Pure Nash Equilibria in Weighted Congestion Games with Complementarities and Beyond
Kenjiro Takazawa
TL;DR
The paper addresses the problem of guaranteeing pure Nash equilibria in a broad class of generalized congestion games that combine weights, complementarities, and mixed/set-functional costs. It introduces a common model with matroid-based strategy spaces, set-functional resource costs, and a weakly monotone aggregation function, proving that pure Nash equilibria exist under simplified monotonicity conditions. The main technical contribution is a unified existence result (and its corollaries) that extends prior results for weighted matroid bottleneck and L^p aggregation, and extends to player-specific and weighted matroid games with mixed costs. This work broadens the set of congestion-game variants with guaranteed equilibria, enabling further algorithmic analysis and applications in network design and resource allocation.
Abstract
Congestion games offer a primary model in the study of pure Nash equilibria in non-cooperative games, and a number of generalized models have been proposed in the literature. One line of generalization includes weighted congestion games, in which the cost of a resource is a function of the total weight of the players choosing that resource. Another line includes congestion games with mixed costs, in which the cost imposed on a player is a convex combination of the total cost and the maximum cost of the resources in her strategy. This model is further generalized to that of congestion games with complementarities. For the above models, the existence of a pure Nash equilibrium is proved under some assumptions, including that the strategy space of each player is the base family of a matroid and that the cost functions have a certain kind of monotonicity. In this paper, we deal with common generalizations of these two lines, namely weighted matroid congestion games with complementarities, and its further generalization. Our main technical contribution is a proof of the existence of pure Nash equilibria in these generalized models under a simplified assumption on the monotonicity, which provide a common extension of the previous results. We also present some extensions on the existence of pure Nash equilibria in player-specific and weighted matroid congestion games with mixed costs.