Consider the two player game described by the payoff matric below. L RU 3,3 2,xD 2,3 1,2What value must x NOT BE for the iterative elimination of strongly dominated strategies to lead to a single outcome? Write your answer as an integer (e.g. 5).
Question
Consider the two player game described by the payoff matric below. L RU 3,3 2,xD 2,3 1,2What value must x NOT BE for the iterative elimination of strongly dominated strategies to lead to a single outcome? Write your answer as an integer (e.g. 5).
Solution 1
The iterative elimination of strongly dominated strategies in a game involves removing strategies that are always worse than another strategy, regardless of what the other player does.
In the given payoff matrix, we can see that for Player 1, the strategy 'U' is always better than 'D' if x < 3. This is because the payoff for 'U' is 3 when Player 2 plays 'L' and 2 when Player 2 plays 'R', while the payoff for 'D' is 2 when Player 2 plays 'L' and 1 when Player 2 plays 'R'.
For Player 2, the strategy 'L' is always better than 'R' if x < 3. This is because the payoff for 'L' is 3 when Player 1 plays 'U' and 3 when Player 1 plays 'D', while the payoff for 'R' is x when Player 1 plays 'U' and 2 when Player 1 plays 'D'.
Therefore, for the iterative elimination of strongly dominated strategies to lead to a single outcome, x must not be less than 3. So, the integer value that x must NOT BE is 2.
Solution 2
The iterative elimination of strongly dominated strategies in a game involves removing strategies that are always worse than another strategy, regardless of what the other player does.
In the given payoff matrix, we can see that for Player 1, the strategy 'U' is always better than 'D' if x < 3. This is because the payoff for 'U' is 3 when Player 2 plays 'L' and 2 when Player 2 plays 'R', while the payoff for 'D' is 2 when Player 2 plays 'L' and 1 when Player 2 plays 'R'.
For Player 2, the strategy 'L' is always better than 'R' if x < 3. This is because the payoff for 'L' is 3 when Player 1 plays 'U' and 3 when Player 1 plays 'D', while the payoff for 'R' is x when Player 1 plays 'U' and 2 when Player 1 plays 'D'.
Therefore, for the iterative elimination of strongly dominated strategies to lead to a single outcome, x must not be less than 3. So, the integer value that x must NOT BE is 2.
Solution 3
To determine the value that x must not be for the iterative elimination of strongly dominated strategies to lead to a single outcome, we need to analyze the payoff matrix.
The matrix shows the payoffs for Player 1 and Player 2 in different strategies. The strategies for Player 1 are represented by the choices "L" and "RU", while the strategies for Player 2 are represented by the choices "D" and "U".
To start the iterative elimination of strongly dominated strategies, we look for any strategies that are always worse than another strategy, regardless of the opponent's choice. In this case, we can see that strategy "RU" for Player 1 is always better than strategy "L" because it has higher payoffs in both scenarios.
After eliminating the strongly dominated strategy "L" for Player 1, we are left with the following reduced payoff matrix:
L RU
D 2,3 1,2
Now, we need to consider the strategies for Player 2. Looking at the reduced matrix, we can see that strategy "D" for Player 2 is always worse than strategy "U" because it has lower payoffs in both scenarios.
Therefore, we eliminate the strongly dominated strategy "D" for Player 2, resulting in the following reduced payoff matrix:
L RU
U 2,x
To ensure that the iterative elimination of strongly dominated strategies leads to a single outcome, we need to have only one remaining strategy for each player. In this case, Player 1 has only one strategy left, which is "RU". However, for Player 2, the value of x must not be equal to 2. If x is equal to 2, then Player 2 would have two strategies left, "U" and "D", which would not result in a single outcome.
Therefore, the value that x must not be for the iterative elimination of strongly dominated strategies to lead to a single outcome is 2.
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