University of Kansas, Fall 2007
Philosophy 666: Rational Choice Theory
Ben Eggleston—eggleston@ku.edu

test on utility theory: answer key

(September 21, 2007)

Instructions:

1. Answer all of the following questions on the answer sheets provided. You can write on this list 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 test, do not leave the room before you are finished taking the test, and be sure to finish the test within this 50-minute testing period—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 testing period.
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 list of questions.

Questions:

1. Consider the following preferences:
   w P x
   x P y
   y P z
   z P w
   And consider the following proof (which may be entirely correct or may contain one or more mistakes) purporting to show that these preferences violate the transitivity condition:

What is the largest number n that makes the following sentence true? “Lines 1 through n are all correct and correctly justified.” (If the proof is entirely correct, give the number of the last line. If the proof has one or more mistakes, give the number of the line immediately preceding the one containing the first mistake.)

Answer: 7

2. Consider the following preferences. (They are numbered only for ease of reference; the numbers do not have anything to do with utilities.)
   1. b I a
   2. i P h
   3. h P c
   4. a I c
   5. g I h
   6. c P i
   7. d P b
   8. f P d
   If they violate the transitivity condition, give an answer of the form ‘violation of transitivity condition (n₁, n₂, n₃, . . .)’, where the numbers in parentheses refer to the line numbers of preferences that form a cycle. List the line numbers in order so that, if the preferences were written out, the option that come first alphabetically would be mentioned first, and the option that is dispreferred in the first preference is the one that is preferred in the second preference, and so on. If the stated preferences violate the transitivity condition, this is all you have to do, regardless of whether they satisfy or violate the completeness condition. If the stated preferences satisfy the transitivity condition but not the completeness condition, provide two or more (but as few as possible) independent preference orderings that represent them. If the stated preferences satisfy both the transitivity condition and the completeness condition, provide a preference ordering that represents the stated preferences.

Answer: violation of transitivity condition (6, 2, 3)

3. Suppose Miss Brooke has to take a trip, and she has to choose between driving and flying. If she travels by car and there is an accident, then her utility will be –50, and if she travels by car and there is no accident, then her utility will be 10. If she travels by plane and there is an accident (which, let us say, has a different probability than a car accident), then her utility will be –800, and if she travels by plane and there is no accident, then her utility will be 30. Set up and fill in a matrix for this situation so that probabilities summing to 1 could be assigned to the states of the world.

Answer:

	no car accident and no plane accident	no car accident, but plane accident	car accident, but no plane accident	car accident and plane accident
driving	10	10	–50	–50
flying	30	–800	30	–800

4. Suppose an option has a utility of 9 in state S₁, a utility of 1 in state S₂, and a utility of 6 in state S₃, with the probabilities of S₁, S₂, and S₃ being unspecified. Suppose also that, for Mr. Lydgate, this option’s value according to the optimism/pessimism rule is the same as its value according to the principle of maximizing expected utility using the principle of insufficient reason. What is Mr. Lydgate’s optimism index?

Answer:

optimism/pessimism value	=	value according to P.M.E. using P.I.R.
(α)(9) + (1 – α)(1)	=	(1/3)(9) + (1/3)(1) + (1/3)(6)
9α + 1 – α	=	9/3 + 1/3 + 6/3
8α + 1	=	16/3
8α	=	13/3
α	=	13/24

5. Suppose Mr. Casaubon prefers more money to less, but also prefers $1,000 to an option giving him a 1/4 chance at $4,100 and a 3/4 chance at $0. Give utility assignments for the three dollar amounts that make the principle of maximizing expected utility agree with Mr. Casaubon’s preferences.

There are infinitely many correct answers; here’s one:

u($0) = 0
u($1,000) = 10
u($4,100) = 11

6. Suppose Mr. Tyke thinks Mr. Farebrother has a 50-percent chance at winning a $10,000 prize for which Mr. Tyke is also competing. Mr. Tyke, thinking that he is sure to win if Mr. Farebrother is out of the contest, offers Mr. Farebrother $4,000 to withdraw from the contest. If Mr. Farebrother accepts Mr. Tyke’s offer, is Mr. Farebrother risk averse, risk neutral, or risk seeking, or do we not have enough information to say for sure? Explain your answer.

