Chapter 10.2, Problem 52E

Corporate farming For a one-time play (investment), you would split your investment proportional to the entries in your optimal strategy matrix. Assume that fate is a very clever player. Then if fate deviates from its optimal strategy, you know you will not do any worse than the value of the game, and you may do better.

$\begin{array}{l}\text{ Weather (fate)}\\ \begin{array}{ccc}\text{Wet}& \text{Normal}& \text{Dry}\end{array}\\ \begin{array}{cc}\begin{array}{l}\text{Corporate}\\ \text{farm}\end{array}& \begin{array}{l}\text{Wheat}\\ \text{Rice}\end{array}\end{array}\left[\begin{array}{c}-2\\ 7\end{array}\begin{array}{c}8\\ 3\end{array}\begin{array}{c}2\\ -3\end{array}\right]\end{array}$

The managers would like to determine the best strategy against the weather's “best strategy" to destroy them. Then, no matter what the weather does, the farm will do no worse than the value of the game and may do a lot better. This information could be very useful to the company when applying for loans.

Note: For each year that the payoff matrix holds, the farm can split the planting between wheat and rice proportional to the size of entries in its optimal strategy matrix.

(A) Find the optimal strategies for the farm and the weather, and the value of the game.

(B) What is the expected value of the game for the farm if the weather (fate) chooses to play the pure strategy “wet" for many years, and the farm continues to play its optimal strategy?

(C) Answer part (B), replacing “wet" with “normal."

(D) Answer part (B), replacing “wet" with “dry.”