Optimal cost sharing protocols for scheduling games

Abstract

We consider the problem of designing cost sharing protocols to minimize the price of anarchy and stability for a class of scheduling games. Here, we are given a set of players, each associated with a job of certain non-negative weight. Any job fits on any machine, and the cost of a machine is a non-decreasing function of the total load on the machine. We assume that the private cost of a player is determined by a cost sharing protocol. We consider four natural design restrictions for feasible protocols: stability, budget balance, separability, and uniformity. While budget balance is self-explanatory, the stability requirement asks for the existence of pure-strategy Nash equilibria. Separability requires that the resulting cost shares only depend on the set of players on a machine. Uniformity additionally requires that the cost shares on a machine are instance-independent, that is, they remain the same even if new machines are added to or removed from the instance. We call a cost sharing protocol basic, if it satisfies only stability and budget balance. Separable and uniform cost sharing protocols additionally satisfy separability and uniformity, respectively. For n-player games we show that among all basic and separable cost sharing protocols, there is an optimal protocol with price of anarchy and stability of precisely the n-th harmonic number. For uniform protocols we present a strong lower bound showing that the price of anarchy is unbounded. Moreover, we obtain several results for special cases in which either the cost functions are restricted, or the job sizes are restricted. As a byproduct of our analysis, we obtain a complete characterization of outcomes that can be enforced as a pure-strategy Nash equilibrium by basic and separable cost sharing protocols.