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.
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The first author is supported by the Research Foundation – Flanders (FWO), grant number 1197622N. This research was supported by funding from the Flemish Government under the “Onderzoeksprogramma Artificiële Intelligentie (AI) Vlaanderen” program.
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Part of this work was carried out by the first author for his thesis  under the supervision of the other authors. Some preliminary results in this article were presented in the Multi-Objective Decision Making Workshop 2021 .
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Röpke, W., Roijers, D.M., Nowé, A. et al. On nash equilibria in normal-form games with vectorial payoffs. Auton Agent Multi-Agent Syst 36, 53 (2022). https://doi.org/10.1007/s10458-022-09582-6