If a 2 × 2 (2 players with 2 actions each) normal form (matrix) game has no pure strategy Nash equilibria then: (A) Neither player has a dominant strategy. (B) Exactly one player has a dominant strategy. (C) At least one player has a dominant strategy. (D) Both players have dominant strategies. (E) None of the above is necessarily true
Question
If a 2 × 2 (2 players with 2 actions each) normal form (matrix) game has no pure strategy Nash equilibria then: (A) Neither player has a dominant strategy. (B) Exactly one player has a dominant strategy. (C) At least one player has a dominant strategy. (D) Both players have dominant strategies. (E) None of the above is necessarily true
Solution
(A) Neither player has a dominant strategy.
This is because a dominant strategy is a strategy that results in the highest payoff for a player no matter what the other player does. If a dominant strategy existed for a player, then that strategy would be a part of a pure strategy Nash equilibrium. However, the question states that no pure strategy Nash equilibria exist in the game. Therefore, it must be the case that neither player has a dominant strategy.
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