the price of anarchy [@koutsoupias1999worst], we define a measurement of the worst-case efficiency of individualistic behaviors in games, compared to the optimum. In the multi-objective case, it is far from trivial, as worst-case equilibria and optima are not uniquely defined. In Section 6, we show how to compute the set of (worst) pure-strategy Pareto-Nash equilibria for several game structures, and provide algorithms to compute and approximate our multi-objective coordination ratio.[^3]

Preliminaries
=============

A *multi-objective game* (MO game, or MOG) is defined by the following tuple $\left(N,\{A^{i}\}_{i\in N},{{\mathcal{D}}},\{\bm{u}^{i}\}_{i\in N}\right)$:

-   The agents set is $N=\lbrace1,\ldots,n\rbrace$. Agent $i$ decides action $a^{i}$ in action-set $A^{i}$.

-   The shared list of objectives is denoted by ${{\mathcal{D}}}=\lbrace1,\ldots,d\rbrace$ and every agent $i\in N$ gets her payoff from function $\bm{u}^{i}:A=A^{1}\times\ldots\times A^{n}\rightarrow\mathbb{R}^{d}$ which maps every overall action to a *vector-valued* payoff; e.g., real $u_{k}^{i}(\bm{a})$ is the payoff of agent $i$ on objective $k$ for action-profile $\bm{a}=(a^{1},\ldots,a^{n})$.

![image](Figure1)

There are five shops (the nodes) in Ocean Shores: $N=\{1,\ldots,5\}$. Each shop/agent $i$ decides between two activities: $A^{i}=\{a^{i},b^{i}\}$; for instance: renting bikes or buggies, selling clams or fruit, etc. That is, agent $i$, in his payoff table $u^i$, decides row $a^{i}$ or $b^{i}$. The edges define neighborhoods around every agent. The 