Question

This question is from " Networks, Crowds, and Markets: Reasoning about a Highly Connected World" textbook in chapter 6 Say whether the following claim is true or false, and provide a brief (1-3 sentence) explanation for your answer. Claim: If player A in a two-person game has a dominant strategy sA, then there is a pure strategy Nash equilibrium in which player A plays sA and player B plays a best response to sA.

Answer 1

If either player had a strictly dominant strategy, then she would have to play that strategy in both Nash equilibria in pure strategies; you can’t play a strictly dominated strategy in a Nash equilibrium. So if player i has a strictly dominant strategy, then she must play it in both Nash equilibria. But since the other player, j, isn’t indifferent between any two outcomes, j cannot be indifferent between the two Nash equilibria in pure strategies. Since j strictly prefers one Nash equilibrium to the other, and since i plays the same strategy in both NE, j would deviate from the NE that she likes less to the one she likes more. Thus the NE that j likes less cannot be a NE, and we’ve reached a contradiction. We conclude that neither player can have a strictly dominant strategy.