every agent $i$ and action $a^{i}\in A^{i}$ if $a^{i}$ is played with positive probability $p^{i}(a^{i})>0$, then it holds that $$\bm{u}^{i}(a^{i},\bm{p}^{-i})\quad\in\quad{\mbox{EFF}}\left[\bm{u}^{i}(A^{i},\bm{p}^{-i})\right].$$

![Single-agent three-actions bi-objective game showing inconsistencies. (The coordinates correspond to the bi-objective valuation $(u_1,u_2)$.)[]{data-label="fig:mix"}](Figure3.png)

 

The three outcomes, $u(A)=\{(1,4),(2,2),(4,1)\}$, are depicted by black dots. With Def. 4, since the mixed outcomes are all convex-combinations of $\{(1,4),(2,2),(4,1)\}$, the Pareto-efficient mixed-strategies are here the convex-combinations of $\{(1,4),(4,1)\}$; and outcome $(2,2)$ is Pareto-dominated. Not every pure-strategy Pareto-Nash equilibrium is a mixed-strategy one, which is a severe inconsistency. Furthermore, since outcome $(2,2)$ is well balanced, it may also be decided with a non-null probability, e.g., if the agent’s utility is concave [@cobb1928theory], or if she is risk-averse [@kahneman1979prospect]. Our *revised* definition considers instead all the convex-combinations of the Pareto-efficient pure actions $\{(1,4),(2,2),(4,1)\}$.

This generalized definition connects in the single-objective case to a less know definition of Nash-equilibria (see [@papadimitriou2007agt], page 30, Theorem 2.1). In this alternative definition, each mixed strategy must be a mixture of pure-strategies that are best-responses. In other words, the support of each mixed strategy must be included in the set of pure-strategy best-responses. Furthermore, concerning existence, since this revised definition contains the former one, (which is guaranteed to exist) the new 