University of Kansas, Fall 2006
Philosophy 666: Rational Choice Theory
Ben Eggleston—eggleston@ku.edu

test on utility theory: answer key

(September 22, 2006)

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 exam, do not leave the room before you are finished taking the exam, and be sure to finish the exam within this 50-minute class 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 this class 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:
   v P w
   w P x
   x P y
   y P z
   z P v
   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 sound, 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.)

The answer is 6.

2. Consider the following preferences. (They are numbered only for each of reference; the numbers do not have anything to do with utilities.)
   1. c P h
   2. f P h
   3. h P g
   4. g I b
   5. b P a
   6. f I a
   7. e P a
   8. d I c
   9. f P b
   10. a P d
   What is the largest number n that makes the following sentence true? “The preferences numbered 1 through n can all be represented by an ordered list.”

The answer is 5.

3. Consider the following preferences:
   c P b
   b I e
   c I f
   a P e
   d P f
   d I a
   And consider the following utility assignments, which are numbered for ease of reference:
   1. u(a) = 9
   2. u(b) = 8
   3. u(c) = 7
   4. u(d) = 6
   5. u(e) = 5
   6. u(f) = 4
   What is the largest number n that makes the following sentence true? “Utility assignments 1 through n could be part or all of the piecewise specification of an ordinal utility function for the preferences specified above.”

The answer is 2.

4. Suppose that, tomorrow afternoon, Hannah can either wash her car or play poker online. If she washes her car, then the outcome will be good if it doesn’t rain right away and bad if it does. If she plays poker online, the outcome will be great if she’s lucky and terrible if she’s unlucky. Set up and fill in a matrix for this situation.

Here’s a correct matrix:

	no rain and lucky	no rain and unlucky	rain and lucky	rain and unlucky
wash car	good	good	bad	bad
play poker online	great	terrible	great	terrible

5. Copy the following matrix to your answer sheet and fill it in with utilities so that A₁ (and nothing else) would be selected by the optimism/pessimism rule with an optimism index of ½, and A₂ (and nothing else) would be selected by the rule of maximizing expected utility using the principle of insufficient reason:

Here’s a correct way to fill in the matrix:

For questions 6 and 7, assume that Isaac prefers more money to less, and also prefers $150 to an option giving him a 40-percent chance at $500 and a 60-percent chance at $0.

6. If u($100) = u($0) + x, and u($500) = u($150) + y (with x and y being positive numbers), what constraint(s) on x and y (in addition to x > 0 and y > 0) imply utility assignments for the three dollar amounts that make the rule of maximizing expected utility agree with Isaac’s preferences?

In class, this question was corrected as follows: ‘u($100)’ should be ‘u($150)’. Here’s the answer:

u($150) = u($0) + x
u($500) = u($150) + y = u($100) + x + y
          EU($150) > EU(40-percent chance at $500 and 60-percent chance at $0)
             u($150) > (40/100)u($500) + (60/100)u($0)
           u($0) + x > (40/100)[u($0) + x + y] + (60/100)u($0)
   100[u($0) + x] > 40[u($0) + x + y] + 60[u($0)]
100u($0) + 100x > 40u($0) + 40x + 40y + 60u($0)
100u($0) + 100x > 100u($0) + 40x + 40y
                 100x > 40x + 40y
                   60x > 40y
                      x > (2/3)y
                      y < (3/2)x

7. Give utility assignments for the three dollar amounts that make the rule of maximizing expected utility agree with Isaac’s preferences. Also, show that those utility assignments do, indeed, make the rule of maximizing expected utility agree with Isaac’s preferences, by computing the relevant expected utilities that are entailed by those utility assignments. (You do not have to provide a formal proof, but you to have to show your calculations.)

Here is one correct answer:

u($0) = 0
u($150) = 1
u($500) = 2

EU($150) > EU(40-percent chance at $500 and 60-percent chance at $0)
   u($150) > (40/100)u($500) + (60/100)u($0)
            1 > (40/100)(2) + (60/100)(0)
            1 > 80/100

For questions 8 and 9, assume that u($20) = u($0) + x, and u($90) = u($20) + y, with x and y being positive numbers.

8. Suppose Julie prefers $20 to an option giving her a 30-percent chance at $90 and a 70-percent chance at $0. What constraint(s) on x and y (in addition to x > 0 and y > 0) imply utility assignments for the three dollar amounts that make the rule of maximizing expected utility agree with Julie’s preferences?

answer:

u($20) = u($0) + x
u($00) = u($20) + y = u($0) + x + y
          EU($20) > EU(30-percent chance at $90 and 70-percent chance at $0)
             u($20) > (30/100)u($90) + (70/100)u($0)
          u($0) + x > (30/100)[u($0) + x + y] + (70/100)u($0)
   100[u($0) + x] > 30[u($0) + x + y] + 70[u($0)]
100u($0) + 100x > 30u($0) + 30x + 30y + 70u($0)
100u($0) + 100x > 100u($0) + 30x + 30y
                 100x > 30x + 30y
                   70x > 30y
                       x > (3/7)y
                       y < (7/3)x

9. Suppose Julie prefers an option giving her a 20-percent chance at $90 and an 80-percent chance at $0 to a simple $20. What constraint(s) on x and y (in addition to x > 0 and y > 0) imply utility assignments for the three dollar amounts that make the rule of maximizing expected utility agree with Julie’s preferences?

answer:

u($20) = u($0) + x
u($00) = u($20) + y = u($0) + x + y
EU(20-percent chance at $90 and 80-percent chance at $0) > EU($20)
                                         (20/100)u($90) + (80/100)u($0) > u($20)
                             (20/100)[u($0) + x + y] + (80/100)u($0) > u($0) + x
                                           20[u($0) + x + y] + 80[u($0)] > 100[u($0) + x]
                                        20u($0) + 20x + 20y + 80u($0) > 100u($0) + 100x
                                                     100u($0) + 20x + 20y > 100u($0) + 100x
                                                                     20x + 20y > 100x
                                                                              20y > 80x
                                                                                 y > 4x

10. Suppose that, when deriving constraints such as the ones you derived in your answers to questions 8 and 9, you found that they could not both, or could not all, be satisfied by any particular assignment of utilities to options. What does the representation theorem imply about such preferences?

answer:

It implies that the preferences must violate one or more of the six conditions satisfaction of which guarantees that one’s preferences can be represented by a utility function that makes one’s preferences over options conform to the rule of maximizing expected utility.

Instructions, revisited:

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