a *multi-objective symmetric game*, individual payoffs are not impacted by the agents’ identities. There is one sole action-set $A^{\ast}$ for every agent $i$. So, when deciding action $a^{\ast}\in A^{\ast}$, the multi-objective reward only depends on the number of agents that decided every action. Consequently, the game is not specified for every action-profile $\bm{a}\in A=\prod_{i\in N}A^{\ast}$ and every agent $i$, but rather for every action $a^{\ast}\in A^{\ast}$ and every *configuration* $c:A^{\ast}\rightarrow\mathbb{N}$, where number $c(a^{\ast})\in\mathbb{N}$ indicates the number of agents deciding action $a^{\ast}$. Therefore, the utility is given by a function $\bm{u}^{\ast}$ such that $\bm{u}^{\ast}(a^{\ast},c)\in{\mathbb{R}}^{d}$ is the payoff for deciding action $a^{\ast}$ when configuration $c$ occurs.

There is a number ${n+\alpha-1 \choose \alpha-1}$ of configurations[^6] to which the MO symmetric game associates MO vectors. As a consequence, generalizing to vectorial payoffs, the representation length is $L=\alpha{n+\alpha-1 \choose \alpha-1}d$, and when the numbers $\alpha$ and $d$ are fixed constant, length is $L(n)\in\Theta\left(\alpha n^{\alpha}d\right)$. Quite simply, for computing ${\mathcal{E}}$, ${\mbox{EFF}}[{\mathcal{E}}]$ and ${\mbox{WST}}[{\mathcal{E}}]$, configurations enumeration already takes polynomial time.

Given a multi-objective symmetric game with fixed $\alpha$,

-   computing ${\mbox{PN}}$ and ${\mathcal{E}}$ takes time $O(n^{\alpha}\alpha^{2}d)=O(L\alpha)$;

-   computing ${\mbox{EFF}}[{\mathcal{E}}]$ and ${\mbox{WST}}[{\mathcal{E}}]$ takes time $O(n^{2\alpha}d)=O(L^{2})$. If $d=2$, this lowers to $O(L(\alpha+\log_{}(L)))$.

Computing MO-CR
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In this subsection, 