University of Kansas, Fall 2007
Philosophy 666: Rational Choice Theory
Ben Eggleston—eggleston@ku.edu
test on social choice theory: answer key
(November 30, 2007)
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 test, do not
leave the room before you are finished taking the test, and be sure to finish
the test within this 50minute 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.
 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:
For questions 1–3, consider the following two profiles, and suppose social welfare function F specifies the
indicated social preference orderings.
profile 1 
society 

profile 2 
society 
A 
B 
C 
D 
A 
B 
C 
D 
c 
a 
b 
d 
b 
c 
b 
b 
a 
b 
d 
b 
a 
a 
a 
b 
d 
d 
b 
d 
b 
c 
d 
b 
d 
d 
a 
a 
d 
a 
a 
d 
c 
c 
c 
a 
c 
c 
c 
c 
 Indicate which one of the following claims is true, and explain why the
claim stated immediately below it is false (and if the correct answer is c,
explain why a is false).
 These profiles, along with the indicated social orderings, entail that F
satisfies condition U.
 These profiles, along with the indicated social orderings, entail that F
violates condition U.
 These profiles, along with the indicated social orderings, neither entail that
F satisfies condition U nor entail that F violates condition U.
Answer:
The answer is c. Claim a is false because even though F specifies a social
preference ordering for profiles 1 and 2, there is no guarantee that F
specifies a social preference ordering for every possible profile consisting
of four alternatives and four people.
 Follow the same instructions as for question 1, replacing all references
to condition U with references to condition ND.
Answer:
The answer is c. Claim a is false because the social preference ordering matches that of
person C for each of the two profiles, thereby preventing us from being sure
it is nondictatorial.
 Follow the same instructions as for question 1, replacing all references
to condition U with references to condition I.
Answer:
The answer is c. Claim a is false because we cannot be sure there are no
two profiles (1) in which a certain pair of alternatives has the same relative
ranking for each person in the two profiles but (2) whose corresponding social
preference orderings have those alternatives in different orders.
 Pairwise majority rule says that one alternative should be socially
preferred to another if and only if the number of people who prefer the first
alternative to the second is greater than the number of people who prefer the
second alternative to the first. Write a fouralternative, threeperson
profile showing that pairwise majority rule violates condition U. You can use
the following framework as a guide for how to proceed, but write the entirety
of your answer on an answer sheet rather than here.
Answer:
There are many correct answers to this question. Perhaps the simplest is
the profile of the Condorcet paradox with an extra alternative added on to the
end of each person’s preference ordering:
A 
B 
C 
a 
c 
b 
b 
a 
c 
c 
b 
a 
d 
d 
d 
 Suppose there are four alternatives and five people, resulting in
approximately 2 billion profiles. Suppose Wilma writes all of them out,
numbers them consecutively from 1 to 2,000,000,000 or so, and writes a social
preference ordering next to each one of them. Wilma calls the thusspecified
social welfare function F. An interlocutor asks Wilma whether F satisfies a
particular one of Arrow’s conditions on social welfare functions, but
background noise prevents you from hearing which condition the interlocutor
mentions. You do clearly hear Wilma’s reply: she says that F’s satisfaction of
that condition is conclusively proved simply by the social preference
orderings corresponding to profiles 68 and 287,348,837. Assuming that Wilma’s
reply is correct, is it possible to know which condition the interlocutor
must have asked her about?
If not, what are the conditions the interlocutor might have asked her about?
Answer:
No, it is not possible to know. The interlocutor might have asked her about
condition ND or condition CS.
 Write two threealternative, fourperson profiles and corresponding social
preference orderings such that, if a social welfare function specified those
social preference orderings for those profiles, we would have conclusive
evidence that it violated condition PA. You can use the following
framework as a guide for how to proceed, but write the entirety of your answer
on an answer sheet rather than here.
profile 1 
society 

profile 2 
society 
A 
B 
C 
D 
A 
B 
C 
D 






























Answer:
There are many correct answers; here is a simple one, in which b
gains standing in D’s preference ordering (and everything else stays the same)
but loses ground (to c) in the social preference ordering:
profile 1 
society 

