ng bananas is u preious one ($0.20, so.1O, and s hot affected by how many pieces
ID: 3303766 • Letter: N
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ng bananas is u preious one ($0.20, so.1O, and s hot affected by how many pieces of candy you ell l vice versa. is the set S1.25f possible actions you can take given your budget or $1.25? ecision tree that is associated with this decision problem Should you spend all your money with a rational choice argument. c. all your money at the cafeteria? Justify your an Now imagine that the price of a piecr many possible the price of a piece of candy increases to S0.30 e actions do you have? Does your answer to () change d. Recall the example in which you needed to gine that your payoff function is given by ea 14 Alcohol Consumption: Recall the examp -4a2 how much to drink. where o is a parameter that depends on your physique, Every person may have a ditferent value of , and it is known that in the population (1) the smallest your is 0.2; (2 ) the largest is 6; and (3) larger people have higher s than small people. aCan you find an amount that no person should drink? How much should you drink if your =1?If =4? 2 7 c. Show that in general smaller people should drink less than Should any person drink more than one 1-liter bottle of wine? people. d. 1.5 Buying a Car: You plan on buying a used car. You have $12,000, and yu not eligible for any loans. The prices of available cars on the lot are follows: Make, model, and year Toyota Corolla 2002 Toyota Camry 2001 Buick LeSabre 2001 Price $9,350 10,500 8,825 Honda Civic 2000 Subaru Impreza 2000 9.215 9,690Explanation / Answer
1.3, (a) You can buy any combination of bananas and candies that sum up to no more than $1.25. If we denote by (b, c) the choice to buy b bananas and c candies, then the set of possible actions is A = {(0, 0),(0, 1),(0, 2),(0, 3),(0, 4),(0, 5), (1, 0),(1, 1),(1, 2),(1, 3),(2, 0),(2, 1)}
b] For each choice you need to calculate the final net value. For example, if you buy one banana and 2 candies then you get 1.2 worth from the banana, 0.4 from the first candy and 0.2 from the second which totals 1.8. To this we need to add the $0.25 you have left (the cost was only $1) so the net final value you have is 2.05. In the figure below payoffs are written in bold. (0,0) 1.25 (0,1) 1.4 (0,2) 1.35 (0,3) 1.2 (0,4) 1 (0,5) 0.775 (1,0) 1.95 (1,1) 2.1 (1,2) 2.05 (1,3) 1.9 (2,0) 2.05 (2,1) 2.2
c}Yes. The highest net final value is from buying two bananas and one candy.
1.4] (a) Since the largest person’s utility is 0 when a = 1.5, no person should choose any a > 1.5.
(b) The optimal solution is obtained by maximizing the payoff function v(a) = a 4a 2 . To maximize such function, one needs to take derivative and equate it with zero. The derivative is 8a, thus: 8a = 0 a = 8 So for = 1, solution is 1 8 ; and for = 4, solution is 1 2 .
(c) This follows from the solution in part (b) above. For every type of person , the solution is a() = 8 which is increasing in , and larger people have higher values of .
(d) No. Even the largest type of person with = 6 should only consume a = 3 4 liters of wine.
1.5} (a) You have two “slots” that can be left empty, or have one of 3 possible trees planted in each slot. Hence, you have 10 possible choices4 . The outcomes will just be the choices of what to plant.
(b) To calculate the payoffs from each choice it is convenient to use a table as follows
: Choice Cost Food Savings Net Payoff nothing 0 0 0 one apple tree 100 130 30 one orange tree 70 90 20 one pear tree 120 145 25 two apple trees 200 260 60 two orange trees 140 180 40 two pear trees 240 290 50 apple and orange 170 220 50 apple and pear 220 275 55 pear and orange 190 235 45
(c) The optimal choice is two apple trees.
(d) An apple tree is still the best choice for the first tree, but now the second tree should be a pear tree
1.6} (a) For this decision maker choosing the hike is always worse than going to the football game, and he should never go on a hike.
(b) The expected payoffs from each of the remaining two choices are given by, v(F ootball) = p × 1 + (1 p) × 2 = 2 p, v(Boxing) = p × 3 + (1 p) × 0 = 3p, which implies that football is a better choice if and only if 2p 3p, or, p 1 2 , and boxing is better otherwise.
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