Consider the three player normal form game described by the payoff matrices below. Players 1 and 2 have two choices each while player 3 has three choices. Player 3 choices correspond to each of the three matrices. The payoffs in each cell are the payoffs of player 1, player 2, and player 3 respectively.Player 3 chooses U: L Rl 1,0,1 0,1,1r 0,1,1 1,0,0Player 3 chooses M: L Rl 1,1,1 -1,-1,1r -1,-1,1 1,1,1Player 3 chooses D: L Rl 0,1,0 1,0,1r 1,0,1 0,1,1Which of the following statements is true? M is a strongly dominant action for player 3. U is a weakly dominant action for player 3. M is a weakly dominant action for player 3. D is a weakly dominant action for player 3.
Question
Consider the three player normal form game described by the payoff matrices below. Players 1 and 2 have two choices each while player 3 has three choices. Player 3 choices correspond to each of the three matrices. The payoffs in each cell are the payoffs of player 1, player 2, and player 3 respectively.Player 3 chooses U: L Rl 1,0,1 0,1,1r 0,1,1 1,0,0Player 3 chooses M: L Rl 1,1,1 -1,-1,1r -1,-1,1 1,1,1Player 3 chooses D: L Rl 0,1,0 1,0,1r 1,0,1 0,1,1Which of the following statements is true? M is a strongly dominant action for player 3. U is a weakly dominant action for player 3. M is a weakly dominant action for player 3. D is a weakly dominant action for player 3.
Solution
To determine which action is dominant for player 3, we need to compare the payoffs for each action (U, M, D) in all possible scenarios.
A strategy is said to be "strongly dominant" if, regardless of what any other players do, the strategy earns a player a higher payoff than any other. A strategy is "weakly dominant" if it is at least as good as any other strategy, given what the other players do, and there is at least one set of other players' actions for which it is the best response.
- If player 3 chooses U, the payoffs are 1, 1, and 0.
- If player 3 chooses M, the payoffs are 1, 1, and 1.
- If player 3 chooses D, the payoffs are 0, 1, and 1.
Comparing these, we can see that M is not a strongly dominant action for player 3 because it does not always result in a higher payoff than the other actions.
However, M is a weakly dominant action for player 3. This is because choosing M always gives at least as high a payoff as choosing U or D, and there are some scenarios (specifically, when players 1 and 2 choose L and r, or R and l) where M gives a higher payoff.
Therefore, the statement "M is a weakly dominant action for player 3" is true.
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