profile 2 
society 
A 
B 
C 
D 
A 
B 
C 
D 
a 
a 
a 
a 
a 
a 
a 
a 
b 
a 
b 
b 
b 
b 
b 
b 
b 
b 
a 
c 
c 
c 
c 
c 
c 
c 
c 
c 
c 
b 
 You are familiar with some of the work of Arrow, including Arrow’s
theorem. Now suppose a person named Brown shows (correctly and
uncontroversially) that any social welfare function violating condition U or
condition I also thereby violates a new condition that Brown formulates and
that comes to be called condition B. Now suppose, further, that a person named
Cook proposes the following theorem, and calls it Cook’s theorem: if the
number of alternatives is at least 3 and the number of people is at least 2,
no social welfare function satisfies conditions B, ND, CS, and PA. Taking
Brown’s result as given, indicate
which of the following is true, and explain why.
 Arrow’s theorem implies Cook’s theorem.
 Cook’s theorem implies Arrow’s theorem.
 Each theorem implies the other.
 Neither theorem implies the other.
Answer:
The answer is a. Arrow’s theorem implies that if a social welfare
function satisfies conditions ND, CS, and PA, then it violates either condition
U or condition I (or both). Brown’s result, we are assuming, shows that if a
social welfare function violates either condition U or condition I (or both),
then it violates condition B. These statements imply that if a social welfare
function satisfies conditions ND, CS, and PA, then it violates condition B.
This, in turn, immediately implies Cook’s theorem. (But no analogous reasoning
is available to prove the converse.)
 Suppose that a particular social welfare function F is defined for a
situation of alternatives a, b, and x and individuals A,
B, C, and D, and let the set of all individuals (i.e., A, B, C, and D) be called S. Assuming that (1)
F satisfies conditions U, I, and P, (2) S is quasidecisive for a over
b, and (3) x P a in the social preference ordering
specified by F for the following profile, prove that there is a proper subset
of S that is quasidecisive for some alternative over another:
A 
B 
C 
D 
a 
x 
a 
x 
b 
a 
b 
a 
x 
b 
x 
b 
Answer:
This question turned out to have more correct answers than realized when I
wrote it. Here is the answer I had in mind:
 Given that S is quasidecisive for a over b, we have a
P b in the social preference ordering corresponding to this profile.
(You could also just apply condition P to arrive at a P b.)
 Given the justderived a P b, and the assumed x P
a, and transitivity (implicit in condition U), we have x P
b in the social preference ordering corresponding to this profile.
 Given that x P b in the social preference ordering
corresponding to this profile, condition I implies that x P b
in the social preference ordering corresponding to every profile in which B
and D rank x above b and A and C rank b above x,
regardless of the position of alternative a.
 This last claim means that the set consisting of B and D is
quasidecisive for x over b. B and D form a proper subset of S, so there is
a proper subset of S that is quasidecisive for x over b.
 Suppose a society has four alternatives (a, b, c, and
d) and two people (1 and 2). Alternatives a and b are in
the personal sphere of person 1, and alternatives c and d are in
the personal sphere of person 2. What is a profile that proves that no social
welfare function can satisfy conditions U, P, and L? You can use the following
framework as a guide for how to proceed, but write the entirety of your answer
on an answer sheet rather than here.
Answer:
There are several correct answers; here is the most straightforward one:
 Many people believe the following, which we’ll call claim Lchoice:
That certain states of affairs rather than others get chosen in accordance
with a person’s preferences is not a sufficient condition for that
person’s liberty to be respected; rather, respecting a person’s liberty
requires that certain states of affairs rather than others come about as a result of
that person’s consciously made choices. Does Sen agree or disagree with claim
Lchoice? What is an argument that he gives in support of his agreement or
disagreement? (Note that this question is not asking whether Sen’s
impossibility result holds if condition L is interpreted along the lines of
claim Lchoice.)
Answer:
Sen disagrees with claim Lchoice. The argument he gives in support of his
disagreement is that if someone is unconscious (e.g., after having been in an
accident) and cannot be involved in the making of a choice that affects him or
her, it is possible for his or her liberty to be respected if those who do
make the choice do so in a way that intentionally reflects his or her
preferences. (For more on this, see section VI of Sen’s “Liberty and Social
Choice.”)
Instructions, revisited:
As stated in item 3 of the instructions, you do not need to turn in this
list of questions.