Represent the situation as a game and find the optimal strategy for each player.

Question

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 400-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 $800 if there is no infestation, but they decrease by $500 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 No pesticides Pesticides r1 = r2 = c1 = c2 = v = 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 acres for pesticide-free apples and use pesticide on the other acres. What will be her expected increase in profits per acre with this strategy? This strategy will increase her expected profits by $ per acre.

Explanation / Answer

Let x be the usual level of profits per acre.

The game

No Pesticide: [+800 -500

Pesticide: 0 +700]

represents the matrix of the increase of decrease of earnings per acre in the pesticide and no pesticide case, the farmers optimal mixed strategy is (a row operation)

r1 = (700 - 0)/((800+700) - (0-500)) = 700/2000 = 7/20

r2 = 1 - 7/20 = 13/20

c1 = (700 - (-500))/((800+700) - (0-500)) = 1200/2000 = 12/20

c2 = 1 - c1 = 8/20

v = (800*700 - 0*(-500))/((800+700) - (0-500)) = 560000/2000 = $280

The farmer should devote 7/20 of the farm to no pesticide zone and 13/20 of the farm to the pesticide zone. With the mixed strategy, the farmer's increase in profits will be:

1) No worms case:

(7/20)*(400)*($800) - (13/20)*(400)($0) = $112,000

2) Worms case:

(7/20)*(400)($-500) + (13/20)*(400)($700) = $112,000

The increase in profit of 40,000 $ represents a security level for the farmer.Thus this strategy will increase his profits by 280$ per acre and the total profits will be 400x + 112,000$, where x is the usual level of profits per acre.

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