Solve the Problem $2 \times 5$ Matrix Game Can Be Found as Follows: Obtain 5 Linear

question 41

Multiple Choice

Solve the problem.
-The optimal strategy for a $2 \times 5$ matrix game can be found as follows: obtain 5 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 5 linear functions on a t-z coordinate system. Then $v ( x ( t ) )$ is the minimum value of the 5 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 $z$ coordinates of points on the 5 lines. The highest point on the path $v ( x ( t ) )$ is M. Suppose that only the lines corresponding to columns 1 and 4 of matrix $A$ pass through the point $\mathrm { M }$ . What can be said about the optimal column strategy $\hat { y }$ ?

• A) \(\hat { y } = \left[ \begin{array} { l } \frac { 1 } { 2 } \\ 0 \\ 0 \\ \frac { 1 } { 2 } \\ 0 \end{array} \right]\)
• B) \(\hat { y } = \left[ \begin{array} { l } 0 \\ c _ { 2 } \\ c _ { 3 } \\ 0 \\ c _ { 5 } \end{array} \right]\) where \(c _ { 2 } + c _ { 3 } + c _ { 5 } = 1\)
• C) \(\hat { \mathrm { y } } = \left[ \begin{array} { l } \mathrm { c } _ { 1 } \\ 0 \\ 0 \\ \mathrm { c } _ { 4 } \\ 0 \end{array} \right]\)
• D) \(\hat { y } = \left[ \begin{array} { l } 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{array} \right]\)

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