Which of the following statements is true? I. If a two-player game has a strictly dominant strategy for one player, then it must have a strictly dominant strategy for the other player. II. If a strategy is strictly dominant, then it is also weakly dominant. III. A Subgame Perfect Nash Equilibrium (SPNE) is always a Nash Equilibrium in the normal form of the game Group of answer choices I and II are correct I and III are correct II and III are correct I, II and III are correct None of the other answers are correct
Question
Which of the following statements is true?
I. If a two-player game has a strictly dominant strategy for one player, then it must have a strictly dominant strategy for the other player.
II. If a strategy is strictly dominant, then it is also weakly dominant.
III. A Subgame Perfect Nash Equilibrium (SPNE) is always a Nash Equilibrium in the normal form of the game
Group of answer choices
I and II are correct
I and III are correct
II and III are correct
I, II and III are correct
None of the other answers are correct
Solution 1
The correct answer is "II and III are correct".
Explanation:
I. This statement is false. A strictly dominant strategy for one player does not necessarily mean there is a strictly dominant strategy for the other player. A strictly dominant strategy is one where one strategy is better than another, regardless of what the other player does. It's possible for one player to have a strictly dominant strategy while the other player does not.
II. This statement is true. If a strategy is strictly dominant, then it is also weakly dominant. A strictly dominant strategy is always better than other strategies, no matter what the other player does. A weakly dominant strategy is at least as good as any other strategy, and sometimes better, depending on what the other player does. Therefore, a strictly dominant strategy, which is always better, is also weakly dominant.
III. This statement is true. A Subgame Perfect Nash Equilibrium (SPNE) is always a Nash Equilibrium in the normal form of the game. A Nash Equilibrium is a set of strategies where no player can do better by unilaterally changing their strategy, given the other players' strategies. A SPNE is a more specific type of Nash Equilibrium that also holds in all subgames of the game. Therefore, a SPNE, which holds in all subgames, also holds in the overall game and is a Nash Equilibrium.
Solution 2
The correct answer is "II and III are correct".
Here's why:
I. This statement is false. A strictly dominant strategy for one player does not necessarily mean there is a strictly dominant strategy for the other player. A strictly dominant strategy is one where one strategy is better than all other strategies, no matter what the other player does. It's possible for one player to have such a strategy while the other player does not.
II. This statement is true. If a strategy is strictly dominant, then it is also weakly dominant. A strictly dominant strategy is always better than all other strategies, no matter what the other player does. A weakly dominant strategy is one that is at least as good as all other strategies, no matter what the other player does. Therefore, a strictly dominant strategy, which is always better, is also at least as good.
III. This statement is true. A Subgame Perfect Nash Equilibrium (SPNE) is always a Nash Equilibrium in the normal form of the game. A Nash Equilibrium is a set of strategies where no player can unilaterally improve their outcome by changing their strategy, given the other players' strategies. A SPNE is a more specific type of Nash Equilibrium that also holds in all subgames of the game. Therefore, a SPNE, which holds in all subgames, also holds in the overall game.
Similar Questions
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
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