On the Existence of Pure Nash Equilibria in Weighted Congestion Games

We study the existence of pure Nash equilibria in weighted congestion games. Let C denote a set of cost functions. We say that C is consistent if every weighted congestion game with cost functions in C possesses a pure Nash equilibrium. Our main contribution is a complete characterization of consistency of cost functions. Specifically, we prove that a nonempty set C of twice continuously differentiable functions is consistent for two-player games if and only if C contains only monotonic functions and for all c1, c2 ∈ C, there are constants a, b ∈ R such that c1(x) = a c2(x) + b for all x ∈ R≥0. For games with at least 3 players, we prove that C is consistent if and only if exactly one of the following cases hold: (a) C contains only affine functions; (b) C contains only exponential functions such that c(x) = ac e x +bc for some ac, bc, φ ∈ R, where ac and bc may depend on c, while φ must be equal for every c ∈ C. The latter characterization is even valid for 3-player games, thus, closing the gap to 2-player games considered above. Finally, we derive several characterizations of consistency of cost functions for games with restricted strategy spaces, such as games with singleton strategies or weighted network congestion games.