$q,m$ are exponentials with respect to the compact size of the game’s representation. For instance, this is the case for multi-objective graphical games. Theorem \[th:approx\] below answers this issue by taking only a small and approximate representation of sets ${\mbox{WST}}[{\mathcal{E}}]$ and ${\mathcal{F}}$, in order to output a guaranteed approximation of sets $\text{MO-CR}$ or $R[{\mbox{WST}}[{\mathcal{E}}],{\mathcal{F}}]$. This suggests the following general method:

1.  Given a compact MOG representation, compute quickly an approximation $E^{(\varepsilon)}$ of ${\mbox{WST}}[{\mathcal{E}}]$ and an approximation $F^{(\varepsilon')}$ of ${\mathcal{F}}$.

2.  Then, given $E^{(\varepsilon)}$ and $F^{(\varepsilon')}$, use Algorithm 1 to approximate the MO-CR.

For this general method to be implemented rigorously, we must specify the precise definitions of the two approximations required in input, for the desired output to be indeed some approximation of the MO-CR.

Firstly, let us specify the [output]{}. The ratios in $R[{\mbox{WST}}[{\mathcal{E}}],{\mathcal{F}}]$ must be represented, even approximately, but only by using valid ratios of efficiency, as below.

\[def:covering\] Given $R\subset{\mathbb{R}^d_{+}}$ and $\varepsilon>0$,    $R^{(\varepsilon)}\subset R$ is a $(1+\varepsilon)$-covering of $R$, if and only if: $$\forall \bm{\rho}\in R,\quad
\exists \bm{\rho}'\in R^{(\varepsilon)}:\quad
(1+\varepsilon)\bm{\rho}' {\succsim}\bm{\rho}$$

For instance, $R[{\mbox{WST}}[{\mathcal{E}}],{\mathcal{F}}]$ is $(1+0)$-covered by $\text{MO-CR}={\mbox{EFF}}[R[{\mbox{WST}}[{\mathcal{E}}],{\mathcal{F}}]]$. Denote $\bm{\varphi}:{\mathbb{R}^d_{+}}\rightarrow{\mathbb{N}^d}$ the discretization into the $(1+\varepsilon)$-logarithmic grid. Given a vector $\bm{x}\in{\mathbb{R}^d_{+}}$, $\bm{\varphi}(x)$ is defined by: $\forall k\in{{\mathcal{D}}},~~\varphi_k(x)=\lfloor\log_{(1+\varepsilon)}(x_k)\rfloor$. A 