University of Kansas, Fall 2006
Philosophy 666: Rational Choice Theory
Ben Eggleston—eggleston@ku.edu
test on social choice theory: answer key
(December 13, 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 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:
 Let swf_{1} be a social welfare function, and suppose
that you are trying to prove that swf_{1} satisfies the
nondictatorship condition. Suppose that there are c citizens, that
there are n profiles for which a social ordering is needed (with n
> c), and that,
for each of those n profiles, swf_{1} generates some social ordering or
other. Which one of the following is true? (This is one multiplechoice
question, not four true/false questions. Just answer a, b, c, or d.)
 It is possible that you could show that swf_{1} satisfies the
nondictatorship condition by pointing to just one profile (along with its
corresponding social ordering determined by swf_{1}).
 You would have to point to at least two profiles (along with their
corresponding social orderings determined by swf_{1}) in order to show that swf_{1}
satisfies the nondictatorship condition; but you would not have to point to
as many as c profiles (along with their corresponding social orderings
determined by swf_{1}).
 You would have to point to at least c profiles (along with their
corresponding social orderings determined by swf_{1}) in order to show that swf_{1}
satisfies the nondictatorship condition; but you would not have to point to
all n profiles (along with their corresponding social orderings
determined by swf_{1}).
 You would have to point to all n profiles (along with their
corresponding social orderings determined by swf_{1}) in order to show that swf_{1}
satisfies the nondictatorship condition.
answer: a
 Let swf_{2} be a social welfare function. If swf_{2}
satisfies the citizens’sovereignty condition, then which one of the following is
true? (This is one multiplechoice question, not four true/false questions.
Just answer a, b, c, or d.)
 For some pair of distinct alternatives x and y, there is some profile for
which swf_{2} yields a social ordering that ranks x above y.
 For some pair of distinct alternatives x and y, each profile is one for
which swf_{2} yields a social ordering that ranks x above y.
 For each pair of distinct alternatives x and y, there is some profile for
which swf_{2} yields a social ordering that ranks x above y.
 For each pair of distinct alternatives x and y, each profile is one for
which swf_{2} yields a social ordering that ranks x above y.
answer: c
Answer the following four questions (‘yes’ or ‘no’ will suffice for each),
based on the assumptions that swf_{3} is a social welfare function and
that Q is the claim that swf_{3} generates the preference orderings
for society that are listed next to the following two profiles.
Jane 
Edward 
Adele 
Bertha 
society 

Jane 
Edward 
Adele 
Bertha 
society 
c 
b 
c 
d 
d 

d 
b 
c 
b 
b 
d 
a 
b 
b 
c 

c 
a 
a 
d 
c 
b 
d 
d 
c 
a 

a 
c 
d 
c 
a 
a 
c 
a 
a 
b 

b 
d 
b 
a 

 If Q is true, does that mean that swf_{3} violates
the unrestricteddomain condition?
Yes. (For the second profile, swf_{3} fails to rank d.)
 If Q is true, does that mean that swf_{3} violates
the Pareto condition?
Yes. (In the first profile, each citizen ranks b above a, but
in the social ordering, a is ranked above b.)
 If Q is true, does that mean that swf_{3} violates
the positiveassociation condition?
No. (The two profiles are not sufficiently similar for a violation of the
positiveassociation condition to be possible.)
 If Q is true, does that mean that swf_{3} violates
the independence condition?
Yes. (Every citizen is consistent in his or her ranking of b and
c in the two profiles, but the relative standing of these two alternatives
is reversed in the corresponding social orderings.)
 Suppose a social welfare function called MR (for ‘majority
rule’) says that society ranks x above y if and only if a
majority of the citizens rank x above y. Write a profile of
three citizens (Diana, Mary, and St. John) and three alternatives (a, b, and c) for
which MR says that society ranks a above b, ranks b above
c, and ranks c above a.
answer (one of several correct ones):
Diana 
Mary 
St. John 
a 
c 
b 
b 
a 
c 
c 
b 
a 
 Let R be the claim that for some profile(s), MR ranks a above b,
ranks b above c, and ranks c above a. Which one of
the following is true? (This is one multiplechoice question, not eight
true/false questions. Just answer a, b, c, d, e, f, g, or h.)
 R entails that MR violates the nondictatorship condition.
 R entails that MR violates the citizens'sovereignty condition.
 R entails that MR violates the unrestricteddomain condition.
 R entails that MR violates the Pareto condition.
 R entails that MR violates the positiveassociation condition.
 R entails that MR violates the independence condition.
 R entails that MR violates two or more of the six conditions mentioned
in answers a through f.
 R does not entail that MR violates any of the six conditions mentioned
in answers a through f.
answer: c. (Failures of transitivity are violations of the unrestricteddomain
condition.)
 What is the best way to complete the following sentence? “The general
strategy for proving the Pareto version of Arrow’s theorem is to prove that
when there are at least three alternative and at least two citizens, any
social welfare function satisfying __________.” (This is one multiplechoice
question, not four true/false questions. Just answer a, b, c, or d.)
 the nondictatorship condition, the Pareto condition, and the
unrestricteddomain condition violates the independence condition
 the independence condition, the Pareto condition, and the
unrestricteddomain condition violates the nondictatorship condition
 the independence condition, the nondictatorship condition, and the
unrestricteddomain condition violates the Pareto condition
 the independence condition, the nondictatorship condition, and the
Pareto condition violates the unrestricteddomain condition
answer: b
 Suppose you have proved that, for some x and some y, it is the case that
when Rosamond ranks x above y and everyone else in society ranks
y above x,
then the social ordering must rank x above y. And suppose you have also
proved that when Rosamond ranks x above a (where a is any alternative
distinct from x and y), then x must be ranked above a in the social
ordering, regardless of how everyone else ranks x and a. (Call this claim XA.)
And suppose you have also proved that when Rosamond ranks a above y, then
a must be ranked above y in the social
ordering, regardless of how everyone else ranks a and y. (Call this claim
AY.) Now suppose you
want to prove that when Rosamond ranks y above a, then y must be ranked
above a in the social ordering, regardless of how everyone else ranks
y and
a. (Call this claim YA.) You might start by setting up a profile in which
Rosamond ranks y above x, and x above a, and in which everyone else ranks
a
and y above x, but does not rank a and y in any particular order.
(Call this profile P_{1}.) Given such
a profile, explain how claim YA can be proved. (Your answer should mention
four distinct steps that one would take in the process of proving claim YA.)
answer:
 You show that because of the Pareto condition, y must be ranked
above x in the social ordering for P_{1}.
 You show that because of claim XA, x must be ranked above a
in the social ordering for P_{1}.
 You show that because of the unrestricteddomain condition (which
includes transitivity), the two things just shown imply that y must
be ranked above a in the social ordering for P_{1}.
 You show that because of the independence condition, x can be
dropped from P_{1}, meaning that y must be ranked above a
in the social ordering for any profile in which Rosamond ranks y
above a.
Instructions, revisited:
As stated in item 3 of the instructions, you do not need to turn in this
list of questions.