million people every year [@world2011report] is hardly a creation of value[^1]. In this work, we assume that agents evaluate outcomes over a finite set of distinct objectives[^2]; hence, agents have vectorial payoffs. For instance, in the case of tobacco consumers, this slightly more informative model would keep the information on these three objectives [@conover2014smoking]: smoking pleasure, cigarette cost and consequences on life expectancy. In literature, this model was called games with vectorial payoffs, multi-objective games or multi-criteria games; and several applications were considered (see e.g. [@zeleny1975games; @wierzbicki1995multiple]). Indeed, behaviors are less assumptively modeled by a partial preference: the Pareto-dominance. Using Pareto-efficiency in place of best-response condition induces Pareto-Nash (PN) equilibria as the solution concept for stability, without even assuming that individuals combine the objectives in a precise manner. Pareto-Nash equilibria encompass the outcomes, even under unknown, uncertain or inconsistent preferences.

This paper more particularly addresses two unexplored issues.(1) The algorithmic aspects of multi-objective games have never been studied. (2) Also, the efficiency of Pareto-Nash equilibria has never been a concern.

*Related literature on mixed-strategies and similar strategy spaces.* Games with vectorial payoffs, or multi-objective games, were firstly introduced in the late fifties by Blackwell and Shapley [@blackwell1956analog; @shapley1959equilibrium]. The former 