Question

Problem 1:

For the following, please answer "True" or "False" and explain why.

1. In simultaneous move game where one player has a dominant strategy, then he is sure that he will get the best possible payoff in a Nash equilibrium.
2. In a simultaneous game where both players prefer doing the opposite of what the opponent does, a Nash equilibrium does not exist.
3. If neither firm has a dominant strategy, a Nash equilibrium cannot exist.

Answer 1

A.

False
Playing dominant strategy doesnt yield Nash equilibrium. The best outcome of each player depends on the best response of the other one.
Nash equilibrium occurs when both players defect. The best solution is to defect and other player to coperate. But this is not a Nash Equilibrium.

B. False

In this situation there a two Nash Equilibria.
In each situation player A and Player B doing the opposite strategy of one another. It is not easy to analyse which equilibrium situation is achieved without the cooperative moves from players.

C.
False

If both players have no dominant strategy, still they have strategy that can attain nash equilibrium. When both firms played the same strategy without cooperation, a Nash equilibrium can exist.
Any dominant strategy can always be nash equilibrium but not all nash equilibria are dominant strategy.