University of Kansas, Fall 2006
Philosophy 666: Rational Choice Theory
Ben Eggleston—eggleston@ku.edu
test on utility theory
(September 22, 2006)
Instructions:
- 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.
- 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.
- 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:
- 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:
# |
claim |
justification |
1 |
v P w |
given |
2 |
w P x |
given |
3 |
x P y |
given |
4 |
y P z |
given |
5 |
z P v |
given |
6 |
v P x |
1 and 2, transitivity condition |
7 |
v P z |
6 and 4, transitivity condition |
8 |
contradiction |
5 and 7, completeness 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.)
- 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.”
- 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.”
- 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.
- 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 ˝, and A2 (and
nothing else) would be selected by the rule of maximizing expected utility
using the principle of insufficient reason:
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.
- 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)’.
- 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.)
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.
- 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?
- 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?
- 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?
Instructions, revisited:
As stated in item 3 of the instructions, you do not need to turn in this
list of questions.