Non-cooperative Facility Location and Covering Games

We study a general class of non-cooperative games coming from combinatorial covering and facility location problems. A game for k players is based on an integer programming formulation. Each player wants to satisfy a subset of the constraints. Variables represent resources, which are available in costly integer units and must be bought. The cost can be shared arbitrarily between players. Once a unit is bought, it can be used by all players to satisfy their constraints. In general the cost of pure-strategy Nash equilibria in this game can be prohibitively high, as both prices of anarchy and stability are in Θ(k). In addition, deciding the existence of pure Nash equilibria is NP-hard. These results extend to recently studied single-source connection games. Under certain conditions, however, cheap Nash equilibria exist: if the integrality gap of the underlying integer program is 1 and in the case of single constraint players. In addition, we present algorithms that compute cheap approximate Nash equilibria in polynomial time.