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180
The complexity of game dynamics: Bgp oscillations, sink equilibria, and beyond
 In SODA ’08: Proceedings of the nineteenth annual ACMSIAM symposium on Discrete algorithms
, 2008
"... We settle the complexity of a wellknown problem in networking by establishing that it is PSPACEcomplete to tell whether a system of path preferences in the BGP protocol [25] can lead to oscillatory behavior; one key insight is that the BGP oscillation question is in fact one about Nash dynamics. W ..."
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Cited by 21 (4 self)
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We settle the complexity of a wellknown problem in networking by establishing that it is PSPACEcomplete to tell whether a system of path preferences in the BGP protocol [25] can lead to oscillatory behavior; one key insight is that the BGP oscillation question is in fact one about Nash dynamics. We show that the concept of sink equilibria proposed recently in [11] is also PSPACEcomplete to analyze and approximate for graphical games. Finally, we propose a new equilibrium concept inspired by game dynamics, unit recall equilibria, which we show to be close to universal (exists with high probability in a random game) and algorithmically promising. We also give a relaxation thereof, called componentwise unit recall equilibria, which we show to be both tractable and universal (guaranteed to exist in every game).
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 20 (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.
Polynomial algorithms for approximating Nash equilibria of bimatrix games
 In: Proceedings of the 2nd Workshop on Internet and Network Economics (WINE’06
, 2006
"... 1 PROBLEM DEFINITION Nash [13] introduced the concept of Nash equilibria in noncooperative games and proved that any game possesses at least one such equilibrium. A wellknown algorithm for computing a Nash equilibrium of a 2player game is the LemkeHowson algorithm [11], however it has exponentia ..."
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Cited by 19 (4 self)
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1 PROBLEM DEFINITION Nash [13] introduced the concept of Nash equilibria in noncooperative games and proved that any game possesses at least one such equilibrium. A wellknown algorithm for computing a Nash equilibrium of a 2player game is the LemkeHowson algorithm [11], however it has exponential worstcase running time in the number of available pure strategies [15]. Recently, Daskalakis et al [4] showed that the problem of computing a Nash equilibrium in a game with 4 or more players is PPADcomplete; this result was later extended to games with 3 players [7]. Eventually, Chen and Deng [2] proved that the problem is PPADcomplete for 2player games as well. This fact emerged the computation of approximate Nash equilibria. There are several versions of approximate Nash equilibria that have been defined in the literature; however the focus of this entry is on the notions of ɛNash equilibrium and ɛwellsupported Nash equilibrium. An ɛNash equilibrium is a strategy profile such that no deviating player could achieve a payoff higher than the one that the specific profile gives her, plus ɛ. A stronger notion of approximate Nash equilibria is the ɛwellsupported Nash equilibria; these are strategy profiles such that each player plays only
Symmetries and the Complexity of Pure Nash Equilibrium
, 2006
"... Strategic games may exhibit symmetries in a variety of ways. A common aspect of symmetry, enabling the compact representation of games even when the number of players is unbounded, is that players cannot (or need not) distinguish between the other players. We define four classes of symmetric games b ..."
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Cited by 18 (3 self)
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Strategic games may exhibit symmetries in a variety of ways. A common aspect of symmetry, enabling the compact representation of games even when the number of players is unbounded, is that players cannot (or need not) distinguish between the other players. We define four classes of symmetric games by considering two additional properties: identical payoff functions for all players and the ability to distinguish oneself from the other players. Based on these varying notions of symmetry, we investigate the computational complexity of pure Nash equilibria. It turns out that in all four classes of games equilibria can be found efficiently when only a constant number of actions is available to each player, a problem that has been shown intractable for other succinct representations of multiplayer games. We further show that identical payoff functions simplify the search for equilibria, while a growing number of actions renders it intractable. Finally, we show that our results extend to wider classes of threshold symmetric games where players are unable to determine the exact number of players playing a certain action.
An approach to bounded rationality
 In Advances in Neural Information Processing Systems 19 (Proc. of NIPS 2006
, 2007
"... A central question in game theory and artificial intelligence is how a rational agent should behave in a complex environment, given that it cannot perform unbounded computations. We study strategic aspects of this question by formulating a simple model of a game with additional costs (computational ..."
