Strong equilibrium in network congestion games: increasing versus decreasing costs

A network congestion game is played on a directed, two-terminal network. Every player chooses a route from his origin to his destination. The cost of a route is the sum of the costs of the arcs on it. The arc cost is a function of the number of players who use it. Rosenthal proved that such a game always has a Nash equilibrium in pure strategies. Here we pursue a systematic study of the classes of networks for which a strong equilibrium is guaranteed to exist, under two opposite monotonicity assumptions on the arc cost functions. Our main results are: (a) If costs are increasing, strong equilibrium is guaranteed on extension-parallel networks, regardless of whether the players’ origins and destinations are the same or may differ. (b) If costs are decreasing, and the players have the same origin but possibly different destinations, strong equilibrium is guaranteed on series-parallel networks. (c) If costs are decreasing, and both origins and destinations may differ, strong equilibrium is guaranteed on multiextension-parallel networks. In each case, the network condition is not only sufficient but also necessary in order to guarantee strong equilibrium. These results extend and improve earlier ones by Holzman and Law-Yone in the increasing case, and by Epstein et al. in the decreasing case.