HW1

Submit a single .pdf (or handcopy) for the written problems to me via email (or in person) by the date of the deadline (11:59 pm)

(1) Please show ALL of your work!!!!

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1) Your Decision: Think of a simple decision you face regularly and formalize it as a decision problem, carefully listing the actions, outcomes, and the preference relation. Then construct a payoff or utility function to the outcomes and draw the decision tree. (5 points)

2) Going to the Movies: There are two movie theaters in your neighborhood: Lincoln Regal 14, which is located one mile from your home, and Lincoln AMC 13, located three miles from your home. Each is showing three films. Regal 14 is showing Aquaman, Escape Room, and Bumblebee. ACM 13 is showing Spiderman, The Mule, and Dr. Seuss' The Grinch. Your problem is to decide which movie to go to.

(a) Imagine that you don't care about the distance and that your preferences for movies are alphabetic (i.e., you like Aquaman the most). Formulate this problem as a decision problem (listing the actions, outcomes, and the preference relation). Then construct a payoff or utility function to the outcomes using payoff values 1 through 6. (5 points)

(b) Now imagine that your car is in the shop and that the cost of walking each mile is equal to one unit of payoff. Update the payoffs to account for the cost. (5 points)

3) Fruit Trees: You have room for two fruit-bearing trees in your garden. The fruit trees that can grow in your garden are either apple, orange, or pear (or nothing). The cost of maintenance is $100 for an apple tree, $70 for an orange tree, and $120 for a pear tree. Your food bill will be reduced by $130 for each apple tree you plant, by $145 for each pear tree you plant, and by $90 for each orange tree you plant. You care only about your total expenditure for making any planting decisions.

(a) What is the set of possible actions and related outcomes (please list them)? (5 points)

(b) What is the payoff of each action/outcome? (5 points)

(d) Now imagine that the reduction in your food bill is half for the second tree of the same kind. (You like variety.) That is, the first apple tree still reduces your food bill by $130, but if you plant two apple trees your food bill will be reduced by $130 + $65 = $195, and similarly for pear and orange trees. What will a rational player choose now? (5 points)

4) Recreation Choices: A player has three possible activities from which to choose: going to a football game, going to a boxing match, or going for a hike. The payoff from each of these alternatives will depend on the weather. The following table gives the agent's payoff in each of the two relevant weather events:

Alternative Payoff if rain Payoff if shine

Football game 1 2

Boxing match 3 0

Hike 0 1

Let p denote the probability of rain.

(a) Is there an alternative that a rational player will never take regardless of p? (5 points)

(b) What is the optimal decision as a function of p? (5 points)

5) Drilling for oil: An oil drilling company must decide whether or not to engage in a new drilling venture before regulators pass a law that bans drilling on that site. The cost of drilling is $1 million. The company will learn whether or not there is oil on the site only after drilling has been completed and all drilling costs have been incurred. If there is oil, operating profits are estimated at $4 million. If there is no oil, there will be no future profits.

(a) Using p to denote the likelihood that drilling results in oil, draw the decision tree for this problem. (5 points)

(b) The company estimates that p = 0.6. What is the expected value of drilling? Should the company go ahead and drill? (5 points)

(c) To be on the safe side, the company hires a specialist to come up with a more accurate estimate of p. What is the minimum value of p for which it would be the company's best response to go ahead and drill? (5 points)

6) Juice: Bozoni is a Swiss maker of fruit and vegetable juice, whose products are sold at specialty stores throughout Western Europe. Bozoni is considering whether to add cherimoya juice to its line of products. “It would be one of our more difficult varieties to produce and distribute,” observes Johann Ziffenboeffel, Bozoni’s CEO. “The cherimoya would be flown in from New Zealand in firm, unripe form, and it would need its own dedicated ripening facility here in Europe.” Three successful steps are absolutely necessary for the new cherimoya variety to be worth producing. The industrial ripening process must be shown to allow the delicate flavors of the cherimoya to be preserved; the testing of the ripening process requires the building of a small-scale ripening facility. Market research in selected limited regions around Europe must show that there is sufficient demand among consumers for cherimoya juice. And cherimoya juice must be shown able to withstand the existing tiny gaps in the cold chain (the temperature-controlled supply chain) between the Bozoni plant and the end consumers (these gaps would be prohibitively expensive to fix). Once these three steps have been completed, there would be about €2,500,000 worth of expenses in launching the new variety of juice. A successful new variety will then yield profits, in expected present-value terms, of €42.5 million.

The three necessary steps can be done in parallel or sequentially, and in any order. Data about these three steps are given in the following table below.

“Probability of success” refers to how likely it is that the step will be successful. If it is not successful, then that means that cherimoya juice cannot be sold at a profit. All probabilities are independent of each other (i.e., whether a given step is successful or not does not affect the probabilities that the other steps will be successful). “Cost” refers to the cost of undertaking the step (regardless of whether it is successful or not).

Suppose Mr. Ziffenboeffel calls you and asks your advice about the project. In particular, he wants to know

(i) should he undertake the three necessary steps in parallel (i.e., all at once) or should he undertake them sequentially, and (5 points)

(ii) if sequentially, what’s the correct order in which the steps should be done? What answers do you give him? (5 points)