Question

for each matrix, state te maximin for player 1(the row player). State the minimax for player 1(the column player). Is there a saddlepoint? does either player do better than their strategy?


[347]
[281]
[455]

Answer 1

Answer on Question #49664 – Math – Other

For each matrix, state the maximin for player 1 (the row player). State the minimax for player 1 (the column player). Is there a saddle point? Does either player do better than their strategy?

[347]

[281]

[455]

Solution

$$A = \left( \begin{array}{ccc} 3 & 4 & 7 \\ 2 & 8 & 1 \\ 4 & 5 & 5 \end{array} \right) \begin{array}{c} 3 \\ 1 \text{ row min} \\ 4 \end{array}$$

4 8 7 column max

The maximin for player 1 (the row player) is

$$\max \{3, 1, 4\} = 4.$$

The minimax for player 1 (the column player) is

$$\min \{4, 8, 7\} = 4.$$

$a_{31} = 4$ is the minimum of the $3^{\text{rd}}$ row, and the maximum of the $1^{\text{st}}$ column, since it is a saddle point. Thus it is optimal for player 2 to choose the third row, and for player 1 to choose the first column. The value of the game is 4, and choosing the $3^{\text{rd}}$ row and $1^{\text{st}}$ column gives optimal strategies for both players.

If either alone departs from saddle point strategy, he will suffer unnecessary loss.

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