Congestion games are a well-studied model for resource sharing among uncoordinated selfish players. Usually, one assumes that the resources in a congestion game do not have any preferences regarding the players that can access them. In typical load-balancing applications, however, different jobs can have different priorities, and jobs with higher priorities get, for example, larger shares of processor time. We extend the classical notion of congestion game and introduce a model in which each resource can assign priorities to the players, and players with higher priorities can displace players with lower priorities. Not only does our model extend classical congestion games, it can also be seen as a model of two-sided markets with ties. Hence it unifies previous results for these two classical models.

We prove that singleton congestion games with priorities are potential games. Furthermore, we show that every player-specific singleton congestion game with priorities possesses a pure Nash equilibrium that can be found in polynomial time. Finally, we extend our results to matroid congestion games, in which the strategy spaces of the players are matroids over the resources.