Solve the Problem $2 \times \mathrm { n }$ Matrix Game Can Be Found as Follows: Obtain

question 32

Multiple Choice

Solve the problem.
-The optimal strategy for a $2 \times \mathrm { n }$ matrix game can be found as follows: obtain $\mathrm { n }$ linear functions by finding the inner product of $x ( t ) = \left[ \begin{array} { c } 1 - t \\ t \end{array} \right]$ with each of the columns of the payoff matrix A. Graph the $\mathrm { n }$ linear functions on a $\mathrm { t } - \mathrm { z }$ coordinate system. Then $v ( \mathrm { x } ( \mathrm { t } ) )$ is the minimum value of the $\mathrm { n }$ linear functions which will be seen on the graph as a polygonal path. The $z$ -coordinate of any point on this path is the minimum of the corresponding $\mathrm { z }$ coordinates of points on the $\mathrm { n }$ lines. The highest point on the path $v ( x ( t ) )$ is M. Suppose that $M$ has coordinates $( a , b )$ . What information is given by these coordinates?

• A) The optimal strategy for \(C\) is \(\left[ \begin{array} { c } a \\ 1 - a \end{array} \right]\) and the value of the game for \(C\) is \(b\).
• B) The optimal strategy for \(R\) is \(\left[ \begin{array} { c } 1 - a \\ a \end{array} \right]\) and the value of the game for \(R\) is \(b\).
• C) The optimal strategy for \(R\) is \(\left[ \begin{array} { c } 1 - a \\ a \end{array} \right]\) and the optimal strategy for \(C\) is \(\left[ \begin{array} { c } 1 - b \\ b \end{array} \right]\).
• D) The optimal strategy for \(R\) is \(\left[ \begin{array} { c } a \\ 1 - a \end{array} \right]\) and the value of the game for \(R\) is \(b\).

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Definitions:

Population Proportion

The fraction or percentage of a population that possesses a particular attribute or characteristic.

Sampling Distribution

The framework of distributed probabilities for an established statistic, gathered from random sampling.

Approximately Normal

Describing a distribution that resembles the bell-shaped curve of a normal distribution, especially when certain statistical conditions are met.

Population Proportion

The ratio of members in a group or category to the total population under study.