as the following set of non-Pareto-dominated vectors: $${\mbox{EFF}}[X]=\{\bm{y}\in X~~|~~\forall \bm{x}\in X,\mbox{~not~}(\bm{x}{\succ}\bm{y})\}.$$ Since Pareto-dominance is a partial order, it induces a multiplicity of Pareto-efficient vectors. These are the best compromises between objectives. Similarly, let ${\mbox{WST}}[X]=\{\bm{y} \in X|\forall \bm{x}\in X,\mbox{not}(\bm{y}{\succ}\bm{x})\}$ denote the worst vectors.

In a multi-objective game, individuals behave according to the Pareto -dominance, inducing the solution concept *Pareto-Nash equilibrium* (${\mbox{PN}}$), formally defined as any action-profile $\bm{a}\in A$ such that for every agent $i\in N$: $$\bm{u}^{i}(a^{i},\bm{a}^{-i})\quad\in\quad{\mbox{EFF}}\left[\quad \{\bm{u}^{i}(b^{i},\bm{a}^{-i})\mid b^{i}\in A^{i}\}\quad\right].$$ We call these conditions *Pareto-efficient responses*. Let ${\mbox{PN}}\subseteq A$ denote the set of Pareto-Nash equilibria. For instance, in Figure 1, action-profile $(b^{1},b^{2},a^{3},b^{4},b^{5})$ is a PN equilibrium, since each action, given the adversary local action profile (column), is Pareto-efficient among the given agent’s two actions (rows). In this example, there are $13$ Pareto-Nash equilibria (depicted in Figure 2).

Such an encompassing solution concept provides the first phase for bounding the efficiency of games. It is well-known that individualistic behaviors can be far from the optimum/maximum in terms of utilitarian evaluation $u(\bm{a})=\sum_{i\in N}u^{i}(\bm{a})$. In single-objective games[^4], this inefficiency is measured by the *Coordination Ratio* $\mbox{CR}=\frac{\min[u(PN)]}{\max[u(A)]}$ [@koutsoupias1999worst], which is more commonly known as the *Price of Anarchy* [@roughgarden2009intrinsic]. However, in the multi-objective case, the utilitarian 