of agents, $\alpha\geq2$ of actions-per-agent and $\beta\leq\alpha$ of Pareto-efficient responses, based on probability distribution ${\mathbb{P}}_{n,\alpha,\beta}$, the number $Z$ of Pareto-Nash equilibria satisfies ${\mathbb{E}}[Z] = \beta^n$ and: $$\begin{aligned}
{\mathbb{P}}\left( (1-\gamma)\beta^n\leq {Z} \leq (1+\gamma)\beta^n \right) & \geq & 1 - \frac{1}{\gamma^2 \beta^{n}},\quad\forall\gamma\in(0,1). \end{aligned}$$

It argues for the existence of numerous Pareto-Nash equilibria when there are enough agents and efficient responses, and follows from the Bienaymé-Tchebychev inequality. For instance, (given $\gamma=1/2$) the probability that the number of Pareto-Nash equilibria $Z$ is between $(1/2)\beta^{n}$ and $(3/2)\beta^{n}$, is at least $1-4\beta^{-n}$, which for $\beta=2$ efficient responses and $n=5$ agents, gives ${\mathbb{P}}(16\leq Z\leq48)\geq7/8$.

Consistent extension to mixed strategies {#sec:mixed}
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To guarantee equilibrium existence by means of fixed-point theorems on compact sets [@vnmAndMorgenstern1944; @nash1950equilibrium], the finite action sets of every agent are expanded to include *mixed strategies*. That is: every agent $i$ decides a probability distribution $p^{i}$ in the set $\Delta(A^{i})$ of probability distributions over his action-set $A^{i}$. Each payoff function $\bm{u}^{i}$ is redefined to be the expected utility $$\begin{aligned}
\bm{u}^{i}(\bm{p}) &=& {\mathbb{E}}_{\bm{a}\sim\bm{p}}[\bm{u}^{i}(\bm{a})],\end{aligned}$$ under the mixed-strategy profile $\bm{p}=(p^{1},\ldots,p^{n})\in\prod_{i\in N}\Delta(A^{i})$. This defines a mixed-extension of the original game. The stability concept induced is called a mixed-strategy Nash equilibrium.

In MOGs, Pareto-Nash equilibria based on their original definition 