Journal of Computer and System Sciences - 26th IEEE Conference on Foundations of Computer Science, October 21-23, 1985
Simple local search problems that are hard to solve
SIAM Journal on Computing
Integer Linear Programs and Local Search for Max-Cut
SIAM Journal on Computing
Algorithms, games, and the internet
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Journal of the ACM (JACM)
Market sharing games applied to content distribution in ad-hoc networks
Proceedings of the 5th ACM international symposium on Mobile ad hoc networking and computing
The complexity of pure Nash equilibria
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Machine Learning
Selfish load balancing and atomic congestion games
Proceedings of the sixteenth annual ACM symposium on Parallelism in algorithms and architectures
Maximizing Quadratic Programs: Extending Grothendieck's Inequality
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
The Price of Routing Unsplittable Flow
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
The price of anarchy of finite congestion games
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Sink Equilibria and Convergence
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Tight approximation algorithms for maximum general assignment problems
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Convergence to approximate Nash equilibria in congestion games
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Inapproximability of pure nash equilibria
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Fast convergence to nearly optimal solutions in potential games
Proceedings of the 9th ACM conference on Electronic commerce
The Speed of Convergence in Congestion Games under Best-Response Dynamics
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Performances of One-Round Walks in Linear Congestion Games
SAGT '09 Proceedings of the 2nd International Symposium on Algorithmic Game Theory
Convergence time to Nash equilibria
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Convergence and approximation in potential games
STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
On the price of anarchy and stability of correlated equilibria of linear congestion games,,
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Uncoordinated load balancing and congestion games in p2p systems
IPTPS'04 Proceedings of the Third international conference on Peer-to-Peer Systems
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We study the speed of convergence to approximately optimal states in two classes of potential games. We provide bounds in terms of the number of rounds, where a round consists of a sequence of movements, with each player appearing at least once in each round. We model the sequential interaction between players by a best-response walk in the state graph, where every transition in the walk corresponds to a best response of a player. Our goal is to bound the social value of the states at the end of such walks. In this paper, we focus on two classes of potential games: selfish routing games, and cut games (or party affiliation games (Fabrikant et al. 2004 [12])). Other than bounding the price of anarchy of selfish routing games (Roughgarden and Tardos, 2002 [25], Awerbuch et al. 2005 [2], Christodoulou and Koutsoupias, 2005 [9]), there are many interesting problems about game dynamics in these games. It is known that exponentially long best-response walks may exist to pure Nash equilibria (Fabrikant et al. 2004 [12]), and random best-response walks converge to solutions with good approximation guarantees after polynomially many best responses (Goemans et al. 2005 [17]). In this paper, we study the speed of convergence on deterministic best-response walks in these games and prove that starting from an arbitrary configuration, after one round of best responses of players, the resulting configuration is a @Q(n)-approximate solution. Furthermore, we show that starting from an empty configuration, the solution after any round of best responses is a constant-factor approximation. We also provide a lower bound for the multi-round case, where we show that for any constant number of rounds t, the approximation guarantee cannot be better than n^@e^(^t^), for some @e(t)0. We also study cut games, that provide an illustrative example of potential games. The convergence of potential games to locally optimum solutions has been studied in the context of local search algorithms (Johnson et al. 1988 [19], Schaffer and Yannakakis, 1991 [28]). In these games, we consider two social functions: the cut (defined as the weight of the edges in the cut), and the total happiness (defined as the weight of the edges in the cut, minus the weight of the remaining edges). For the cut social function, we prove that the expected social value after one round of a random best-response walk is at least a constant factor approximation to the optimal social value. We also exhibit exponentially long best-response walks with poor social value. For the unweighted version of this cut game, we prove @W(n) and O(n) lower and upper bounds on the number of rounds of best responses to converge to a constant-factor cut. In addition, we suggest a way to modify the game to enforce a fast convergence in any fair best-response walk. For the total happiness social function, we show that for unweighted graphs of sufficiently large girth, starting from a random configuration, greedy behavior of players in a random order converges to an approximate solution after one round.