Congestion Games with Player-Specific Costs Revisited

We study the existence of pure Nash equilibria in congestion games with player-specific costs. Specifically, we provide a thorough characterization of the maximal sets of cost functions that guarantee the existence of a pure Nash equilibrium. For the case that the players are unweighted, we show that it is necessary and sufficient that for every resource and for every pair of players the corresponding cost functions are affine transformations of each other. For weighted players, we show that in addition one needs to require that all cost functions are affine or all cost functions are exponential. Finally, we construct a four-player singleton weighted congestion game where the cost functions are identical among the resources and differ only by an additive constant among the players and show that it does not have a pure Nash equilibrium. This answers an open question by Mavronicolas et al. [15] who showed that such games with at most three players always have a pure Nash equilibrium.