Approximation Algorithm for the Binary-Preference Capacitated Selfish Replication Game and a Tight Bound on its Price of Anarchy

TL;DR

This paper analyzes the capacitated selfish replication game with binary preferences, proving the existence of Nash equilibria, bounding the price of anarchy, and developing algorithms for equilibrium and near-optimal solutions.

Contribution

It introduces a potential function for the game, provides bounds on the price of anarchy, and presents polynomial and quasi-polynomial algorithms for finding equilibria and near-optimal allocations.

Findings

Existence of pure-strategy Nash equilibrium due to potential function.

Price of anarchy is bounded above by 3, with some instances at least 2.

A distributed quasi-polynomial algorithm approximates optimal allocations within a constant factor.

Abstract

We consider the capacitated selfish replication (CSR) game with binary preferences, over general undirected networks. We first show that such games have an associated ordinary potential function, and hence always admit a pure-strategy Nash equilibrium (NE). Further, when the minimum degree of the network and the number of resources are of the same order, there exists an exact polynomial time algorithm which can find a NE. Following this, we study the price of anarchy of such games, and show that it is bounded above by 3; we further provide some instances for which the price of anarchy is at least 2. We develop a quasi-polynomial algorithm O(n^2D^{ln n}), where n is the number of players and D is the diameter of the network, which can find, in a distributed manner, an allocation profile that is within a constant factor of the optimal allocation, and hence of any pure-strategy NE of the game. Proof of this result uses a novel potential function.