Abstract
We introduce multidimensional congestion games, that is, congestion games whose set of players is partitioned into d+1 clusters C0,C1,…,Cd. Players in C0 have full information about all the other participants in the game, while players in Ci, for any 1≤i≤d, have full information only about the members of C0∪Ci and are unaware of all the others. This model has at least two interesting applications: (i) it is a special case of graphical congestion games induced by an undirected social knowledge graph with independence number equal to d, and (ii) it represents scenarios in which players have a type and the level of competition they experience on a resource depends on their type and on the types of the other players using it. We focus on the case in which the cost function associated with each resource is affine and bound the price of anarchy and stability as a function of d with respect to two meaningful social cost functions and for both weighted and unweighted players. We also provide refined bounds for the special case of d=2 in presence of unweighted players.
Highlights
Congestion games [1,2,3,4] are, perhaps, the most famous class of non-cooperative games due to their capability to model several interesting competitive scenarios, while maintaining nice properties.In these games, there is a set of players sharing a set of resources
In order to bound the price of stability with respect to the social cost function Pres, we consider a pure Nash equilibrium that minimizes the potential function Φ defined in (1), which leads to the following upper bound
They can be reinterpreted as a particular subclass of that of graphical congestion games defined by an undirected social knowledge graph whose independence number is equal to d
Summary
Congestion games [1,2,3,4] are, perhaps, the most famous class of non-cooperative games due to their capability to model several interesting competitive scenarios, while maintaining nice properties. This existence result makes congestion games appealing especially in all those applications in which pure Nash equilibria are elected as the ideal solution concept In these contexts, the study of inefficiency due to selfish and non-cooperative behavior has affirmed as a fervent research direction. In order to deal with significative bounds on the prices of anarchy and stability, some kind of regularity needs to be imposed on the cost functions associated with the resources To this aim, lot of research attention has been devoted to the case of polynomial cost functions [8,9,10,11,12,13,14,15,16,17], and more general latency functions verifying some mild assumptions [10,18,19]. As shown in References [20,21,22], they represent the only case, together with that (perhaps not meaningful) of exponential cost functions, for which weighted congestion games [20], that is the generalization of congestion games in which each player has a weight and the congestion of a resource becomes the sum of the weights of its users, still admit a potential function
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