Finding Social Optima in Congestion Games with Positive Externalities

Abstract

We consider a variant of congestion games where every player i expresses for each resource e and player j a positive externality, i.e., a value for being on e together with player j. Rather than adopting a game-theoretic perspective, we take an optimization point of view and consider the problem of optimizing the social welfare.

We show that this problem is NP-hard even for very special cases, notably also for the case where the players’ utility functions for each resource are affine (contrasting with the tractable case of linear functions [3]). We derive a 2-approximation algorithm by rounding an optimal solution of a natural LP formulation of the problem. Our rounding procedure is sophisticated because it needs to take care of the dependencies between the players resulting from the pairwise externalities. We also show that this is essentially best possible by showing that the integrality gap of the LP is close to 2.

Small adaptations of our rounding approach enable us to derive approximation algorithms for several generalizations of the problem. Most notably, we obtain an (r + 1)-approximation when every player may express for each resource externalities on player sets of size r. Further, we derive a 2-approximation when the strategy sets of the players are restricted and a $\frac32$-approximation when these sets are of size 2.