On Nash Equilibria in Normal-Form Games With Vectorial Payoffs

TL;DR

This paper studies Nash equilibria in multi-objective normal form games with vector payoffs, exploring existence conditions, differences between criteria, and algorithms for pure equilibria, with a focus on quasiconvex utilities.

Contribution

It introduces conditions for Nash equilibrium existence, compares equilibrium sets under different criteria, and develops an algorithm for pure equilibria in MONFGs with quasiconvex utilities.

Findings

Nash equilibria existence is guaranteed under certain conditions.

Equilibria sets can differ between the two optimisation criteria.

An algorithm for computing pure strategy Nash equilibria is proposed.

Abstract

We provide an in-depth study of Nash equilibria in multi-objective normal form games (MONFGs), i.e., normal form games with vectorial payoffs. Taking a utility-based approach, we assume that each player's utility can be modelled with a utility function that maps a vector to a scalar utility. In the case of a mixed strategy, it is meaningful to apply such a scalarisation both before calculating the expectation of the payoff vector as well as after. This distinction leads to two optimisation criteria. With the first criterion, players aim to optimise the expected value of their utility function applied to the payoff vectors obtained in the game. With the second criterion, players aim to optimise the utility of expected payoff vectors given a joint strategy. Under this latter criterion, it was shown that Nash equilibria need not exist. Our first contribution is to provide a sufficient condition under which Nash equilibria are guaranteed to exist. Secondly, we show that when Nash equilibria do exist under both criteria, no equilibrium needs to be shared between the two criteria, and even the number of equilibria can differ. Thirdly, we contribute a study of pure strategy Nash equilibria under both criteria. We show that when assuming quasiconvex utility functions for players, the sets of pure strategy Nash equilibria under both optimisation criteria are equivalent. This result is further extended to games in which players adhere to different optimisation criteria. Finally, given these theoretical results, we construct an algorithm to compute all pure strategy Nash equilibria in MONFGs where players have a quasiconvex utility function.