On the Convergence Time of the Best Response Dynamics in Player-specific Congestion Games

📝 Abstract

We study the convergence time of the best response dynamics in player-specific singleton congestion games. It is well known that this dynamics can cycle, although from every state a short sequence of best responses to a Nash equilibrium exists. Thus, the random best response dynamics, which selects the next player to play a best response uniformly at random, terminates in a Nash equilibrium with probability one. In this paper, we are interested in the expected number of best responses until the random best response dynamics terminates. As a first step towards this goal, we consider games in which each player can choose between only two resources. These games have a natural representation as (multi-)graphs by identifying nodes with resources and edges with players. For the class of games that can be represented as trees, we show that the best-response dynamics cannot cycle and that it terminates after O(n^2) steps where n denotes the number of resources. For the class of games represented as cycles, we show that the best response dynamics can cycle. However, we also show that the random best response dynamics terminates after O(n^2) steps in expectation. Additionally, we conjecture that in general player-specific singleton congestion games there exists no polynomial upper bound on the expected number of steps until the random best response dynamics terminates. We support our conjecture by presenting a family of games for which simulations indicate a super-polynomial convergence time.

💡 Analysis

We study the convergence time of the best response dynamics in player-specific singleton congestion games. It is well known that this dynamics can cycle, although from every state a short sequence of best responses to a Nash equilibrium exists. Thus, the random best response dynamics, which selects the next player to play a best response uniformly at random, terminates in a Nash equilibrium with probability one. In this paper, we are interested in the expected number of best responses until the random best response dynamics terminates. As a first step towards this goal, we consider games in which each player can choose between only two resources. These games have a natural representation as (multi-)graphs by identifying nodes with resources and edges with players. For the class of games that can be represented as trees, we show that the best-response dynamics cannot cycle and that it terminates after O(n^2) steps where n denotes the number of resources. For the class of games represented as cycles, we show that the best response dynamics can cycle. However, we also show that the random best response dynamics terminates after O(n^2) steps in expectation. Additionally, we conjecture that in general player-specific singleton congestion games there exists no polynomial upper bound on the expected number of steps until the random best response dynamics terminates. We support our conjecture by presenting a family of games for which simulations indicate a super-polynomial convergence time.

📄 Content

arXiv:0805.1130v1 [cs.GT] 8 May 2008 On the Convergence Time of the Best Response Dynamics in Player-specific Congestion Games∗ Heiner Ackermann Heiko R¨oglin Department of Computer Science RWTH Aachen, D-52056 Aachen, Germany {ackermann, roeglin}@cs.rwth-aachen.de November 9, 2018 Abstract We study the convergence time of the best response dynamics in player-specific singleton congestion games. It is well known that this dynamics can cycle, although from every state a short sequence of best responses to a Nash equilibrium exists. Thus, the random best response dynamics, which selects the next player to play a best response uniformly at random, terminates in a Nash equilibrium with probability one. In this paper, we are interested in the expected number of best responses until the random best response dynamics terminates. As a first step towards this goal, we consider games in which each player can choose between only two resources. These games have a natural representation as (multi-)graphs by identifying nodes with resources and edges with players. For the class of games that can be represented as trees, we show that the best-response dynamics cannot cycle and that it terminates after O(n2) steps where n denotes the number of resources. For the class of games represented as cycles, we show that the best response dynamics can cycle. However, we also show that the random best response dynamics terminates after O(n2) steps in expectation. Additionally, we conjecture that in general player-specific singleton congestion games there exists no polynomial upper bound on the expected number of steps until the random best response dynamics terminates. We support our conjecture by presenting a family of games for which simulations indicate a super-polynomial convergence time. ∗Parts of the results presented here already appeared in the Proceedings of the 4th Symposium on Stochastic Algorithms, Foundations, and Applications (SAGA) in 2007 [1]. 1 Introduction We study the convergence time of the best response dynamics to pure Nash equilibria1 in player- specific singleton congestion games. In such games, we are given a set of resources and a set of players. Each player is equipped with a set of non-decreasing, player-specific delay functions which measure the delay the player experiences from allocating a particular resource and sharing it with a certain number of other players. A player’s goal is to allocate a single resource with minimum delay given fixed choices of the other players. Milchtaich [12], who was the first to consider player- specific singleton congestion games, proves that every such game possesses a Nash equilibrium which can be computed efficiently. However, he also observes that these games are no potential games [14], that is, the best response dynamics, in which players consecutively change to resources with minimum delay, can cycle. This is in contrast to congestion games with common delay functions in which all players sharing a resource observe the same delay. In the following, we refer to congestion games with common delay functions as standard congestion games. Rosenthal [15], who introduces standard congestion games, proves that they always admit a potential function guaranteeing the existence of Nash equilibria and that the best response dynamics cannot cycle. Ieong et al. [9] consider the convergence time of the best response dynamics to Nash equilibria in standard singleton congestion games. They observe that the delay values can be replaced by their ranks in the sorted list of theses values without affecting the best responses dynamics. By applying Rosenthal’s potential functions to these new delay functions they observe that after at most n2m best responses a Nash equilibrium is reached, where n equals the number of players and m the number of resources. This result is independent of any assumption on the ordering according to which players change their strategies. Since the best response dynamics in player-specific singleton congestion games can cycle, we propose to study random best response dynamics in such games. This approach is motivated by the following observation due to Milchtaich [12]: From every state of a player-specific singleton congestion game there exists a polynomially long sequence of best responses leading to a Nash equilibrium. Thus, the random best response dynamics selecting the next player to play a best response at random terminates with probability one after a finite number of steps. Milchtaich’s analysis leaves open the question how long it takes until the random best response dynamics terminates in expectation. In this paper, we address this question as we think that it is a natural and interesting one. Currently, we are not able to analyze the convergence time in arbitrary player-specific singleton congestion games. However, our experimental results support the following conjecture. Conjecture 1. There exist player-specific singleton congestion games in which the expected n

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