We present a detailed analysis of Nash equilibria in multi-objective normal-form games, which are normal-form games with vectorial payoffs.Our approach is based on modelling each player's utility using a utility function that maps a vector to a scalar utility.For mixed strategies, we can apply the utility function before calculating the expectation of the payoff vector as well as after, resulting in two distinct optimisation criteria.We show that when computing the utility from the expected payoff, a Nash equilibrium can be guaranteed when players have quasiconcave utility functions.In addition, we show that when players have quasiconvex utility functions, pure strategy Nash equilibria are equal under both optimisation criteria.We extend this to settings where some players optimise for one criterion, while others optimise for the second.We combine these results and formulate an algorithm that computes all pure strategy Nash equilibria given quasiconvex utility functions.
Willem Röpke (Sun,) studied this question.