efficient outcomes ${\mathcal{F}},{\mathcal{F}}'\subset{\mathbb{R}^d_{+}}$ reflect onto ratio $\text{MO-CR}[{\mathcal{E}},{\mathcal{F}}]$.

Secondly, in MOGs, on average, there are many Pareto-Nash equilibria. An efficiency *guarantee* $\bm{\rho}\in{\mathbb{R}^d}$ should hold for every equilibrium outcome. It induces this conjunctive definition of the set of guaranteed vectorial ratios: $$\begin{aligned}
R[{\mathcal{E}},{\mathcal{F}}]\quad=\quad\bigcap_{\bm{y}\in{\mathcal{E}}} R[\bm{y},{\mathcal{F}}]. \end{aligned}$$

In fact, because of the conjunction on equilibria outcomes, the set $R[{\mathcal{E}},{\mathcal{F}}]$ only depends on sets ${\mbox{WST}}[{\mathcal{E}}]$ (instead of set ${\mathcal{E}}$) and ${\mathcal{F}}$.

Finally, if two bounds on efficiencies $\bm{\rho}$ and $\bm{\rho}'$ are such that $\bm{\rho}{\succ}\bm{\rho}'$ (e.g. the former guarantees fraction $\bm{\rho}=(0.75,0.75)$ of efficiency and the later fraction $\bm{\rho}'=(0.5,0.5)$), then $\bm{\rho}'$ brings no more information; hence, MO-CR is defined using ${\mbox{EFF}}$ on the guaranteed efficiency ratios $R[{\mbox{WST}}[{\mathcal{E}}],{\mathcal{F}}]$. These points are summed up in the following definition:

\[def:MOCR\] Given an MO game, vector $\bm{\rho}\in{\mathbb{R}^d}$ bounds its inefficiency (i.e. $\bm{\rho}\in R[{\mathcal{E}},{\mathcal{F}}]$) if and only if the following holds (see Fig. 4) : $$\forall \bm{y}\in{\mathcal{E}},\quad\exists \bm{z}\in{\mathcal{F}},\quad \bm{y}/\bm{z}{\succsim}\bm{\rho}.$$ The multi-objective coordination ratio $\text{MO-CR}[{\mathcal{E}},{\mathcal{F}}]$ is then defined as: $$\text{MO-CR}[{\mathcal{E}},{\mathcal{F}}]\quad=\quad{\mbox{EFF}}[R[{\mbox{WST}}[{\mathcal{E}}],{\mathcal{F}}]].$$

![Didactic depiction of a guaranteed vectorial ratio $\bm{\rho}$ from $\text{MO-CR}[{\mathcal{E}},{\mathcal{F}}]$.](Figure4)

 

The most famous results of the coordination ratio (or price of anarchy) are stated analytically on families of games, for instance on congestion games [@christodoulou2005price; @roughgarden2009intrinsic]. Such results would also be desirable in 