find a way to predict a person’s cards by analysing his actions. 1
CASPER: DESIGN AND DEVELOPMENT OF A CASEBASED POKER PLAYER
"... Poker provides a challenging domain for Artificial Intelligence research due to the game’s properties such as hidden information (the other player’s cards) and nondeterminism (random shuffling of the deck). Recent approaches to Poker research have required intensive knowledge engineering efforts. Th ..."
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Poker provides a challenging domain for Artificial Intelligence research due to the game’s properties such as hidden information (the other player’s cards) and nondeterminism (random shuffling of the deck). Recent approaches to Poker research have required intensive knowledge engineering efforts. This thesis discusses the design and development of a CASebased Poker playER (CASPER) that uses the CaseBased Reasoning methodology to make betting decisions at the poker table. The results suggest it is possible to record instances of games played between strong poker players and then reuse these to obtain a similar performance therefore bypassing the need for the initial, intensive knowledge engineering process. An investigation into deriving optimal feature weights using evolutionary algorithms has also been conducted. Casper has been extensively evaluated by challenging various sets of opponents, including both computerised opponents and real opponents. ii Acknowledgements Thank you to my supervisor, Ian Watson, for the opportunities you provided for me and the time and effort you devoted to me. I also need to thank the University of Alberta
Electronic Colloquium on Computational Complexity, Report No. 52 (2005) The Computational Complexity of Nash Equilibria in Concisely Represented Games ∗
, 2005
"... Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blo ..."
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Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blowup as the number of agents grows. We study two models of concisely represented games: circuit games, where the payoffs are computed by a given boolean circuit, and graph games, where each agent’s payoff is a function of only the strategies played by its neighbors in a given graph. For these two models, we study the complexity of four questions: determining if a given strategy is a Nash equilibrium, finding a Nash equilibrium, determining if there exists a pure Nash equilibrium, and determining if there exists a Nash equilibrium in which the payoffs to a player meet some given guarantees. In many cases, we obtain tight results, showing that the problems are complete for various complexity classes. 1