  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.
