Approximating pure nash equilibrium in cut, party affiliation, and satisfiability games

Bhalgat, Anand; Chakraborty, Tanmoy; Khanna, Sanjeev

doi:10.1145/1807342.1807353

Cited by 37 publications

(54 citation statements)

References 18 publications

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“…This justifies the idea of resorting to mildly greedy players who can give life to solutions having a more permissive computational complexity. To this aim, Bhalgat et al (2010) give a polynomial time algorithm to compute a (1 + )-approximate pure Nash equilibrium, for any > 2.…”

Section: Our Contributionmentioning

confidence: 99%

“…For sufficiently high values of , the problem of computing a (1 + )-approximate pure Nash equilibrium becomes polynomial time solvable in several games of interest. In particular, there exist polynomial time algorithms for computing such an equilibrium in several special cases of congestion games (Bhalgat et al 2010;Caragiannis et al , 2012Chien and Sinclair 2001).…”

Section: Introductionmentioning

confidence: 99%

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On the performance of mildly greedy players in cut games

Bilò

Paladini

2015

J Comb Optim

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We continue the study of the performance of mildly greedy players in cut games initiated by Christodoulou et al. (Theoret Comput Sci 438:13-27, 2012), where a mildly greedy player is a selfish agent who is willing to deviate from a certain strategy profile only if her payoff improves by a factor of more than 1 + , for some given ≥ 0. Hence, in presence of mildly greedy players, the classical concepts of pure Nash equilibria and best-responses generalize to those of (1 + )-approximate pure Nash equilibria and (1 + )-approximate best-responses, respectively. We first show that the -approximate price of anarchy, that is the price of anarchy of (1 + )-approximate pure Nash equilibria, is at least 1 2+ and that this bound is tight for any ≥ 0. Then, we evaluate the approximation ratio of the solutions achieved after a (1 + )-approximate one-round walk starting from any initial strategy profile, where a (1 + )-approximate one-round walk is a sequence of (1 + )-approximate bestresponses, one for each player. We improve the currently known lower bound on this ratio from min 1 4+2 , 4+2 up to min 1 2+ , 2 (1+ )(2+ ) and show that this is again tight for any ≥ 0. An interesting and quite surprising consequence of our results is that the worst-case performance guarantee of the very simple solutions generated after a (1 + )-approximate one-round walk is the same as that of (1 + )-approximate pure Nash equilibria when ≥ 1 and of that of subgame perfect equilibria (i.e., Nash equilibria for greedy players with farsighted, rather than myopic, rationality) when = 1.

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Section: Our Contributionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the performance of mildly greedy players in cut games

Bilò

Paladini

2015

J Comb Optim

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“…As a result, many recent studies have focused on the complexity of finding Nash equilibria (e.g. [1,4,5,7,11,12,13]). For the complexity problem to be meaningful, however, the game, particularly its payoffs, should allow a compact representation [23].…”

Section: Overviewmentioning

confidence: 99%

“…The parity affiliation game [12] with all edge weights −1 and the cut game [4] with all edge weights +1 correspond to the pairwiseinteraction game with payoff matrix P = 0 1 1 0 . Any pure Nash equilibrium in these games is a STABLE-CONFIGURATION and a MAX-CUT, in the sense of Schäffer and Yannakakis [25].…”

Section: Related Workmentioning

confidence: 99%

Pairwise-Interaction Games

Dyer

Mohanaraj

2011

Automata, Languages and Programming

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Abstract. We study the complexity of computing Nash equilibria in games where players arranged as the vertices of a graph play a symmetric 2-player game against their neighbours. We call this a pairwiseinteraction game. We analyse this game for n players with a fixed number of actions and show that (1) a mixed Nash equilibrium can be computed in constant time for any game, (2) a pure Nash equilibrium can be computed through Nash dynamics in polynomial time for games with a symmetrisable payoff matrix, (3) determining whether a pure Nash equilibrium exists for zero-sum games is NP-complete, and (4) counting pure Nash equilibria is #P-complete even for 2-strategy games. In proving (3), we define a new defective graph colouring problem called Nash colouring, which is of independent interest, and prove that its decision version is NP-complete. Finally, we show that pairwise-interaction games form a proper subclass of the usual graphical games.

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“…For symmetric unweighted congestion games, Chien and Sinclair [2011] prove that the (1 + )-improvement dynamics converges to a (1 + )-approximate equilibrium after a polynomial number of steps; this result holds under mild assumptions on the latency functions and the participation of the players in the dynamics. Efficient algorithms for approximate equilibria have been recently obtained for other classes of games such as constraint satisfaction [Bhalgat et al 2010;Nguyen and Tardos 2009], anonymous games [Daskalakis and Papadimitriou 2007], network formation [Anshelevish and Caskurlu 2009], and facility location games [Cardinal and Hoefer 2010].…”

Section: Introductionmentioning

confidence: 99%

Approximate Pure Nash Equilibria in Weighted Congestion Games

CaragiannisIoannis

Fanelli

GravinNick

et al. 2015

ACM Trans. Econ. Comput.

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We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by a potential function argument. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria in such games is a PLS-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. A ρ-approximate pure Nash equilibrium is a state of a (weighted congestion) game from which no player has any incentive to deviate in order to improve her cost by a multiplicative factor higher than ρ. Do such equilibria exist for small values of ρ? And if so, can we compute them efficiently?We provide positive answers to both questions for weighted congestion games with polynomial latency functions by exploiting an "approximation" of such games by a new class of potential games that we call -games. This allows us to show that these games have d!-approximate pure Nash equilibria, where d is the maximum degree of the latency functions. Our main technical contribution is an efficient algorithm for computing O(1)-approximate pure Nash equilibria when d is a constant. For games with linear latency functions, the approximation guarantee is 3+The running time is polynomial in the number of bits in the representation of the game and 1/γ . As a byproduct of our techniques, we also show the following interesting structural statement for weighted congestion games with polynomial latency functions of maximum degree d ≥ 2: polynomially-long sequences of best-response moves from any initial state to a d O(d 2 ) -approximate pure Nash equilibrium exist and can be efficiently identified in such games as long as d is a constant.To the best of our knowledge, these are the first positive algorithmic results for approximate pure Nash equilibria in weighted congestion games. Our techniques significantly extend our recent work on unweighted congestion games through the use of -games. The concept of approximating nonpotential games by potential ones is interesting in itself and might have further applications.

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Approximating pure nash equilibrium in cut, party affiliation, and satisfiability games

Cited by 37 publications

References 18 publications

On the performance of mildly greedy players in cut games

On the performance of mildly greedy players in cut games

Pairwise-Interaction Games

Approximate Pure Nash Equilibria in Weighted Congestion Games

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