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39
The Complexity of Pure Nash Equilibria
, 2004
"... We investigate from the computational viewpoint multiplayer games that are guaranteed to have pure Nash equilibria. We focus on congestion games, and show that a pure Nash equilibrium can be computed in polynomial time in the symmetric network case, while the problem is PLScomplete in general. ..."
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Cited by 145 (6 self)
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We investigate from the computational viewpoint multiplayer games that are guaranteed to have pure Nash equilibria. We focus on congestion games, and show that a pure Nash equilibrium can be computed in polynomial time in the symmetric network case, while the problem is PLScomplete in general. We discuss implications to nonatomic congestion games, and we explore the scope of the potential function method for proving existence of pure Nash equilibria.
Complexity Results about Nash Equilibria
, 2002
"... Noncooperative game theory provides a normative framework for analyzing strategic interactions. ..."
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Cited by 130 (10 self)
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Noncooperative game theory provides a normative framework for analyzing strategic interactions.
Playing Large Games using Simple Strategies
, 2003
"... We prove the existence of #Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payo#s to all players in any (exact) Nash equilibrium can be #approximated by the payo#s to the players in some such logarithmic support #Nash equilibrium. These ..."
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Cited by 90 (1 self)
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We prove the existence of #Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payo#s to all players in any (exact) Nash equilibrium can be #approximated by the payo#s to the players in some such logarithmic support #Nash equilibrium. These strategies are also uniform on a multiset of logarithmic size and therefore this leads to a quasipolynomial algorithm for computing an #Nash equilibrium. To our knowledge this is the first subexponential algorithm for finding an #Nash equilibrium. Our results hold for any multipleplayer game as long as the number of players is a constant (i.e., it is independent of the number of pure strategies). A similar argument also proves that for a fixed number of players m, the payo#s to all players in any mtuple of mixed strategies can be #approximated by the payo#s in some mtuple of constant support strategies.
Run the GAMUT: A comprehensive approach to evaluating gametheoretic algorithms
 In AAMAS04
, 2004
"... We present GAMUT 1, a suite of game generators designed for testing gametheoretic algorithms. We explain why such a generator is necessary, offer a way of visualizing relationships between the sets of games supported by GAMUT, and give an overview of GAMUT’s architecture. We highlight the importanc ..."
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Cited by 65 (8 self)
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We present GAMUT 1, a suite of game generators designed for testing gametheoretic algorithms. We explain why such a generator is necessary, offer a way of visualizing relationships between the sets of games supported by GAMUT, and give an overview of GAMUT’s architecture. We highlight the importance of using comprehensive test data by benchmarking existing algorithms. We show surprisingly large variation in algorithm performance across different sets of games for two widelystudied problems: computing Nash equilibria and multiagent learning in repeated games. 2 1.
Finding equilibria in large sequential games of imperfect information
 In ACM Conference on Electronic Commerce
, 2006
"... Information ∗ ..."
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.
GAMES OF FIXED RANK: A HIERARCHY OF BIMATRIX GAMES
, 2007
"... We propose and investigate a hierarchy of bimatrix games (A, B), whose (entrywise) sum of the payoff matrices of the two players is of rank k, where k is a constant. We will say the rank of such a game is k. For every fixed k, the class of rank kgames strictly generalizes the class of zerosum ga ..."
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Cited by 19 (1 self)
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We propose and investigate a hierarchy of bimatrix games (A, B), whose (entrywise) sum of the payoff matrices of the two players is of rank k, where k is a constant. We will say the rank of such a game is k. For every fixed k, the class of rank kgames strictly generalizes the class of zerosum games, but is a very special case of general bimatrix games. We study both the expressive power and the algorithmic behavior of these games. Specifically, we show that even for k = 1 the set of Nash equilibria of these games can consist of an arbitrarily large number of connected components. While the question of exact polynomial time algorithms to find a Nash equilibrium remains open for games of fixed rank, we present polynomial time algorithms for finding an εapproximation.
