can rule out comparisons of equilibria outcomes to ideal vector $\mathcal{I}=(\max_{z\in{\mathcal{F}}}\lbrace z_{k}\rbrace|k\in{{\mathcal{D}}})$ does not satisfy requirement (5) to have $\bm{1}\in\text{MO-CR}[{\mathcal{E}},{\mathcal{F}}]$ when ${\mathcal{E}}\subseteq{\mathcal{F}}$. By starting from a social welfare $f:{\mathbb{R}^d_{+}}\rightarrow{\mathbb{R}}_+$, taking ratio $\min f({\mathcal{E}})/\max f({\mathcal{F}})$, induces the same problem.

This measurement should also be non-dictatorial, in the sense that no point of view should be imposed on what the overall efficiency is: no prior choice must be done on the set of efficient outcomes. Formally, if two sets of efficient outcomes ${\mathcal{F}},{\mathcal{F}}'\subset{\mathbb{R}^d_{+}}$ differ even slightly, then this must reflect at least for some numerator set ${\mathcal{E}}$ onto ratio $\text{MO-CR}[{\mathcal{E}},{\mathcal{F}}]$. This amounts to a disjunction on efficient outcomes. Finally $\text{MO-CR}[{\mathcal{E}},{\mathcal{F}}]$ must provide guaranteed efficiency ratios that hold for every equilibrium outcome $\bm{y}\in{\mathcal{E}}$, which amounts to a conjunction on equilibria outcomes. The definition below follows from these requirements.

Firstly, the efficiency of *one* equilibrium $\bm{y}\in{\mathcal{E}}$ is quantified without prior choices on what efficient outcome should we compare it to, as required: $$\begin{aligned}
R[\bm{y},{\mathcal{F}}]\quad =\quad \bigcup_{\bm{z}\in{\mathcal{F}}}{\mathcal{C}}(\bm{y}/\bm{z}), \end{aligned}$$ The idea is that we do not take sides with any efficient outcome. Instead, we define with flexibility and without a dictatorship a disjunctive set of guaranteed efficiency ratios, which lets the differences between two sets of 