of a utility function $v^{i}:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}$, which is monotonic with respect to the Pareto-dominance, that is: $$\begin{aligned}
\bm{x}{\succ}\bm{y} &\Rightarrow& v^{i}(\bm{x})> v^{i}(\bm{y})\end{aligned}$$ Such functions are called *Pareto-monotonic*. For instance, these include positive weighted sums, Cobb-Douglas utilities, and utility functions in general as assumed by the Arrow-Debreu theorem.

A straightforward consequence is that the set of Pareto-efficient vectors contains the optima of any Pareto-monotonic utility function. Formally, given a MOG $\Gamma$, from Pareto-monotonic utility functions $V=(v^{i}:{\mathbb{R}^d}\rightarrow{\mathbb{R}}|i\in N)$ the single-objective game $V\circ\Gamma=(N,\{A^{i}\}_{i\in N},\{v^{i}\circ \bm{u}^{i}\}_{i\in N})$ results from the given utilities, and one has: ${\mbox{PN}}(V\circ\Gamma)\subseteq{\mbox{PN}}(\Gamma).$ In other words, Pareto-Nash equilibria encompass the game’s outcome, regardless of the unknown preferences.

Also, inclusion ${\mbox{PN}}(V\circ\Gamma)\subseteq{\mbox{PN}}(\Gamma)$ argues for the guaranteed existence of numerous PN equilibria in MO games, under the following assumptions:

1.  the structure of the SO game on every objective is the same,

2.  equilibria are guaranteed in that structure of SO game,

3.  and a positive linear combination of the MO game induces that SO game.

This remark is the canonical argument used in previous results (e.g. [@shapley1959equilibrium; @patrone2007multicriteria]).

Multi-objective potentials
--------------------------

We now explore potential games, as introduced for congestion games by Robert Rosenthal [@rosenthal1973class; @monderer1996potential] and recently generalized to MO games [@patrone2007multicriteria]. The existence 