Can a Nash equilibrium fail to be Pareto optimal? Can a Pareto optimal outcome fail to be a Nash equilibrium? Justify your answers using the Prisoners' Dilemma game.

What will be an ideal response?

A Nash equilibrium need not be Pareto optimal, and a Pareto optimal outcome need not be a Nash equilibrium. In the Prisoners' Dilemma game, confessing is the dominant strategy for both players, so this outcome is the only Nash equilibrium. It is not, however, Pareto optimal, because both players get a better payoff in the situation where neither confesses. If at least one player chooses not to confess, the result is Pareto optimal because it is impossible to improve one player's welfare without harming the other player. However, these situations are not Nash equilibria, because one or both players is not choosing the dominant strategy of confessing.