Sharing Non-Anonymous Costs of Multiple Resources Optimally

TL;DR

This paper analyzes the efficiency of pure Nash equilibria in resource cost sharing games using uniform protocols, providing tight bounds on inefficiency measures for various cost function classes.

Contribution

It precisely quantifies the inefficiency bounds of Nash equilibria under uniform cost sharing protocols for different resource cost functions.

Findings

Tight bounds on prices of stability and anarchy for submodular and supermodular costs.

Upper bounds are achieved by the Shapley cost sharing protocol.

Lower bounds are valid for arbitrary uniform protocols and games with anonymous costs.

Abstract

In cost sharing games, the existence and efficiency of pure Nash equilibria fundamentally depends on the method that is used to share the resources' costs. We consider a general class of resource allocation problems in which a set of resources is used by a heterogeneous set of selfish users. The cost of a resource is a (non-decreasing) function of the set of its users. Under the assumption that the costs of the resources are shared by uniform cost sharing protocols, i.e., protocols that use only local information of the resource's cost structure and its users to determine the cost shares, we exactly quantify the inefficiency of the resulting pure Nash equilibria. Specifically, we show tight bounds on prices of stability and anarchy for games with only submodular and only supermodular cost functions, respectively, and an asymptotically tight bound for games with arbitrary set-functions. While all our upper bounds are attained for the well-known Shapley cost sharing protocol, our lower bounds hold for arbitrary uniform cost sharing protocols and are even valid for games with anonymous costs, i.e., games in which the cost of each resource only depends on the cardinality of the set of its users.