Results 1  10
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17
Settling the complexity of twoplayer Nash equilibrium
 In Proc. 47th FOCS
, 2006
"... We prove that the problem of finding a Nash equilibrium in a twoplayer game is PPADcomplete. 1 ..."
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Cited by 105 (5 self)
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We prove that the problem of finding a Nash equilibrium in a twoplayer game is PPADcomplete. 1
Efficient algorithms for constant well supported approximate equilibria in bimatrix games
 In: Proc. of the 34th Int. Col. on Aut., Lang. and Progr (ICALP ’07). LNCS
, 2007
"... Abstract. In this work we study the tractability of well supported approximate Nash Equilibria (SuppNE in short) in bimatrix games. In view of the apparent intractability of constructing Nash Equilibria (NE in short) in polynomial time, even for bimatrix games, understanding the limitations of the a ..."
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Cited by 8 (4 self)
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Abstract. In this work we study the tractability of well supported approximate Nash Equilibria (SuppNE in short) in bimatrix games. In view of the apparent intractability of constructing Nash Equilibria (NE in short) in polynomial time, even for bimatrix games, understanding the limitations of the approximability of the problem is of great importance. We initially prove that SuppNE are immune to the addition of arbitrary real vectors to the rows (columns) of the row (column) player’s payoff matrix. Consequently we propose a polynomial time algorithm (based on linear programming) that constructs a 0.5−SuppNE for arbitrary win lose games. We then parameterize our technique for win lose games, in order to apply it to arbitrary (normalized) bimatrix games. Indeed, this new technique leads to a weaker φ−SuppNE for win lose games, where φ = √ 5−1 2 the golden ratio. Nevertheless, this parameterized technique extends nicely to a technique for arbitrary [0, 1]−bimatrix games, which assures a 0.658−SuppNE in polynomial time. To our knowledge, these are the first polynomial time algorithms providing ε−SuppNE of normalized or win lose bimatrix games, for some nontrivial constant ε ∈ [0, 1), bounded away from 1.
Computing equilibria: A computational complexity perspective
, 2009
"... Computational complexity is the subfield of computer science that rigorously studies the intrinsic difficulty of computational problems. This survey explains how complexity theory defines “hard problems”; applies these concepts to several equilibrium computation problems; and discusses implications ..."
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Cited by 7 (2 self)
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Computational complexity is the subfield of computer science that rigorously studies the intrinsic difficulty of computational problems. This survey explains how complexity theory defines “hard problems”; applies these concepts to several equilibrium computation problems; and discusses implications for computation, games, and behavior. We assume
An algorithmic game theory primer
, 2008
"... We give a brief and biased survey of the past, present, and future of research on the interface of theoretical computer science and game theory. 1 ..."
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Cited by 5 (0 self)
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We give a brief and biased survey of the past, present, and future of research on the interface of theoretical computer science and game theory. 1
Well supported approximate equilibria in bimatrix games: A graph theoretic approach
 Project DELIS – Dynamically Evolving Large Scale Information Systems
, 2007
"... Abstract. We study the existence and tractability of a notion of approximate equilibria in bimatrix games, called well supported approximate Nash Equilibria (SuppNE in short). We prove existence of ε−SuppNE for any constant ε ∈ (0,1), with only logarithmic support sizes for both players. Also we pro ..."
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Cited by 4 (2 self)
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Abstract. We study the existence and tractability of a notion of approximate equilibria in bimatrix games, called well supported approximate Nash Equilibria (SuppNE in short). We prove existence of ε−SuppNE for any constant ε ∈ (0,1), with only logarithmic support sizes for both players. Also we propose a polynomial–time construction of SuppNE, both for win lose and for arbitrary (normalized) bimatrix games. The quality of these SuppNE depends on the girth of the Nash Dynamics graph in the win lose game, or a (rounded–off) win lose image of the original normalized game. Our constructions are very successful in sparse win lose games (ie, having a constant number of (0,1)−elements in the bimatrix) with large girth in the Nash Dynamics graph. The same holds also for normalized games whose win lose image is sparse with large girth. Finally we prove the simplicity of constructing SuppNE both in random normalized games and in random win lose games. In the former case we prove that the uniform full mix is an o(1) −SuppNE, while in the case of win lose games, we show that (with high probability) there is either a PNE or a 0.5SuppNE with support sizes only 2. 1
Approximation guarantees for fictitious play
 In Proceedings of the 47th Annual Allerton Conference on Communication, Control, and Computing
, 2009
"... Abstract—Fictitious play is a simple, wellknown, and oftenused algorithm for playing (and, especially, learning to play) games. However, in general it does not converge to equilibrium; even when it does, we may not be able to run it to convergence. Still, we may obtain an approximate equilibrium. I ..."
