Willem Röpke, Diederik M. Roijers, , Roxana Radulescu

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.

Röpke, W , Roijers, DM , Nowe, A & Radulescu, R 2022, ' On Nash Equilibria in Normal-Form Games With Vectorial Payoffs ', Autonomous Agents and Multi-Agent Systems , vol. 36, no. 2, 53.

Röpke, W. , Roijers, D. M. , Nowe, A. , & Radulescu, R. (2022). On Nash Equilibria in Normal-Form Games With Vectorial Payoffs . Autonomous Agents and Multi-Agent Systems , 36 (2), [53].

@article{c1a40259ac8b4a0ebe9bcbf7db2c4b4e,

title = " On Nash Equilibria in Normal-Form Games With Vectorial Payoffs " ,

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{ extquoteright}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. " ,

keywords = " Game theory, Nash equilibrium, Multi-objective " ,

author = " Willem R{ " o}pke and Roijers, {Diederik M.} and Ann Nowe and Roxana Radulescu " ,

note = " Funding Information: 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 Artifici{ " e}le Intelligentie (AI) Vlaanderen” program. Publisher Copyright: { extcopyright} 2022, Springer Science+Business Media, LLC, part of Springer Nature. Copyright: Copyright 2022 Elsevier B.V., All rights reserved. " ,

year = " 2022 " ,

month = oct,

day = " 10 " ,

doi = " 10.1007/s10458-022-09582-6 " ,

language = " English " ,

volume = " 36 " ,

journal = " Autonomous Agents and Multi-Agent Systems " ,

issn = " 1387-2532 " ,

publisher = " Springer Netherlands " ,

number = " 2 " ,

}