On the impact of strategy and utility structures on congestion-averse games

Abstract

Recent results regarding games with congestion-averse utilities (or, congestion-averse games-CAGs) have shown they possess some very desirable properties. Specifically, they have pure strategy Nash equilibria, which may be found in polynomial time. However, these results were accompanied by a very limiting assumption that each player is capable of using any subset of its available set of resources. This is often unrealistic-for example, resources may have complementarities between them such that a minimal number of resources is required for any to be useful. To remove this restriction, in this paper we prove the existence and tractability of a pure strategy equilibrium for a much more general setting where each player is given a matroid over the set of resources, along with the bounds on the size of a subset of resources to be selected, and its strategy space consists of all elements of this matroid that fit in the given size range. Moreover, we show that if a player strategy space in a given CAG does not satisfy these matroid properties, then a pure strategy equilibrium need not exist, and in fact the determination of whether or not a game has such an equilibrium is NP-complete. We further prove analogous results for each of the congestion-averse conditions on utility functions, thus showing that current assumptions on strategy and utility structures in this model cannot be relaxed anymore. © 2009 Springer-Verlag Berlin Heidelberg.