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Cited by 18 (0 self)
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A central question in game theory and artificial intelligence is how a rational agent should behave in a complex environment, given that it cannot perform unbounded computations. We study strategic aspects of this question by formulating a simple model of a game with additional costs (computational or otherwise) for each strategy. First we connect this to zerosum games, proving a counterintuitive generalization of the classic minmax theorem to zerosum games with the addition of strategy costs. We then show that potential games with strategy costs remain potential games. Both zerosum and potential games with strategy costs maintain a very appealing property: simple learning dynamics converge to equilibrium. 1 The Approach and Basic Model How should an intelligent agent play a complicated game like chess, given that it does not have unlimited time to think? This question reflects one fundamental aspect of “bounded rationality, ” a term coined by Herbert Simon [1]. However, bounded rationality has proven to be a slippery concept to formalize (prior work has focused largely on finite automata playing simple repeated games such
On the Complexity of PureStrategy Nash Equilibria in Congestion and LocalEffect Games
 In Proc. of the 2nd Int. Workshop on Internet and Network Economics (WINE
, 2006
"... doi 10.1287/moor.1080.0322 ..."
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
A generalized strategy eliminability criterion and computational methods for applying it
 In Proceedings of the National Conference on Artificial Intelligence (AAAI
, 2005
"... We define a generalized strategy eliminability criterion for bimatrix games that considers whether a given strategy is eliminable relative to given dominator & eliminee subsets of the players ’ strategies. We show that this definition spans a spectrum of eliminability criteria from strict dominance ..."
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Cited by 14 (6 self)
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We define a generalized strategy eliminability criterion for bimatrix games that considers whether a given strategy is eliminable relative to given dominator & eliminee subsets of the players ’ strategies. We show that this definition spans a spectrum of eliminability criteria from strict dominance (when the sets are as small as possible) to Nash equilibrium (when the sets are as large as possible). We show that checking whether a strategy is eliminable according to this criterion is coNPcomplete (both when all the sets are as large as possible and when the dominator sets each have size 1). We then give an alternative definition of the eliminability criterion and show that it is equivalent using the Minimax Theorem. We show how this alternative definition can be translated into a mixed integer program of polynomial size with a number of (binary) integer variables equal to the sum of the sizes of the eliminee sets, implying that checking whether a strategy is eliminable according to the criterion can be done in polynomial time, given that the eliminee sets are small. Finally, we study using the criterion for iterated elimination of strategies. Categories and Subject Descriptors
Approximating Nash equilibria using smallsupport strategies
 ACM Conference on Electronic Commerce
, 2007
"... We study the problem of finding approximate Nash equilibria of two player games. We show that for any 0 < ǫ < 1, there is no 1approximate equilibrium with strategies of 1+ǫ log n support O( ǫ2). Categories and Subject Descriptors ..."
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Cited by 14 (0 self)
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We study the problem of finding approximate Nash equilibria of two player games. We show that for any 0 < ǫ < 1, there is no 1approximate equilibrium with strategies of 1+ǫ log n support O( ǫ2). Categories and Subject Descriptors
Learning and Approximating the Optimal Strategy to Commit To ∗
"... Abstract. Computing optimal Stackelberg strategies in general twoplayer Bayesian games (not to be confused with Stackelberg strategies in routing games) is a topic that has recently been gaining attention, due to their application in various security and law enforcement scenarios. Earlier results c ..."
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Cited by 14 (9 self)
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Abstract. Computing optimal Stackelberg strategies in general twoplayer Bayesian games (not to be confused with Stackelberg strategies in routing games) is a topic that has recently been gaining attention, due to their application in various security and law enforcement scenarios. Earlier results consider the computation of optimal Stackelberg strategies, given that all the payoffs and the prior distribution over types are known. We extend these results in two different ways. First, we consider learning optimal Stackelberg strategies. Our results here are mostly positive. Second, we consider computing approximately optimal Stackelberg strategies. Our results here are mostly negative. 1