Improving approximate pure Nash equilibria in congestion games

TL;DR

This paper improves the approximation factor for pure Nash equilibria in congestion games by modifying existing algorithms, extending to non-decreasing cost functions, and analyzing universal taxes using linear programming.

Contribution

It introduces a simple modification to existing algorithms that achieves better approximation factors and extends these results to more general cost functions and taxation methods.

Findings

Approximation factor improved to (1.61+ε) for linear cost functions.

Extension of algorithms to arbitrary non-decreasing resource cost functions.

Optimal computation of load-dependent taxes and tight bounds on Price of Anarchy.

Abstract

Congestion games constitute an important class of games to model resource allocation by different users. As computing an exact or even an approximate pure Nash equilibrium is in general PLS-complete, Caragiannis et al. (2011) present a polynomial-time algorithm that computes a ($2 + \epsilon$)-approximate pure Nash equilibria for games with linear cost functions and further results for polynomial cost functions. We show that this factor can be improved to $(1.61+\epsilon)$ and further improved results for polynomial cost functions, by a seemingly simple modification to their algorithm by allowing for the cost functions used during the best response dynamics be different from the overall objective function. Interestingly, our modification to the algorithm also extends to efficiently computing improved approximate pure Nash equilibria in games with arbitrary non-decreasing resource cost functions. Additionally, our analysis exhibits an interesting method to optimally compute universal load dependent taxes and using linear programming duality prove tight bounds on PoA under universal taxation, e.g, 2.012 for linear congestion games and further results for polynomial cost functions. Although our approach yield weaker results than that in Bil\`{o} and Vinci (2016), we remark that our cost functions are locally computable and in contrast to Bil\`{o} and Vinci (2016) are independent of the actual instance of the game.