Finite Mathematics - Games and Decisions

Consider the following payoff matrix for a game:

$\begin{bmatrix} 300 & 200 & 700 \ 600 & 500 & 850 \400 & 350 & 250 \end{bmatrix}$

Give the value of the saddle point entry of the matrix, if it exists.

The matrix has no saddle point.

Explanation

If a payoff matrix has a saddle point, it must be both the minimum element of its row and the maximum element of its column. The minimum element of each row is shown in red:

$\begin{bmatrix} 300 & {\color{Red} 200} & 700 \ 600 & {\color{Red} 500} & 850 \400 & 350 & {\color{Red} 250} \end{bmatrix}$

Of the three, only 500 is the maximum of its column, making 500 the saddle point entry.

Consider the following payoff matrix for a game:

$\begin{bmatrix} 75 & 80 & 150 \ 40 & 50 & 90 \ 65 & 100 & 160\end{bmatrix}$

Give the value of the saddle point entry of the matrix, if it exists.

The matrix has no saddle point.

Explanation

If a payoff matrix has a saddle point, it must be both the minimum element of its row and the maximum element of its column. The minimum element of each row is shown in red:

$\begin{bmatrix} \textbf{{\color{Red} 75}} & 80 & 150 \ \textbf{{\color{Red} 40}} & 50 & 90 \ \textbf{{\color{Red} 65}} & 100 & 160\end{bmatrix}$

Of these three entries, only 75 is the maximum of its column. This is the correct choice.

Consider the following payoff matrix for a game:

$\begin{bmatrix} 2 & -1 & -4 \ - 1 & 3 & -3 \end{bmatrix}$

Give the value of the saddle point entry of the matrix, if it exists.

The matrix has no saddle point.

Explanation

If a payoff matrix has a saddle point, it must be both the minimum element of its row and the maximum element of its column. The minmum element of each row is shown in red:

$\begin{bmatrix} 2 &-1& \textbf{ {\color{Red} -4 }} \ -1 & 3 & \textbf{ {\color{Red} -3}} \end{bmatrix}$

Both saddle point entries are in Column 3, so we select the greater of the two, , as the saddle point entry.

Consider the following payoff matrix for a game:

$\begin{bmatrix} 60 & 80 & 90 \ 120 & 100 & 110 \ 160 & 40 & 130 \end{bmatrix}$

Give the value of the saddle point entry of the matrix, if it exists.

The matrix has no saddle point.

Explanation

If a payoff matrix has a saddle point, it must be both the minimum element of its row and the maximum element of its column. The minimum element of each row is shown in red:

$\begin{bmatrix} {\color{Red} 60} & 80 & 90 \ 120 & {\color{Red} 100} & 110 \ 160 & {\color{Red} 40} & 130 \end{bmatrix}$

Of the three, only 100 is the maximum element of its column. This is the saddle point entry.

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