Equilibria in Multiclass and Multidimensional Atomic Congestion Games

This paper studies the existence of pure Nash equilibria in atomic congestion games with different user classes where the cost of each resource depends on the aggregated demand of each class. A set of cost functions is called consistent for this class if all games with cost functions from the set have a pure Nash equilibrium. We give a complete characterization of consistent sets of cost functions showing that the only consistent sets of cost functions are sets of certain affine functions and sets of certain exponential functions. This characterization is also extended to a larger class of games where each atomic player may control flow that belongs to different classes.