Approximate Pure Nash Equilibria in Weighted Congestion Games

Abstract

We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by a potential function argument. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria in such games is a PLS-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. A ρ-approximate pure Nash equilibrium is a state of a (weighted congestion) game from which no player has any incentive to deviate in order to improve her cost by a multiplicative factor higher than ρ. Do such equilibria exist for small values of ρ? And if so, can we compute them efficiently?We provide positive answers to both questions for weighted congestion games with polynomial latency functions by exploiting an "approximation" of such games by a new class of potential games that we call -games. This allows us to show that these games have d!-approximate pure Nash equilibria, where d is the maximum degree of the latency functions. Our main technical contribution is an efficient algorithm for computing O(1)-approximate pure Nash equilibria when d is a constant. For games with linear latency functions, the approximation guarantee is 3+The running time is polynomial in the number of bits in the representation of the game and 1/γ . As a byproduct of our techniques, we also show the following interesting structural statement for weighted congestion games with polynomial latency functions of maximum degree d ≥ 2: polynomially-long sequences of best-response moves from any initial state to a d O(d 2 ) -approximate pure Nash equilibrium exist and can be efficiently identified in such games as long as d is a constant.To the best of our knowledge, these are the first positive algorithmic results for approximate pure Nash equilibria in weighted congestion games. Our techniques significantly extend our recent work on unweighted congestion games through the use of -games. The concept of approximating nonpotential games by potential ones is interesting in itself and might have further applications.

Computing Approximate Pure Nash Equilibria in Shapley Value Weighted Congestion Games

“…We can bound the potential value of an arbitrary q-approximate equilibrium with the minimal potential value (using the stretch). Compared to the approach in [7], we directly work with the exact potential function of the game which significantly improves the results, but also requires a more involved analysis. We show that the potential of the sub game in one phase is significantly smaller than b r .…”

Congestion Games with Complementarities

“…In contrast to that, Skopalik and Vöcking [33] show that it is in general even PLS-hard to compute approximate pure Nash equilibria for any polynomially computable approximation factor. However, if the cost functions are restricted to linear or constant degree polynomials, approximate pure Nash equilibria can be computed in polynomial time as shown by Caragiannis et al [7], even for weighted games [9] and some other variants [8,15]. Hansknecht et al [19] use the concept of approximate potential functions to examine approximate pure Nash equilibria in weighted congestion games under different restrictions on the cost functions.…”

Congestion games with mixed objectives

“…Chien and Sinclair [7] study the convergence towards approximate pure Nash equilibria in symmetric congestion games. Skopalik and Vöcking [21] show inapproximability in asymmetric congestion games, which is complemented by approximation algorithms for linear and polynomial delay functions [5,10], even for weighted games [6]. Hansknecht et al [13] use the concept of approximate potential functions to examine of approximate pure Nash equilibria in weighted congestion games under different restrictions on the cost functions.…”