shows the existence of a mixed-strategy Pareto-Nash equilibrium in finite two-player zero-sum multi-objective games. The later generalizes this existence result to finite multi-objective games. Both use a definition of mixed-strategy Pareto-Nash equilibria that suffers an inconsistency: pure-strategy Pareto-Nash equilibria are not included in the set of mixed-strategy Nash equilibria (see Sec. \[sec:mixed\]). Nonetheless, there is an established literature on games with vector payoffs that uses this definition. Deep formal works generalized known existence results [@shapley1959equilibrium] to individual action-sets being compact convex subsets of a normed space [@wang1993existence]. Weak Pareto-Nash equilibria can be approximated [@morgan2005approximations]. *Works related to pure strategies and algorithms.* [@wierzbicki1995multiple] achieves to characterize the entire set of Pareto-Nash equilibria by mean of augmented Tchebycheff norms. However, the number of dimensions that parameterize these Tchebycheff norms is algorithmically prohibitive. [@patrone2007multicriteria] shows that a MO potential function guarantees that a Pareto-Nash equilibrium exists in finite MO games.

In Section 3, we show in three different settings that pure-strategy Pareto-Nash equilibria are guaranteed to exist, or very likely to be numerous. In Section 4, we show an inconsistency in the current concept of mixed-strategy PN equilibrium, and propose an extension to solve this flaw. In Section 5, in the fashion of 