New Maximal Numbers of Equilibria in Bimatrix Games
, 1999
"... This paper presents a new lower bound of 2.414 d / √ d on the maximal number of Nash equilibria in d × d bimatrix games, a central concept in game theory. The proof uses an equivalent formulation of the problem in terms of pairs of polytopes with 2d facets in dspace. It refutes a recent conjecture ..."
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Cited by 18 (3 self)
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This paper presents a new lower bound of 2.414 d / √ d on the maximal number of Nash equilibria in d × d bimatrix games, a central concept in game theory. The proof uses an equivalent formulation of the problem in terms of pairs of polytopes with 2d facets in dspace. It refutes a recent conjecture that 2 d −1 is an upper bound, which was proved for d ≤ 4. The first counterexample is a 6×6 game with 75 equilibria. The case d = 5 remains open. The result carries the lower bound closer to the previously known upper bound of 2.6 d / √ d.
Interestbased negotiation in multiagent Systems
, 2004
"... Software systems involving autonomous interacting software entities (or agents) present new challenges in computer science and software engineering. A particularly challenging problem is the engineering of various forms of interaction among agents. Interaction may be aimed at enabling agents to coor ..."
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Cited by 17 (10 self)
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Software systems involving autonomous interacting software entities (or agents) present new challenges in computer science and software engineering. A particularly challenging problem is the engineering of various forms of interaction among agents. Interaction may be aimed at enabling agents to coordinate their activities, cooperate to reach common objectives, or exchange resources to better achieve their individual objectives. This thesis is concerned with negotiation: a process through which multiple selfinterested agents can reach agreement over the exchange of scarce resources. In particular, I focus on settings where agents have limited or uncertain information, precluding them from making optimal individual decisions. I demonstrate that this form of boundedrationality may lead agents to suboptimal negotiation agreements. I argue that rational dialogue based on the exchange of arguments can enable agents to overcome this problem. Since agents make decisions based on particular underlying reasons, namely their interests, beliefs and planning knowledge, then rational dialogue over these reasons can enable agents to refine their individual decisions and consequently reach better agreements. I refer to this form of interaction as “interestedbased negotiation.” The contributions of the thesis begin with a conceptual and formal framework for interestbased negotiation among computational agents. Then, an exploration of the differences between this approach and the more traditional bargainingbased approaches is presented. Strategic issues are then explored and a methodology for designing negotiation strategies is developed. Finally, the applicability of the framework is explored through a pilot application that makes use of interestbased negotiation in order to support cooperative activity among mobile users. iii Declaration This is to certify that 1. the thesis comprises only my original work towards the PhD except where indicated in the Preface 2. due acknowledgement has been made in the text to all other material used, 3. the thesis is less than 100,000 words in length, exclusive of tables, bibliographies and appendices. Iyad Rahwan v To my parents... with love and gratitude vii
The game world is flat: The complexity of Nash equilibria in succinct games
 Proc. ICALP
, 2006
"... Abstract. A recent sequence of results established that computing Nash equilibria in normal form games is a PPADcomplete problem even in the case of two players [11,6,4]. By extending these techniques we prove a general theorem, showing that, for a far more general class of families of succinctly r ..."
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Cited by 15 (4 self)
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Abstract. A recent sequence of results established that computing Nash equilibria in normal form games is a PPADcomplete problem even in the case of two players [11,6,4]. By extending these techniques we prove a general theorem, showing that, for a far more general class of families of succinctly representable multiplayer games, the Nash equilibrium problem can also be reduced to the twoplayer case. In view of empirically successful algorithms available for this problem, this is in essence a positive result — even though, due to the complexity of the reductions, it is of no immediate practical significance. We further extend this conclusion to extensive form games and network congestion games, two classes which do not fall into the same succinct representation framework, and for which no positive algorithmic result had been known. 1