# Pareto Efficiency Outcomes in Games

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 we check the Pareto efficiency outcomes?

Is there a better, more efficient way to compute the efficiency of different outcomes of a game? Maybe the price of anarchy? I am not sure.

I'd appreciate any help or hint. Thank you

• Do you mean to find the entire set of Pareto efficient outcomes using an algorithm? Feb 14 '19 at 3:57
• @HerrK. Yes. An algorithm that will work for an $n$-player Prisoner's dilemma game, where $n\geq 2$. Feb 14 '19 at 18:01
• The Pareto set of payoff profiles is the "northeast" (in the 2-dimensional sense) boundary of the convex hull of all possible payoff profiles. Thus you should in principle be able to use a convex hull algorithm to map out such a set. But how exactly this can be done is way out of my league, as I'm not a computer scientist. You'd probably have better luck asking on stackoverflow.com or cs.stackexchange.com Feb 14 '19 at 19:35
• @HerrK.Do you know any reference for this type of thing? Textbook or paper. I am more interested in the theoretical aspect. Feb 15 '19 at 5:35
• What do you mean by "this type of thing"? Feb 15 '19 at 17:59