Question

4. Consider a two-player game with strategy sets S1 = {α1, . . . , αm} and S2 = {β1, . . . , βn}.

(a) Suppose that (αi , βj ) is a Nash equilibrium. Is it possible that a strategy αj strictly dominates αi?

(b) Assume again that (αi , βj ) is a Nash equilibrium. Is it possible that another strategy αj weakly dominates αi?

Answer 1

(a) No.
If a strategy αj strictly dominates αi, then the player would not played αi. If he had not played αi, it would not have become the Nash equilibrium.

(b) Yes.
It is possible that αj weakly dominates αi, because for the other player, βj must have been a dominant strategy (strong or weak), which he chose to play. When both the players chose (αi , βj ), singly or along with the respective weakly dominated strategies, (αi , βj ) became the Nash equilibrium.