The following is the payoff matrix of a 2 * 2 zero-sumtwo-person game: 3 −4−3 1 (a) What randomized strategy should Player A use so asto minimize his maximum expected loss?(b) What randomized strategy should Player B use so asto maximize her minimum expected gain?(c) What is the value of the game?
The following is the payoff matrix of a 2 * 2 zero-sum
two-person game:
3 −4
−3 1
(a) What randomized strategy should Player A use so as
to minimize his maximum expected loss?
(b) What randomized strategy should Player B use so as
to maximize her minimum expected gain?
(c) What is the value of the game?
(a) To minimize his maximum expected loss, Player A should use a minimax strategy. This means that he should choose a strategy that minimizes the maximum possible loss that he could incur, assuming that Player B is also playing optimally.
To find the optimal strategy for Player A, we need to calculate the expected payoffs for each of his possible strategies, assuming that Player B will choose her optimal strategy in response. The two strategies available to Player A are to play the first row with probability p and the second row with probability 1-p. The expected payoff for each of these strategies is given by:
Expected payoff for playing first row: 3p - 3(1-p) = 6p - 3
Expected payoff for playing second row: -4p + 1(1-p) = -5p + 1
Player A wants to choose the value of p that minimizes the maximum possible expected loss. In other words, he wants to choose the value of p that makes the expected payoffs of the two strategies as close to each other as possible. To do this, we can set the two expected payoffs equal to each other and solve for p:
6p - 3 = -5p + 1
11p = 4
p = 4/11
Therefore, Player A should play the first row with probability 4/11 and the second row with probability 7/11.
(b) To maximize her minimum expected gain, Player B should also use a minimax strategy. This means that she should choose a strategy that maximizes the minimum possible gain that she could receive, assuming that Player A is also playing optimally.
Again, we need to calculate the expected payoffs for each of Player B's possible strategies, assuming that Player A will choose his optimal strategy in response. The two strategies available to Player B are to play the first column with probability q and the second column with probability 1-q. The expected payoff for each of these strategies is given by:
Expected payoff for playing first column: 3q - 3(1-q) = 6q - 3
Expected payoff for playing second column: -4q + 1(1-q) = -5q + 1
Player B wants to choose the value of q that maximizes the minimum possible expected gain. In other words, she wants to choose the value of q that makes the expected payoffs of the two strategies as far apart from each other as possible. To do this, we can compute the difference between the expected payoffs of the two strategies:
(6q - 3) - (-5q + 1) = 11q - 4
Player B wants to maximize the value of this difference. We can see that this difference is maximized when q = 4/11 (the same value that Player A chose in part (a)). Therefore, Player B should play the first column with probability 4/11 and the second column with probability 7/11.
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