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264
Decentralized charging control for large populations of plugin electric vehicles,” Submitted to
 IEEE Transactions on Control Systems Technology
"... Abstract—The paper develops a novel decentralized charging control strategy for large populations of plugin electric vehicles (PEVs). We consider the situation where PEV agents are rational and weakly coupled via their operation costs. At an established Nash equilibrium, each of the PEV agents reac ..."
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Cited by 28 (1 self)
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Abstract—The paper develops a novel decentralized charging control strategy for large populations of plugin electric vehicles (PEVs). We consider the situation where PEV agents are rational and weakly coupled via their operation costs. At an established Nash equilibrium, each of the PEV agents reacts optimally with respect to the average charging strategy of all the PEV agents. Each of the average charging strategies can be approximated by an infinite population limit which is the solution of a fixed point problem. The control objective is to minimize electricity generation costs by establishing a PEV charging schedule that fills the overnight demand valley. The paper shows that under certain mild conditions, there exists a unique Nash equilibrium that almost satisfies that goal. Moreover, the paper establishes a sufficient condition under which the system converges to the unique Nash equilibrium. The theoretical results are illustrated through various numerical examples.
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 24 (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).
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 20 (3 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
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.
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 17 (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
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 17 (4 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.
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 ..."
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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 domina ..."
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Cited by 16 (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
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