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Cited by 3 (0 self)
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Abstract—Fictitious play is a simple, wellknown, and oftenused algorithm for playing (and, especially, learning to play) games. However, in general it does not converge to equilibrium; even when it does, we may not be able to run it to convergence. Still, we may obtain an approximate equilibrium. In this paper, we study the approximation properties that fictitious play obtains when it is run for a limited number of rounds. We show that if both players randomize uniformly over their actions in the first r rounds of fictitious play, then the result is an ǫequilibrium, where ǫ = (r + 1)/(2r). (Since we are examining only a constant number of pure strategies, we know that ǫ < 1/2 is impossible, due to a result of Feder et al.) We show that this bound is tight in the worst case; however, with an experiment on random games, we illustrate that fictitious play usually obtains a much better approximation. We then consider the possibility that the players fail to choose the same r. We show how to obtain the optimal approximation guarantee when both the opponent’s r and the game are adversarially chosen (but there is an upper bound R on the opponent’s r), using a linear program formulation. We show that if the action played in the ith round of fictitious play is chosen with probability proportional to: 1 for i = 1 and 1/(i −1) for all 2 ≤ i ≤ R+1, this gives an approximation guarantee of 1 − 1/(2 + ln R). We also obtain a lower bound of 1 − 4/ln R. This provides an actionable prescription for how long to run fictitious play. I.
Herding in queues with waiting costs: Rationality and regret
 Manufacturing & Service Operations Management
, 2011
"... We study how consumers with waiting cost disutility choose between two congested services of unknown service value. Consumers observe an imperfect private signal indicating which service facility may provide better service value as well as the queue lengths at the service facilities before making th ..."
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Cited by 2 (1 self)
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We study how consumers with waiting cost disutility choose between two congested services of unknown service value. Consumers observe an imperfect private signal indicating which service facility may provide better service value as well as the queue lengths at the service facilities before making their choice. If more consumers choose the same service facility because of their private information, longer queues will form at that facility and indicate higher quality. On the other hand, a long queue also implies more waiting time. We characterize the equilibrium queuejoining behavior of arriving consumers and the extent of their learning from the queue information in the presence of such positive and negative externalities. We find that when the arrival rates are low, utilitymaximizing rational consumers herd and join the longer queue, ignoring any contrary private information. We show that even when consumers treat queues as independently evolving, herd behavior persists with consumers joining longer queues above a threshold queue difference. However, if the consumers seek to minimize ex post regret when making their decisions, herd behavior may be dampened. Key words: herd behavior; queueing games; learning; regret; bounded rationality
From Imitation Games to Kakutani
, 2006
"... We give a full proof of the Kakutani (1941) fixed point theorem that is brief, elementary, and based on game theoretic concepts. This proof points to a new family of algorithms for computing approximate fixed points that have advantages over simplicial subdivision methods. An imitation game is a fin ..."
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We give a full proof of the Kakutani (1941) fixed point theorem that is brief, elementary, and based on game theoretic concepts. This proof points to a new family of algorithms for computing approximate fixed points that have advantages over simplicial subdivision methods. An imitation game is a finite two person normal form game in which the strategy spaces for the two agents are the same and the goal of the second player is to choose the same strategy as the first player. These appear in our proof, but are also interesting from other points of view. They give new insights into the “long” paths of the LemkeHowson algorithm. They provide a rich class of games with “short” LemkeHowson paths. They are useful for studying the complexity of a number of problems in computational economics. K
Computing Equilibria in Bimatrix Games by Parallel Vertex Enumeration
"... Abstract—Equilibria computation is of great importance to many areas such as economics, control theory, and recently computer science. We focus on the computation of Nash equilibria in twoplayer generalsum normal form games, also called bimatrix games. One efficient method to compute these equilib ..."
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Abstract—Equilibria computation is of great importance to many areas such as economics, control theory, and recently computer science. We focus on the computation of Nash equilibria in twoplayer generalsum normal form games, also called bimatrix games. One efficient method to compute these equilibria is based on enumerating the vertices of the best response polyhedrons of the two players and checking the equilibrium conditions for every pair of vertices. We design and implement a parallel algorithm for computing Nash equilibria in bimatrix games based on vertex enumeration. We analyze the performance of the proposed algorithm by performing extensive experiments on a grid computing system. Keywordsgame theory; Nash equilibrium; parallel algorithm; bimatrix game