A farmer has a choice of growing wheat, barley, or rice. Her success will depend on the weather, which could be dry, average, or wet. Her payoff matrix is as follows. $\begin{array}{|l|lrcr|}
\hline & {\text { Weather }} \
\hline & & \text { Dry } & \text { Average } & \text { Wet } \
\text { Crop Choices } & \text { Wheat } & 20 & 20 & 10 \
& \text { Barley } & 10 & 15 & 20 \
& \text { Rice } & 10 & 20 & 20\\hline
\end{array}$ If the probability that the weather will be dry is 50%, the probability that it will be average is 10%, and the probability that it will be wet is 40%, what is the farmer's best choice of crop

Question

A farmer has a choice of growing wheat, barley, or rice. Her success will depend on the weather, which could be dry, average, or wet. Her payoff matrix is as follows. $\begin{array}{|l|lrcr|}
\hline & {\text { Weather }} \
\hline & & \text { Dry } & \text { Average } & \text { Wet } \
\text { Crop Choices } & \text { Wheat } & 20 & 20 & 10 \
& \text { Barley } & 10 & 15 & 20 \
& \text { Rice } & 10 & 20 & 20\\hline
\end{array}$   If the probability that the weather will be dry is 50%, the probability that it will be average is 10%, and the probability that it will be wet is 40%, what is the farmer's best choice of crop

Quizplus · Accepted Answer

The expected payoff for each crop can be calculated by multiplying the payoff for each weather condition by the probability of that condition and summing these products. For wheat, the expected payoff is $0.5 \times 20 + 0.1 \times 20 + 0.4 \times 10 = 14$. For barley, it is $0.5 \times 10 + 0.1 \times 15 + 0.4 \times 20 = 13$. For rice, it is $0.5 \times 10 + 0.1 \times 20 + 0.4 \times 20 = 14$. However, the question asks for the best choice, and since wheat has a consistently high payoff across more conditions (dry and average) compared to rice which drops significantly in a dry condition, wheat is considered the safer and thus better choice when considering risk and variability, despite the same mathematical expectation as rice.

A Farmer Has a Choice of Growing Wheat, Barley, or Rice