Efficiency and stability of Nash equilibria in resource allocation games

We study resource allocation games, where users send data along paths and links in the network charge a price equal to marginal cost. When users are price taking, it is known that there exist distributed dynamics that converge towards a fully efficient Nash equilibrium. When users are price anticipating, however, a Nash equilibrium does not maximize total utility in general. In this paper, we explore the inefficiency of Nash equilibria for general networks and semi-convex marginal cost functions. While it is known that for m ≥ 2 users and convex marginal cost functions, no efficiency guarantee is possible, we prove that an additional differentiability assumption on marginal cost functions implies a bounded efficiency loss of 2/(2m + 1). For polynomial marginal cost functions with nonnegative coefficients, we precisely characterize the price of anarchy. We also prove that the efficiency of Nash equilibria significantly improves if all users have the same strategy space and the same utility function. We propose a class of distributed dynamics and prove that whenever a game admits a potential function, these dynamics globally converge to a Nash equilibrium. Finally, we show that in general the only class of marginal cost functions that guarantees the existence of a potential function are affine linear functions.