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Represent the situation as a game and find the optimal strategy for each player. State your final answer in the terms of the original question. A farmer grows apples on her 600-acre farm and must cope with occasional infestations of worms. If she refrains from using pesticides, she can get a premium for "organically grown" produce and her profits per acre increase by $900 if there is no infestation, but they decrease by $400 if there is. If she does use pesticides and there is an infestation, her crop is saved and the resulting apple shortage (since other farms are decimated) raises her profits by $700 per acre. Otherwise, her profits remain at their usual levels. No worms Worms 900 -400 No pesticides Pesticides 700 r = r2 = C1 =

Represent the situation as a game and find the optimal strategy for each player. State your final answer in the terms of the original question. A farmer grows apples on her 600-acre farm and must cope with occasional infestations of worms. If she refrains from using pesticides, she can get a premium for "organically grown" produce and her profits per acre increase by $900 if there is no infestation, but they decrease by $400 if there is. If she does use pesticides and there is an infestation, her crop is saved and the resulting apple shortage (since other farms are decimated) raises her profits by $700 per acre. Otherwise, her profits remain at their usual levels. No worms Worms 900 -400 No pesticides Pesticides 700 r = r2 = C1 =

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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**Game Theory and Farming Strategy** **Scenario:** A farmer grows apples on her 600-acre farm and must manage occasional worm infestations. If she refrains from using pesticides, she can earn a premium for "organically grown" produce, increasing her profits by $900 per acre if there is no infestation. However, if worms infest her crops, her profits decrease by $400 per acre. In contrast, using pesticides saves her crop from infestation, resulting in a profit increase of $700 per acre due to apple shortages. Without infestation, her profits remain constant. **Game Payoff Matrix:** | | No Worms | Worms | |----------------|----------|--------| | No Pesticides | 900 | -400 | | Pesticides | 0 | 700 | **Additional Information:** - \( r_1 = \) - \( r_2 = \) - \( c_1 = \) - \( c_2 = \) - \( v = 315 \) **Optimal Strategy Determination:** - Divide the farm into "pesticide-free" and "pesticide-use" zones. - Set aside 300 acres for pesticide-free apples. - Use pesticides on the other 300 acres. **Expected Profit Increase:** With this strategy, the expected profit increase is $1200 per acre.
Transcribed Image Text:**Game Theory and Farming Strategy** **Scenario:** A farmer grows apples on her 600-acre farm and must manage occasional worm infestations. If she refrains from using pesticides, she can earn a premium for "organically grown" produce, increasing her profits by $900 per acre if there is no infestation. However, if worms infest her crops, her profits decrease by $400 per acre. In contrast, using pesticides saves her crop from infestation, resulting in a profit increase of $700 per acre due to apple shortages. Without infestation, her profits remain constant. **Game Payoff Matrix:** | | No Worms | Worms | |----------------|----------|--------| | No Pesticides | 900 | -400 | | Pesticides | 0 | 700 | **Additional Information:** - \( r_1 = \) - \( r_2 = \) - \( c_1 = \) - \( c_2 = \) - \( v = 315 \) **Optimal Strategy Determination:** - Divide the farm into "pesticide-free" and "pesticide-use" zones. - Set aside 300 acres for pesticide-free apples. - Use pesticides on the other 300 acres. **Expected Profit Increase:** With this strategy, the expected profit increase is $1200 per acre.
For the game and mixed strategies, find the expected value. Let \( G = \begin{pmatrix} 5 & -7 \\ 4 & 2 \end{pmatrix} \), \( r = \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \end{pmatrix} \), and \( c = \begin{pmatrix} \frac{1}{2} \\ \frac{1}{2} \end{pmatrix} \).
Transcribed Image Text:For the game and mixed strategies, find the expected value. Let \( G = \begin{pmatrix} 5 & -7 \\ 4 & 2 \end{pmatrix} \), \( r = \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \end{pmatrix} \), and \( c = \begin{pmatrix} \frac{1}{2} \\ \frac{1}{2} \end{pmatrix} \).
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