Sensitivity of Equilibrium in Atomic Congestion Games with Player-specific Cost Functions

Consider an atomic splittable congestion game played on a parallel-link graph with player-specific cost functions. Richman and Shimkin proved in 2007 that the equilibrium is unique when the cost functions are continuous, increasing, and strictly convex. It allows to define a function ~e(·) mapping any demand vector to the unique corresponding equilibrium. The general question we address in this paper is about the behavior of ~e(·): how does the equilibrium change when the demand vector changes? By standard arguments regarding the solutions of variational inequalities, we prove that ~e(·) is continuous. Our main results concern the case when there are only two players or only two arcs. We show that if the cost functions are twice continuously differentiable, ~e(·) is differentiable at any point on a neighborhood of which the supports of the player strategies remain constant. We are able to describe precisely what happens to the support of the strategies at equilibrium when a player transfers a part of his demand to another player with more demand. It allows to recover previous results for this kind of game regarding the impact of coalitions on the equilibrium and to discuss the impact of such transfers on the social cost. We show moreover that most of these results do not hold when there are at least three players and three arcs.