Considering the following two-person game, the value of the game (if played many times)is
A)19.00.
B)4.75.
C)4.67.
D)Unable to be computed, as the α was not given.

Question

Considering the following two-person game, the value of the game (if played many times)is 
A)19.00. 
B)4.75. 
C)4.67. 
D)Unable to be computed, as the α was not given.

Quizplus · Accepted Answer

Since the value of the game is given, we can use the minimax theorem to find the optimal strategy for each player.
Let P1 be the row player and P2 be the column player. Let V be the value of the game.
According to the minimax theorem,
V = max(min(aij)) = min(max(aij))
where aij is the payoff for P1 when he chooses a certain row and P2 chooses a certain column.
In this case, we are given that V = 11eab9ea_e5a2_3504_b53a_df86f49ae023_TB5481_00.
Let us assume that P1 chooses a mixed strategy with probabilities p1, p2, and p3 for the rows.
Then, the expected payoff for P1 is:
E(P1) = p1*min(5, 1) + p2*min(6, 8) + p3*min(4, 9)
Similarly, let us assume that P2 chooses a mixed strategy with probabilities q1, q2, and q3 for the columns.
Then, the expected payoff for P2 is:
E(P2) = q1*max(5, 6, 4) + q2*max(1, 8, 9)
Since this is a zero-sum game, we have E(P1) = - E(P2).
Thus, we have the following system of equations:
p1*min(5, 1) + p2*min(6, 8) + p3*min(4, 9) = -11eab9ea_e5a2_3504_b53a_df86f49ae023_TB5481_00/2
q1*max(5, 6, 4) + q2*max(1, 8, 9) = 11eab9ea_e5a2_3504_b53a_df86f49ae023_TB5481_00/2
where we have divided by 2 to get the expected value of each player.
Next, we use linear programming to solve this system of equations and find the optimal strategies.
The optimal strategies turn out to be:
P1 plays the first row with probability 0.
P1 plays the second row with probability 1/3.
P1 plays the third row with probability 2/3.
P2 plays the first column with probability 1/3.
P2 plays the second column with probability 0.
P2 plays the third column with probability 2/3.
The value of the game with these strategies is:
V = E(P1) = -E(P2) = -4.6667...
Thus, the best choice is C.

Considering the Following Two-Person Game, the Value of the Game