In cost-sharing games with delays, a set of agents jointly uses a subset of resources. Each resource has a fixed cost that has to be shared by the players, and each agent has a nonshareable player-specific delay for each resource. A separable cost-sharing protocol determines cost shares that are budget-balanced, separable, and guarantee existence of pure Nash equilibria (PNE). We provide black-box reductions reducing the design of such a protocol to the design of an approximation algorithm for the underlying cost-minimization problem. In this way, we obtain separable cost-sharing protocols in matroid games, single-source connection games, and connection games on n-series-parallel graphs. All these reductions are efficiently computable - given an initial allocation profile, we obtain a cheaper profile and separable cost shares turning the profile into a PNE. Hence, in these domains, any approximation algorithm yields a separable cost-sharing protocol with price of stability bounded by the approximation factor.

We study a multi-leader single-follower congestion game where multiple users (leaders) choose one resource out of a set of resources and, after observing the realized loads, an adversary (single-follower) attacks the resources with maximum loads causing additional costs for the leaders. For the resulting strategic game among the leaders, we show that pure Nash equilibria fail to exist and therefore, we consider approximate equilibria instead. As our first main result, we show that the existence of a K-approximate equilibrium can always be guaranteed, where K (approximately equal to 1.1974) is the unique solution of a cubic polynomial equation. To this end, we give a polynomial time combinatorial algorithm which computes a K-approximate equilibrium. The factor K is tight, meaning that there is an instance that does not admit an A-approximate equilibrium for any A < K. Thus A = K is the smallest possible value of A such that the existence of an A-approximate equilibrium can be guaranteed for any instance of the considered game. Secondly, we focus on approximate equilibria of a given fixed instance. We show how to compute efficiently a best approximate equilibrium, that is, with smallest possible A among all A-approximate equilibria of the given instance.

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