Answer:

We do not have enough information to say for sure. It might appear that Mr. Farebrother is risk averse, since he’d rather have $4,000 than an option that apparently has an expected monetary value of $5,000. But whether the option has this high an expected monetary value for Mr. Farebrother depends on what he thinks the probability of his winning the prize is, not what Mr. Tyke thinks that probability is. (If Mr. Farebrother thinks the probability is less than 40 percent, then his preference for the $4,000 is consistent with his being risk seeking, not indicative of his being risk averse.) And we have no basis for attributing to Mr. Farebrother any particular belief about the probability of his winning the prize.

7. If (1) Mr. Bulstrode’s insurance company sells an insurance policy to Mr. Vincy, (2) both parties have the same estimates of the probability and dollar amount associated with the insured event, and (3) each party regards the transaction as maximizing his or her or its expected utility, can we conclude that (4) Mr. Vincy’s utility function for money (e.g., a function of the form u($x) = f(x)) reflects greater risk aversion than is reflected by Mr. Bulstrode’s insurance company’s utility function for money? Why or why not?

Answer:

No, we cannot conclude that. The insurance company and Mr. Vincy might both have the same utility function, and the selling of the insurance policy could be expected-utility maximizing for the insurance company because of the large amount of its current assets, while the buying of the insurance policy could be expected-utility maximizing for Mr. Vincy because of the small amount of his current assets.

For questions 8 and 9, assume that u($50) = u($40) + x, u($60) = u($50) + y, and u($90) = u($60) + z, with x, y, and z being positive numbers.

8. Suppose Mr. Raffles prefers an option giving him a 1/3 chance at $90 and a 2/3 chance at $40 to an option giving him a 1/3 chance at $50 and a 2/3 chance at $60. What constraint(s) on x, y, and z (in addition to x > 0, y > 0, and z > 0) imply utility assignments for the four dollar amounts that make the principle of maximizing expected utility agree with these preferences? Any equation or inequality in your answer should have z on the left and x and/or y on the right. Show your work.

Answer:

(⅓)u($90) + (⅔)u($40)	>	(⅓)u($50) + (⅔)u($60)
u($90) + 2u($40)	>	u($50) + 2u($60)
u($40) + x + y + z + 2u($40)	>	u($40) + x + 2[u($40) + x + y]
3u($40) + x + y + z	>	u($40) + x + 2u($40) + 2x + 2y
x + y + z	>	3x + 2y
z	>	2x + y

9. Suppose Mr. Raffles prefers $60 to an option giving him a 1/2 chance at $90 and a 1/2 chance at $50. What constraint(s) on x, y, and z (in addition to x > 0, y > 0, and z > 0) imply utility assignments for the four dollar amounts that make the principle of maximizing expected utility agree with these preferences? Any equation or inequality in your answer should have z on the left and x and/or y on the right. Show your work.

Answer:

u($60)	>	(½)u($90) + (½)u($50)
2u($60)	>	u($90) + u($50)
2[u($40) + x + y]	>	u($40) + x + y + z + u($40) + x
2u($40) + 2x + 2y	>	2u($40) + 2x + y + z
2x + 2y	>	2x + y + z
y	>	z
z	<	y

10. Is it possible for all of Mr. Raffles’s preferences to satisfy the conditions referred to in the representation theorem? Why or why not?

Answer:

No, it is not. The answer to question 8 shows that if we try to make the principle of maximizing expected utility agree with the preferences stated there, then we must have z > 2x + y; and the answer to question 9 shows that if we try to make the principle of maximizing expected utility agree with the preferences stated there, then we must have z < y. Since x > 0, it is not possible for both of these inequalities to be true. So, the principle of maximizing expected utility cannot be made to agree with all of Mr. Raffles’s preferences. Based on this, we can infer from the representation theorem that it must be the case that Mr. Raffles’s preferences do not all satisfy the conditions referred to there.

Instructions, revisited:

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