I am looking for an example of a finite 2 player game that has infinitely many Nash equilibria of which none are Pareto optimal.
I have found a few cases of infinitely many Nash equilibria, but I am struggling to find one which is non-Pareto optimal.
I'm wondering about the special case of all zeros in the bimatrix - would this be "non Pareto-optimal"?
Any help is appreciated! Thank you very much in advance!
Consider the following $3 \times 3$ game $$\begin{array}{c|ccc} & L & C & R \\ \hline T & 0,0 & 0,0 & 3,-1 \\ M & 0,0 & 0,0 & 2,-1 \\ B & -1,3 & -1,2 & 1,1\\ \end{array}$$ A strategy profile is a Nash equilibrium if and only if it puts positive probability only on $\{L,C\} \times \{T,M\}$; so there are infinitely many Nash equilibria, all with (expected) payoffs $(0,0)$.
However, the payoff $(0,0)$ is pareto-dominated by the payoff $(1,1)$ corresponding to the (non-equilibrium) strategy profile $(B,R)$.