8.9 **Game Theory and Agricultural Strategy**This exercise presents a game theory scenario for optimizing decision-making in an agricultural context. Here, a farmer is managing a 600-acre apple farm and faces the challenge of occasional worm infestations.### Decision Context:- **No Pesticides Strategy:** - **Outcome without Worms:** Profits increase by $900 per acre from premiums on "organically grown" produce. - **Outcome with Worms:** Profits decrease by $400 per acre due to infestations.- **Pesticides Strategy:** - **Outcome with Worms:** Profits increase by $700 per acre because crop loss is avoided, and the resulting apple shortage raises prices. - **Outcome without Worms:** Profits remain unchanged.### Payoff Matrix:| | No Worms | Worms ||----------------|----------|--------|| **No Pesticides** | $900 | -$400 || **Pesticides** | $0 | $700 |### Variables:- $ r_1 $, $ r_2 $, $ c_1 $, $ c_2 $ are game theoretical values to be calculated.- $ v = 315 $ represents the value decision boundary.### Strategy Optimization:The objective is to determine how to allocate the farm into "pesticide-free" and "pesticide-use" zones. The optimal division is:- **300 acres** for pesticide-free apple production.- **300 acres** for pesticide use.### Expected Outcome:By adopting this strategy, the farmer's expected increase in profits per acre is $1200.This model highlights how game theory can be applied in practical scenarios to maximize outcomes through strategic decision-making. 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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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 C1 = C2 = V = 315 How should she divide her farm into a "pesticide-free" zone and a "pesticide-use" zone? (Round your answers to two decimal places.) The farmer should set aside 300 acres for pesticide-free apples and use pesticide on the other 300 acres. What will be her expected increase in profits per acre with this strategy? This strategy will increase her expected profits by $ 1200 per acre.

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 C1 = C2 = V = 315 How should she divide her farm into a "pesticide-free" zone and a "pesticide-use" zone? (Round your answers to two decimal places.) The farmer should set aside 300 acres for pesticide-free apples and use pesticide on the other 300 acres. What will be her expected increase in profits per acre with this strategy? This strategy will increase her expected profits by $ 1200 per acre.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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