Congestion Games and Coordination Mechanisms

Abstract

In recent years we witness a fusion of ideas coming from the fields of Game Theory, Networks, and Algorithms. One of the central ideas in this new area is the notion of the price of anarchy [5,7] which is an attempt to measure the deterioration in performance of a system due to selfish behavior of its users or components. The study of the price of anarchy in general games is pointless; it can be arbitrary even for simple 2× 2 games. But it becomes interesting when we restrict our attention to congestion games and to their natural generalizations and variants.

Ideally when users share the resources of a system they behave in a way that optimizes the performance of the system. However, when the users are selfish, they will act in a way that optimizes their own individual and usually conflicting objectives. As an example, consider a set of users that compete for the links of a network. The situation results in a network congestion game [8]: In a congestion game there are users/players and facilities (links in the case of networks). To each user we associate some collections of facilities (paths from a source to a destination in the case of networks); these are the pure strategies of the user. The cost of each facility depends on the number of users that use it; the cost of a user is the sum of the cost of the facilities in its selected strategy. One nice property of congestion games is the not-so-trivial fact that they all have at least one pure Nash equilibrium. Special case of congestion games of particular interest are the singleton games when each strategy is a singleton set (each user uses only one facility) and when costs are linear (the cost of each facility is proportional to the number of users using it). This is essentially the selfish version of the Graham’s classical scheduling problem of identical tasks. Also of particular interest are symmetric network congestion games with a continuum of users; these are the games studied by Roughgarden and Tardos [9].

Given a set of strategies of a congestion game, we can define the social cost (or system cost) as either the maximum or the average cost among the users. Then the (pure) price of anarchy of a congestion game is the maximum ratio among all (pure) Nash equilibria of the social cost of a Nash equilibrium over the social cost of the optimal set of strategies. What is the pure and mixed price of anarchy of congestion games? of singleton and linear games? We know the answers to most of these questions but not to all of them [5,3,4].

There are two natural generalizations of congestion games: when users have weights—in this case, the cost of each facility depends on the total weight of the users using it—and when facilities differentiate between users. Of particular interest is the class of the singleton games with weights, which corresponds essentially to selfish scheduling. Again we know the price of anarchy of some of these cases but many cases are still open.

Given that in many cases the price of anarchy is high, one can ask how to redesign a game to improve its price of anarchy. There are interesting suggestions such as adding taxes [1] or more choices to players [6]. A natural way to address the problem of redesigning a congestion game is the notion of coordination mechanisms: Given a congestion game, consider the class of congestion games when we are allowed to increase the cost on the facilities and when we can differentiate among players (with player-specific costs). The idea is that in many practical situations, increasing the cost of a facility corresponds to “slowing down” which can be implemented easily; similarly, differentiating between players can be naturally done in many cases. A coordination mechanism then simply selects among these games one that has small price of anarchy.

The situation becomes more interesting when we consider congestion games with weights. In this case, we can naturally assume that when we redesign a game we don’t know the weights of the users. For example, in the special case of selfish scheduling we want to have a scheduling policy on each facility which is fixed in advance (i.e., independent of the weights of the jobs/users). The situation resembles very much the framework of competitive analysis: We, the designers, select a scheduling policy for each facility. Then an adversary selects the weights of jobs and we compute the price of anarchy.

The immediate problem is to estimate or bound the optimal price of anarchy achieved by coordination mechanisms. For most classes of congestion games, their special cases, and their variants, the problem is open. We know only partial answers to special cases, such as the case of singleton games [2].

Supported in part by the IST (FLAGS, IST-2001-33116) program.