Efficient Black-Box Reductions for Separable Cost Sharing

Tobias Harks,Manuel Surek,Anja Schedel,Martin Hoefer

doi:10.1287/moor.2020.1050

Abstract

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.

Highlights

Cost sharing is a fundamental task in networks with strategic agents and has attracted a large amount of interest in algorithmic game theory
We present several new polynomial-time black-box reductions for separable cost sharing protocols with small price of stability (PoS)
We devise an efficient black-box reduction that takes as input an arbitrary strategy profile and computes a new profile of no larger cost together with a separable cost sharing protocol inducing the new profile as a pure Nash equilibrium (PNE)

Summary

Introduction

Cost sharing is a fundamental task in networks with strategic agents and has attracted a large amount of interest in algorithmic game theory. Even in connection games on undirected networks, it can be PLS-hard to find a PNE [51] and the PoS (the total cost of the best Nash equilibrium compared to the cost of the optimal allocation) is not known to be constant This contrasts the fact that there are polynomial time approximation algorithms with low approximation factors, see, e.g. Our domains represent broad generalizations of UFL – arbitrary, player-specific matroids, singlesource connection games without delays, and connection games on undirected n-series-parallel graphs with delays In each of these domains, we take as input an arbitrary profile and efficiently turn it into a profile having no larger cost and a sharing of the shareable costs such that it is a Nash equilibrium. If S is polynomial-time computable, both protocol and Nash equilibrium S are both polynomial-time computable and polynomial-space representable

Our Results

Preliminaries and Related Work

Separable Cost Sharing Protocols

Matroid Games

Single-Source Connection Games without Delays

Connection Games and Graph Structure