Exact Price of Anarchy for Weighted Congestion Games with Two Players

TL;DR

This paper provides exact, parametric bounds on the price of anarchy for two-player weighted congestion games with affine costs, revealing how game primitives influence equilibrium quality and how sequential play impacts efficiency.

Contribution

It introduces a duality-based linear programming method to precisely compute worst-case equilibria and establishes how weights and play order affect the price of anarchy.

Findings

Exact PoA bounds depend on players' weights.

Sequential play improves PoA, especially with similar weights.

Worst-case equilibria occur with slight weight differences.

Abstract

This paper gives a complete analysis of worst-case equilibria for various versions of weighted congestion games with two players and affine cost functions. The results are exact price of anarchy bounds which are parametric in the weights of the two players, and establish exactly how the primitives of the game enter into the quality of equilibria. Interestingly, some of the worst-cases are attained when the players' weights only differ slightly. Our findings also show that sequential play improves the price of anarchy in all cases, however, this effect vanishes with an increasing difference in the players' weights. Methodologically, we obtain exact price of anarchy bounds by a duality based proof mechanism, based on a compact linear programming formulation that computes worst-case instances. This mechanism yields duality-based optimality certificates which can eventually be turned into purely algebraic proofs.