University of Kansas, Fall 2005
Philosophy 666: Rational Choice Theory
Ben Eggleston—eggleston@ku.edu

Final exam

(December 16, 2005)

Instructions:

1. Answer all of the following questions on the answer sheets provided. You can write on this sheet of questions, but credit will be awarded only for answers written on answer sheets.
2. Do not access any book, notebook, newspaper, calculator, computer, cell phone, or other possible source of inappropriate aid during the exam, do not leave the room before you are finished taking the exam, and be sure to finish the exam within the 90 minutes allotted—no credit will be given for any work done after you access any possible source of inappropriate aid, after you leave the room for any reason, or after the end of the 90 minutes allotted.
3. When you are finished, be sure your name is written on each of your answer sheets, and turn them in. You do not need to turn in this sheet of questions.

Questions:

1. (10 points) Either prove that t(x) = x + 3 is an ordinal transformation or give a counter-example showing that it’s not.

Question 2 refers to two of the three conditions defined as follows (in the glossary in Allingham’s Choice Theory):

• contraction condition: If you choose some item from a menu and this item remains available in a more restricted menu, then you also choose it from the restricted menu.
• expansion condition: If you choose some item in pairwise choices with every other item on the menu, then you choose it, though not necessarily alone, from the full menu.
• revelation condition: If you ever choose one item when a second item is available, then whenever you choose the second and the first is available, you also choose the first.

2. (5 points) Specify a set of no more than five choices violating the revelation condition but not the expansion condition (or, if there is no such set, say so).

Question 3 refers to the following decision table:

3. (5 points) Which act(s) would be selected by the maximin rule?
4. (5 points) Give an example of an ordinal transformation that is not a linear transformation, by giving a short list of utilities and a corresponding list that is an ordinal transformation of the first list but is not a linear transformation of the first list.
5. (5 points) Suppose Bobby has $1,000 to invest. He believes that one investment opportunity has a 3/5 chance of increasing his investment to $1,500 and a 2/5 chance of reducing it to $500. He believes that a second investment opportunity—a savings account insured by the U.S. government—has a probability p of increasing his investment to $1,200 and a probability 1 – p of reducing it to $0. What number must p be greater than in order for the second investment opportunity to have a higher expected monetary value (EMV) than the first?
6. (5 points) Suppose Helen is deciding whether buy (and ultimately sell) some property that might or not not contain oil that can be extracted and sold for a profit. Unfortunately, the current owners will not let her test the property for the presence of oil before buying, so Helen is deciding between (A₁) buying the property, looking for oil (possibly finding that there is none), extracting any oil that is there, and selling it and (A₂) just skipping the whole thing. Regarding strategy A₁, she believes (A) that there’s a 2/5 chance that there’s oil on the property and that she can buy the property, find any oil that might be there, extract the oil, sell the property, and end up with a total profit of $50,000, and (B) that there’s a 3/5 chance that there’s no oil on the property and that buying it, finding that there is no oil, and selling the property will cost a total of $10,000. She also believes (C) that strategy A₂ will result neither in a gain nor a loss, regardless of whether there is oil on the property. What is the EMV of strategy A₁?
7. (10 points) Now suppose Helen (from the previous question) learns that a geologist is willing to sneak onto the property without the consent of the current owners, test for oil, and tell Helen the results, so that she can buy if the test is positive and not buy if the test is negative. Helen believes A, B, and C from above; in addition, she believes this rogue geologist is astonishingly bad. Specifically, she believes (D) that if the property does contain oil that she can extract and sell (leading to that total profit of $50,000), there is only a 1/4 chance that the geologist will say so, and a 3/4 chance that he’ll mistakenly report the absence of oil, and (E) that if the property does not contain oil, then there is no chance that the geologist will say so, and a 100-percent chance that he’ll mistakenly report the presence of oil. How much should Helen be willing to pay the geologist for his services?
8. (10 points) Describe a game (1) each of whose prizes is finite but (2) whose EMV is infinite (8 points). (You do not have to show that its EMV is infinite.) If the game you describe is often referred to by a particular name, provide that name (2 points).
9. (5 points) Let F be the proposition that Lindsey will go to trial, let G be the proposition that Lindsey will impress her client, and let H be the proposition that Lindsey will either go to trial or impress her client (or both). (So, H is the disjunction of F and G.) Suppose Jimmy believes the probability of F is 30 percent, the probability of G is 40 percent, and the probability of H is 80 percent. If Eugene wants to make a Dutch book against Jimmy, what proposition(s) should he bet for and/or against? (Do not worry about the stakes that should be assigned to any proposition(s).)

