LP-based Covering Games with Low Price of Anarchy

TL;DR

This paper introduces a new class of covering games based on linear programming relaxations, achieving constant-factor bounds on the price of anarchy and efficient convergence to near-optimal equilibria.

Contribution

It presents covering games with provably bounded price of anarchy matching centralized approximation guarantees, a novel connection between game rules and LP relaxations, and linear-time dynamics for linear costs.

Findings

Price of anarchy for the vertex cover game is exactly 2.

Games' bounds match the best known centralized approximation factors.

Linear response dynamics converge efficiently to near-optimal Nash equilibria.

Abstract

We present a new class of vertex cover and set cover games. The price of anarchy bounds match the best known constant factor approximation guarantees for the centralized optimization problems for linear and also for submodular costs -- in contrast to all previously studied covering games, where the price of anarchy cannot be bounded by a constant (e.g. [6, 7, 11, 5, 2]). In particular, we describe a vertex cover game with a price of anarchy of 2. The rules of the games capture the structure of the linear programming relaxations of the underlying optimization problems, and our bounds are established by analyzing these relaxations. Furthermore, for linear costs we exhibit linear time best response dynamics that converge to these almost optimal Nash equilibria. These dynamics mimic the classical greedy approximation algorithm of Bar-Yehuda and Even [3].