  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.
