The following is a payoff matrix for a two-person game. Decide whether the game is strictly determined. If it is, find the saddle point, value, and solution for the strictly determined game, (If an answer does not exist, enter DNE.)
\[
\left[\begin{array}{rrr}
-5 & 4 & 0 \\
-1 & 2 & 9 \\
-4 & 5 & -2
\end{array}\right]
\]
The game is strictly determined.
The game is not strictly determined.
saddle point $(r, c)=(\square)$
value
solution
Row
and Column
Answer
\(\boxed{\text{The game is strictly determined. The saddle point is at } (r, c) = (2, 1), \text{ the value of the game is } -1, \text{ and the solution for the strictly determined game is Row 2 and Column 1.}}\)
Step 1 ：The game is strictly determined if there is a saddle point in the payoff matrix. A saddle point is a point in the matrix which is both the smallest value in its row and the largest value in its column, or the largest value in its row and the smallest value in its column. To find the saddle point, we need to find the minimum of each row and the maximum of each column, and see if there is a common element. If there is, that's the saddle point. If not, the game is not strictly determined.
Step 2 ：The given matrix is \(\left[\begin{array}{rrr} -5 & 4 & 0 \ -1 & 2 & 9 \ -4 & 5 & -2 \end{array}\right]\). The minimum of each row is \([-5, -1, -4]\) and the maximum of each column is \([-1, 5, 9]\).
Step 3 ：The common element between the row minimums and column maximums is \(-1\). Therefore, the saddle point is \(-1\). This means the game is strictly determined.
Step 4 ：The saddle point is located at the position where the row minimum and column maximum intersect. We need to find the position \((r, c)\) of this saddle point in the matrix.
Step 5 ：The position of the saddle point is \((1, 0)\). This is the row and column index in the matrix where the saddle point is located.
Step 6 ：The value of the game is the value at the saddle point, which is \(-1\). The solution for the strictly determined game is the strategies that correspond to the row and column of the saddle point. In this case, the row strategy is 2 (index 1 + 1) and the column strategy is 1 (index 0 + 1).
Step 7 ：\(\boxed{\text{The game is strictly determined. The saddle point is at } (r, c) = (2, 1), \text{ the value of the game is } -1, \text{ and the solution for the strictly determined game is Row 2 and Column 1.}}\)
