we address the problem of computing the set $\text{MO-CR}[{\mathcal{E}},{\mathcal{F}}]$, given sets of worst equilibria outcomes ${\mbox{WST}}[{\mathcal{E}}]$ and efficient outcomes ${\mathcal{F}}$. Algorithm 1 (below) computes such set. In the algorithm, set $D^{t}$ denotes a set of vectors. Given two vectors, $\bm{x},\bm{y}\in{\mathbb{R}^d_{+}}$, let $\bm{x}\wedge \bm{y}$ denote the vector defined by $\forall k\in{{\mathcal{D}}},~(\bm{x}\wedge \bm{y})_{k}=\min\{x_{k},y_{k}\}$, let $\bm{x}^{\bm{y}}\in{\mathbb{R}^d_{+}}$ be the vector defined by $\forall k\in{{\mathcal{D}}}, (\bm{x}^{\bm{y}})_k=(x_k)^{y_k}$, and recall that $\forall k\in{{\mathcal{D}}},~(\bm{x}/\bm{y})_{k}=x_{k}/y_{k}$.

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\[-1.5ex\] [**create** ]{} $D^1\leftarrow \{\bm{y}^1/\bm{z}\in{\mathbb{R}^d_{+}}~|~\bm{z}\in{\mathcal{F}}\}$\
$D^q$

Algorithm 1 outputs $\text{MO-CR}[{\mathcal{E}},{\mathcal{F}}]$ in poly-time $O((qm)^{2d-1}d),$ where $q=|{\mbox{WST}}[{\mathcal{E}}]|$ and $m=|{\mathcal{F}}|$ denote the size of the inputs, and $d$ is fixed.

Algorithm 1 calculates product $\cap_{\bm{y}\in{\mbox{WST}}[{\mathcal{E}}]}\cup_{\bm{z}\in{\mathcal{F}}}{\mathcal{C}}(\bm{y}/\bm{z})$, where there could be $m^q$ terms in the output. This set-algebra of cone-unions is compact.

A decisive corollary is that given an MO game with length $L$ that satisfies $q=O(\text{poly}(L))$, $m=O(\text{poly}(L))$ and both sets ${\mbox{WST}}[{\mathcal{E}}]$ and ${\mathcal{F}}$ are computable in time $O(\text{poly}(L))$, then one can compute $\text{MO-CR}$ in polynomial time $O(\text{poly}(L))$. For instance, it is the case with MO normal forms or MO symmetric games. So this approach is not intractable in the most basic cases.

Approximation of the MO-CR for MO compact representations
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Unfortunately, Algorithm 1 is not practical when the MO game has a compact form and cardinalities 