by Blackwell [@blackwell1956analog] and Shapley [@shapley1959equilibrium] (below) are those usually considered [@borm1988pareto; @corley1985games; @voorneveld1999potential; @zeleny1975games].

Given finite MO game $\Gamma=\left(N,\{A^{i}\}_{i\in N},\{{{\mathcal{D}}}\},\{\bm{u}^{i}\}_{i\in N}\right)$, a mixed-strategy profile $\bm{p}=(p^{1},\ldots,p^{n})\in\prod_{i\in N}\Delta(A^{i})$ is a mixed-strategy Pareto-Nash equilibrium if and only if it satisfies for every agent $i$: $$\bm{u}^{i}(p^{i},\bm{p}^{-i})\in{\mbox{EFF}}\left[\left\{ \bm{u}^{i}(q^{i},\bm{p}^{-i})\in{\mathbb{R}}^{d}\mid q^{i}\in\Delta(A^{i})\right\} \right]$$

The rational behind this first definition is the following. For every agent $i$, mixed-strategy $p^{i}\in\Delta(A^{i})$ acts as a convex-combination of set of vectorial payoffs $\bm{u}^{i}(A^{i},\bm{p}^{-i})$ and the best-response condition is replaced by the fact that mixed-strategy $p^{i}$ should have a Pareto-efficient evaluation $\bm{u}^{i}(p^{i},\bm{p}^{-i})$ among the elements of this convex set of evaluations $\{\bm{u}^{i}(q^{i},\bm{p}^{-i})\in{\mathbb{R}}^{d}\mid q^{i}\in\Delta(A^{i})\}$. That is, a mixed-strategy Pareto-Nash equilibrium is a pure-strategy Pareto-Nash equilibrium in finite game $\Gamma$’s mixed extension. However, as depicted in Figure \[fig:mix\], Definition 1 fails to fulfill two fundamental requirements:

1.  Pure-strategy equilibria must be included in mixed-strategy equilibria.

2.  Mixed-strategies also enable to model a risk-averse agent.

Figure 3 demonstrates these side effects.

To fulfill the two requirements, instead of efficient mixed actions, we consider mixtures of efficient pure-actions. As in Figure \[fig:mix\], it corrects both side effects.

Given a finite multi-objective game $\left(N,\{A^{i}\}_{i\in N},\{{{\mathcal{D}}}\},\{\bm{u}^{i}\}_{i\in N}\right)$, a mixed-strategy Pareto-Nash equilibrium is a mixed-strategy profile$\bm{p}=(p^{1},\ldots,p^{n})\in\prod_{i\in N}\Delta(A^{i})$, such that for 