Nash equilibria in discrete routing games with convex latency functions

In a discrete routing game, each of n selfish users employs a mixed strategy to ship her (unsplittable) traffic over m parallel links. The (expected) latency on a link is determined by an arbitrary non-decreasing, non-constant and convex latency function ϕ. In a Nash equilibrium, each user alone is minimizing her (Expected) Individual Cost, which is the (expected) latency on the link she chooses. To evaluate Nash equilibria, we formulate Social Cost as the sum of the users' (Expected) Individual Costs. The Price of Anarchy is the worst-case ratio of Social Cost for a Nash equilibrium over the least possible Social Cost. A Nash equilibrium is pure if each user deterministically chooses a single link; a Nash equilibrium is fully mixed if each user chooses each link with non-zero probability. We obtain:For the case of identical users, the Social Cost of any Nash equilibrium is no more than the Social Cost of the fully mixed Nash equilibrium, which may exist only uniquely. Moreover, instances admitting a fully mixed Nash equilibrium enjoy an efficient characterization.For the case of identical users, we derive two upper bounds on the Price of Anarchy: For the case of identical links with a monomial latency function ϕ(x)=xd, the Price of Anarchy is the Bell number of order d+1. For pure Nash equilibria, a generic upper bound from the Wardrop model can be transfered to discrete routing games. For polynomial latency functions with non-negative coefficients and degree d, this yields an upper bound of d+1.For the case of identical users, a pure Nash equilibrium (and thereby an optimum pure assignment) can be computed in time O(mlogmlogn). For the general case, computing the best or the worst pure Nash equilibrium is NP-complete, even for identical links with an identity latency function.