finite and sets $A^i$ are just compact.

Likelihood of equilibrium in random games
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Another manner to study whether a ${\mbox{PN}}$-equilibrium exists is to provide a probability distribution on a family of finite games and then discuss the probability of ${\mbox{PN}}$-equilibrium existence. A similar methodology was successfully applied [@goldberg1968probability; @dresher1970probability; @rinott2000number] to SO games in several settings where every SO payoff $u^{i}(\bm{a})$ is independently and identically distributed by a uniform distribution on continuous intervals $[0,1]$. At the heart of this subsection, let random variable $Z$ denote the number of pure Nash-equilibria action-profiles in the game. In the SO case, there is almost surely only one best response. However, when considering MO games, a main technical difference lies in the average number of “best responses” (or here, Pareto-efficient responses), which in most cases exceeds $1$, due to the surface-like shape of the Pareto-efficient set in ${\mathbb{R}}^{d}$, surface which is $(d-1)$ dimensional. Here, we assume a probability distribution ${\mathbb{P}}_{n,\alpha,\beta}$, that builds randomly the Pareto-efficient response tables of an $n$-agent normal form game with $\alpha$ actions-per-agent: for every agent $i$ and every adversary action-profile $\bm{a}^{-i}\in\prod_{j\neq i}A^{j}$, there is a fixed number $\beta:1<\beta\leq\alpha$ of Pareto-efficient responses, for the sake of simplicity.

Given numbers $n\geq2$ 