On Existence, Mixtures, Computation and Efficiency in Multi-objective Games

TL;DR

This paper investigates the existence and computation of Pareto-Nash equilibria in multi-objective games, introduces a measure for their efficiency, and provides algorithms for their calculation.

Contribution

It demonstrates the existence of pure and mixed-strategy Pareto-Nash equilibria, proposes a multi-objective efficiency measure, and develops algorithms for equilibrium and efficiency computation.

Findings

Numerous pure-strategy Pareto-Nash equilibria exist.

A multi-objective coordination ratio effectively measures efficiency.

Algorithms are provided for computing equilibria and efficiency metrics.

Abstract

In a multi-objective game, each individual's payoff is a \emph{vector-valued} function of everyone's actions. Under such vectorial payoffs, Pareto-efficiency is used to formulate each individual's best-response condition, inducing Pareto-Nash equilibria as the fundamental solution concept. In this work, we follow a classical game-theoretic agenda to study equilibria. Firstly, we show in several ways that numerous pure-strategy Pareto-Nash equilibria exist. Secondly, we propose a more consistent extension to mixed-strategy equilibria. Thirdly, we introduce a measurement of the efficiency of multiple objectives games, which purpose is to keep the information on each objective: the multi-objective coordination ratio. Finally, we provide algorithms that compute Pareto-Nash equilibria and that compute or approximate the multi-objective coordination ratio.