Conflicting congestion effects in resource allocation games

We consider resource allocation games with heterogeneous users and identical resources. Most of the previous work considered cost structures with either negative or positive congestion effects. We study a cost structure that encompasses both the resource's load and the job's share in the resource's activation cost. We consider the proportional sharing rule, where the resource's activation cost is shared among its users proportionally to their lengths. We also challenge the assumption regarding the existence of a fixed set of resources, and consider settings with an unlimited supply of resources. We provide results with respect to equilibrium existence, computation, convergence and quality. We show that if the resource's activation cost is shared equally among its users, a pure Nash equilibrium (NE) might not exist. In contrast, under the proportional sharing rule, a pure NE always exists, and can be computed in polynomial time. Yet, starting at an arbitrary profile of actions, best-response dynamics might not converge to a NE. Finally, we prove that the price of anarchy is unbounded and the price of stability is between 18/17 and 5/4.