definition is guaranteed to exist too.

Multi-objective coordination ratio 
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In the single-objective case, the coordination ratio measures the efficiency loss of equilibria compared to the optimum. In MO games, we claim that it is critical to study efficiency with respect to every objective. Even after the actions, the game analyst still has access to the vectorial payoffs. In this section, we follow the agenda outlined in the introduction, to define a *multi-objective coordination ratio* $\text{MO-CR}[{\mathcal{E}},{\mathcal{F}}]$ of the set of equilibria outcomes ${\mathcal{E}}$ to the set of efficient outcomes ${\mathcal{F}}$, that fills the critical purpose to keep information on each objective.

First, we state the list of desirable properties that we want the ratio to satisfy. For the purpose of having meaningful divisions and ratios, some vectors are positive in this section. Given vectors $\bm{\rho},\bm{y}\in{\mathbb{R}^d}$ and $\bm{z}\in{\mathbb{R}^d_{+}}$, vector $\bm{\rho}\star \bm{y}\in{\mathbb{R}^d}$ is defined by $\forall k\in{{\mathcal{D}}},(\bm{\rho}\star \bm{y})_{k}=\rho_{k}y_{k}$. Vector $\bm{y}/\bm{z}\in{\mathbb{R}^d}$ is defined by $\forall k\in{{\mathcal{D}}},(\bm{y}/\bm{z})_{k}=y_{k}/z_{k}$. Given vector $\bm{r}\in{\mathbb{R}^d}$ and set of vectors $Y$, set $\bm{r}\star Y$ is defined by $\{\bm{r}\star \bm{y}\in{\mathbb{R}^d_{+}}|\bm{y}\in Y\}$ and for $\bm{r}\in{\mathbb{R}^d_{+}}$, set $Y/\bm{r}$ is defined by $\lbrace \bm{y}/\bm{r}\in{\mathbb{R}^d}|\bm{y}\in Y\rbrace$. Given $\bm{x}\in{\mathbb{R}^d}$, cone ${\mathcal{C}}(\bm{x})$ denotes $\{\bm{y}\in{\mathbb{R}^d}~|~\bm{x}{\succsim}\bm{y}\}$, and given $X\subset{\mathbb{R}^d}$, cone-union ${\mathcal{C}}(X\text{)}$ is defined by $\cup_{\bm{x}\in X}{\mathcal{C}}(\bm{x})$. Vector $\bm{0}$ denotes 