Game Efficiency through Linear Programming Duality

TL;DR

This paper introduces a linear programming duality approach to analyze and bound the efficiency of various games, providing a unified framework that captures existing techniques and improves bounds on the price of anarchy.

Contribution

The paper develops a duality-based method for analyzing game efficiency, offering a general recipe that encompasses and enhances existing frameworks like smoothness.

Findings

Duality approach yields near-optimal PoA bounds for multiple game classes

Method captures and extends smoothness and non-smooth techniques

Applicable to diverse environments from congestion to Bayesian games

Abstract

The efficiency of a game is typically quantified by the price of anarchy (PoA), defined as the worst ratio of the objective function value of an equilibrium --- solution of the game --- and that of an optimal outcome. Given the tremendous impact of tools from mathematical programming in the design of algorithms and the similarity of the price of anarchy and different measures such as the approximation and competitive ratios, it is intriguing to develop a duality-based method to characterize the efficiency of games. In the paper, we present an approach based on linear programming duality to study the efficiency of games. We show that the approach provides a general recipe to analyze the efficiency of games and also to derive concepts leading to improvements. The approach is particularly appropriate to bound the PoA. Specifically, in our approach the dual programs naturally lead to competitive PoA bounds that are (almost) optimal for several classes of games. The approach indeed captures the smoothness framework and also some current non-smooth techniques/concepts. We show the applicability to the wide variety of games and environments, from congestion games to Bayesian welfare, from full-information settings to incomplete-information ones.