On nash equilibria in normal-form games with vectorial payoffs

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