The equilibrium existence of a robust routing game under interval uncertainty

Abstract

We study an atomic routing game in a network with interval uncertainty, where the cost of each edge is load-dependent, and can be any value in a given interval whose lower and upper limits are expressed as functions of the edge load. Each player would select a path from his source to his terminal in the network that minimizes his maximum regret, where given a path (strategy) profile of the game, the maximum regret of a player is the worst-case difference between his total cost (in the case of sum-type cost) or her bottleneck cost (in the case of bottleneck-type cost) and the optimum he could attain given the choices of other players in the profile and a priori knowledge about the actual realization of edge costs. A NE of this game, termed as robust Nash equilibrium (RNE), is a path profile under which no player can reduce his maximum regret by unilateral deviation. On the negative side, we show that the problem of deciding whether a given 3-player game with the sum-type costs has an RNE is NP-hard, even if the game is symmetric and all intervals have unit length. On the positive side, we characterize network topologies, for the game with either type of costs, that guarantee the RNE existence regardless of source-terminal locations and interval settings.