Question 10 refers to the following table:

10. (10 points) Suppose there is some proposition p and some real numbers a and b such that Jimmy believes that the probability of p is a, that the probability of not p is b, and that these probabilities sum to some number less 1. Using the foregoing table, explain why the strategy of betting for p and for not-p is a way for Eugene to make a Dutch Book against Jimmy. (You do not have to give a proper proof, with numbered lines and a separate justification for each line. But your explanation should convey essentially the same information as a proper proof would.)
11. (5 points) Consider the lottery L(a, (3/4, X, B), (4/5, B, W)), where 0 ≤ a ≤ 1. What inequality must a satisfy in order for the lottery just specified to yield more than a 70-percent chance at B?
12. (5 points) Confining yourself to the basic prizes A, B, and C, along with any lotteries containing no other basic prizes than A, B, and C,
    1. give an example of a set of preferences violating the better-chances condition.
    2. give an example of a set of preferences violating the reduction-of-compound-lotteries condition.
13. (5 points) Assuming that B is defined as a prize than which there is none better, prove (using the rationality conditions) that there is no number a or basic prize x for which L(a, x, B) P L(a, B, B).
14. (10 points) Prove (using the rationality conditions) that if xPy and yPz and a > b, then L(a, x, z) P L(b, y, z).
15. (5 points) What is the positive linear transformation that converts {70, 80, 100} to {70, 75, 85}?
16. (5 points) Suppose Lucy prefers torts to contracts, contracts to civil procedure, and civil procedure to criminal procedure. Lucy prefer torts to contracts twice as strongly as she prefers contracts to civil procedure, which she prefers one third as strongly as she prefers civil procedure to criminal procedure. What is an interval scale that can be used to represent Lucy’s preferences?
17. (10 points) If we measure an agent’s preferences on an interval scale, does it make sense to say that the agent prefers one prize twice as strongly as another? (You may answer this question by explaining why positive linear transformations do, or why they do not, preserve the relationships among the utilities assigned to a couple of prizes that might lead one to think, in a specific case, that an agent prefers one prize twice as strongly as another.)
18. (10 points) Suppose Ellenor is an expected-utility maximizer who prefers more money to less, but who is risk averse. Suppose also that we are representing Ellenor’s preferences with a utility function, and that we begin by assigning the utilities 70 and 100 to the prizes $400 and $500. Suppose, finally, that we want to assign a utility to the prize $480. What is the range within which the utility we assign to $480 must fall?
19. (5 points) Suppose (A) that Rebecca prefers arguments to continuances and continuances to sentences and (B) that she prefers continuances to L(1/2, arguments, sentences). If we want to represent Rebecca's preferences using a utility function with the expected-utility property, and we begin by assigning a utility of 0 to sentences and a utility of 1 to continuances, what is the range (of the form x < u(arguments) < y) within which our assignment of a utility to arguments must fall?
20. (5 points) Suppose (A, same as in question 19) that Rebecca prefers arguments to continuances and continuances to sentences and (C) that she prefers L(2/5, arguments, sentences) to continuances. If we want to represent Rebecca's preferences using a utility function with the expected-utility property, and we begin by assigning a utility of 0 to sentences and a utility of 1 to continuances, what inequality (of the form u(arguments) > z) must our assignment of a utility to arguments satisfy?
21. (5 points) If A and B (from question 19) imply that u(arguments) falls within a range that is entirely below the lower bound on u(arguments) implied by A and C (from question 20), then is it possible that Rebecca is an expected-utility maximizer, or can we conclude that she is not? Why or why not?
22. (10 points) How is Newcomb’s problem a situation in which the dominance principle and the principle of expected-utility maximization give conflicting recommendations?

Instructions, revisited:

As stated in item 3 of the instructions, you do not need to turn in this sheet of questions.