social welfare $\bm{u}(\bm{a})=\sum_{i\in N}\bm{u}^{i}(\bm{a})$ is a vector-valued function $\bm{u}:A\rightarrow{\mathbb{R}}^d$ with respect to $d$ objectives. To study the efficiency of Pareto-Nash equilibria, we introduce:

-   set of *equilibria outcomes* $\quad{\mathcal{E}}\quad=\quad\bm{u}({\mbox{PN}})\quad(\subset{\mathbb{R}^d}),$

-   set of *efficient outcomes* $\quad{\mathcal{F}}\quad=\quad{\mbox{EFF}}[\bm{u}(A)]\quad(\subset{\mathbb{R}^d})$.

![Biobjective set of utilitarian outcomes $\bm{u}(A)\subset{\mathbb{R}}^2$ in Ocean Shores.](Figure2.png)

 

The utilitarian outcomes are a set of vectors, depicted above. Worst case equilibria and optima are not uniquely defined. The ratio of set of equilibria outcomes ${\mathcal{E}}$ ($\Diamond$) to set of efficient outcomes ${\mathcal{F}}$ ($\times$) would be a ratio of sets, which remains undefined. It would be crucial that such a definition keeps information for every objective. E.g., we want to remember that a car pollutes, or that a cigarette kills, not just that it makes some economic agents happy.

Numerous pure strategy Pareto-Nash equilibria exist.
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This section demonstrates the existence of pure strategy Pareto-Nash equilibria. Firstly, we write how the existence results from single-objective (SO) games can be retrieved in MO games. Secondly, we generalize the equilibria existence results of single-objective potential games to multi-objective potential games. Thirdly, we show that on average, numerous Pareto-Nash equilibria exist.

Reductions from MO games to SO games
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In the literature, most rationalities are constructed by means 