On the existence of pure nash equilibria inweighted congestion games

• Authors: Tobias Harks;Max Klimm
• Venue: ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
• Year: 2010

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. We say that C is FIP-consistent if every weighted congestion game with cost functions in C has the Finite Improvement Property. Our main results are structural characterizations of consistency for twice continuously differentiable cost functions. More specifically, we show that C 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 = ac2 + b. For games with at least 3 players we show that C is consistent if and only if exactly one of the following cases hold: (i) C contains only affine functions; (ii) C contains only exponential functions such that c(l) = aceφl + bc for some ac, bc, φ ∈ R, where ac and bc may depend on c, while φ must be equal for every c ∈ C. This characterization is even valid for 3-player games, thus, closing the gap to 2-player games considered above. Finally, we derive various results regarding consistency and FIP-consistency for weighted network congestion games.