I have started studying the Pareto efficiency notion in Game theory. The definition I am familiar with is this:

Strategy profile $\mathbf{s}$ Pareto dominates strategy $\mathbf{s}'$ if for all $i\in\mathcal{N}$, $u_i(\mathbf{s})\geq u_i(\mathbf{s}')$, and there exists some $j\in\mathcal{N}$ for which $u_j(\mathbf{s})>u_j(\mathbf{s}')$. Strategy profile $\mathbf{s}$ is Pareto optimal, or strictly Pareto efficient, if there does not exist another strategy profile $\mathbf{s}'\in S$ that Pareto dominates $\mathbf{s}$.

I am interested in finite normal form games, for example, the $n$-player Prisoner's dilemma. Clearly, for $n=2$ we have three Pareto outcomes and it's not too difficult to derive them.

But my concern is with a large number of players and non-constant utilities. How do can we find the Pareto efficiency outcomes? If you are aware of a paper that does this, please share it with me.

Is there a better, more efficient way to compute the efficiency of different outcomes of a game? Maybe the price of anarchy?

I'd appreciate any help or hint. Thank you.


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