Utility Design for Distributed Resource Allocation -- Part I: Characterizing and Optimizing the Exact Price of Anarchy

TL;DR

This paper develops a framework to precisely measure and optimize the worst-case efficiency loss (price of anarchy) in distributed resource allocation games by designing agents' utility functions, enabling better system-level performance guarantees.

Contribution

It introduces a novel method to characterize and optimize the price of anarchy for any utility design in resource allocation games, using a tractable linear program.

Findings

Derived tight bounds on the price of anarchy for various utility functions.

Provided a linear program to optimize utility functions for improved system performance.

Established performance certificates applicable to learning algorithms reaching equilibrium.

Abstract

Game theory has emerged as a fruitful paradigm for the design of networked multiagent systems. A fundamental component of this approach is the design of agents' utility functions so that their self-interested maximization results in a desirable collective behavior. In this work we focus on a well-studied class of distributed resource allocation problems where each agent is requested to select a subset of resources with the goal of optimizing a given system-level objective. Our core contribution is the development of a novel framework to tightly characterize the worst case performance of any resulting Nash equilibrium (price of anarchy) as a function of the chosen agents' utility functions. Leveraging this result, we identify how to design such utilities so as to optimize the price of anarchy through a tractable linear program. This provides us with a priori performance certificates applicable to any existing learning algorithm capable of driving the system to an equilibrium. Part II of this work specializes these results to submodular and supermodular objectives, discusses the complexity of computing Nash equilibria, and provides multiple illustrations of the theoretical findings.