of an MO potential function guarantees that at least one Pareto-Nash equilibrium exists [@patrone2007multicriteria]. We go further and completely characterize the set of PN equilibria.

An MO game $\Gamma=\left(N,\{A^{i}\}_{i\in N},{{\mathcal{D}}},\{\bm{u}^{i}\}_{i\in N}\right)$ admits *(exact) potential function* $\bm{\Phi}:A\rightarrow{\mathbb{R}}^{d}$ if and only if for every action-profile $\bm{a}\in A$, for every agent $i\in N$ and for every action $b^{i}\in A^{i}$, one has: $$\forall k\in{{\mathcal{D}}},\quad
\Phi_k(b^{i},\bm{a}^{-i})-\Phi_k(\bm{a})\quad=\quad u^{i}_k(b^{i},\bm{a}^{-i})-u^{i}_k(\bm{a}).$$

That is, function $\bm{\Phi}$ additively accumulates the vectorial values of each deviation.

Given a vector valued function $\bm{\Phi}:A\rightarrow{\mathbb{R}}^{d}$, let the set of *locally efficient* action-profiles ${\mbox{LOC}}(\bm{\Phi})$ be the set of action-profiles $\bm{a}\in A$ such that: $$\bm{\Phi}(\bm{a})\quad\in\quad{\mbox{EFF}}[\{\bm{\Phi}(b^{i},\bm{a}^{-i})\in{\mathbb{R}}^{d}\mid i\in N,b^{i}\in A^{i}\}].$$

Set ${\mbox{LOC}}(\bm{\Phi})$ corresponds to a generalization of local optima for function $\bm{\Phi}$, and is non-empty if sets $N$, ${{\mathcal{D}}}$ and $A$ are finite. Moreover, due to the loose requirement for local efficiency, set ${\mbox{LOC}}(\bm{\Phi})$ is likely to contain numerous action-profiles.

\[th:1\] Let $\Gamma=\left(N,\{A^{i}\}_{i\in N},{{\mathcal{D}}},\{\bm{u}^{i}\}_{i\in N}\right)$ be a *finite multi-objective game*[^5] that admits potential function $\bm{\Phi}$. Then, it holds that: $${\mbox{PN}}(\Gamma)\quad=\quad{\mbox{LOC}}(\bm{\Phi})\quad\neq\quad\emptyset.$$

This theorem completely characterizes the set of Pareto-Nash equilibria as the set of locally efficient action-profiles for function $\bm{\Phi}$, which is a non-empty set with numerous action-profiles. More generally, Theorem \[th:1\] also holds when sets $N$ and ${{\mathcal{D}}}$ are 