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

test on social choice theory

(December 13, 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 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. Let swf₁ be a social welfare function, and suppose that you are trying to prove that swf₁ satisfies the non-dictatorship 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, swf₁ generates some social ordering or other. Which one of the following is true? (This is one multiple-choice question, not four true/false questions. Just answer a, b, c, or d.)
   1. It is possible that you could show that swf₁ satisfies the non-dictatorship condition by pointing to just one profile (along with its corresponding social ordering determined by swf₁).
   2. You would have to point to at least two profiles (along with their corresponding social orderings determined by swf₁) in order to show that swf₁ satisfies the non-dictatorship condition; but you would not have to point to as many as c profiles (along with their corresponding social orderings determined by swf₁).
   3. You would have to point to at least c profiles (along with their corresponding social orderings determined by swf₁) in order to show that swf₁ satisfies the non-dictatorship condition; but you would not have to point to all n profiles (along with their corresponding social orderings determined by swf₁).
   4. You would have to point to all n profiles (along with their corresponding social orderings determined by swf₁) in order to show that swf₁ satisfies the non-dictatorship condition.
2. Let swf₂ be a social welfare function. What does it mean to say that swf₂ satisfies the citizens’-sovereignty condition? (This is one multiple-choice question, not four true/false questions. Just answer a, b, c, or d.)
   1. For some pair of distinct alternatives x and y, there is some profile for which swf₂ yields a social ordering that ranks x above y.
   2. For some pair of distinct alternatives x and y, each profile is one for which swf₂ yields a social ordering that ranks x above y.
   3. For each pair of distinct alternatives x and y, there is some profile for which swf₂ yields a social ordering that ranks x above y.
   4. For each pair of distinct alternatives x and y, each profile is one for which swf₂ yields a social ordering that ranks x above y.

Answer the following four questions (‘yes’ or ‘no’ will suffice for each), based on the assumptions that swf₃ is a social welfare function and that Q is the claim that swf₃ generates the preference orderings for society that are listed next to the following two profiles.

3. If Q is true, does that mean that swf₃ violates the unrestricted-domain condition?
4. If Q is true, does that mean that swf₃ violates the Pareto condition?
5. If Q is true, does that mean that swf₃ violates the positive-association condition?
6. If Q is true, does that mean that swf₃ violates the independence condition?
7. 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.
8. 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 multiple-choice question, not eight true/false questions. Just answer a, b, c, d, e, f, g, or h.)
   1. R entails that MR violates the non-dictatorship condition.
   2. R entails that MR violates the citizens'-sovereignty condition.
   3. R entails that MR violates the unrestricted-domain condition.
   4. R entails that MR violates the Pareto condition.
   5. R entails that MR violates the positive-association condition.
   6. R entails that MR violates the independence condition.
   7. R entails that MR violates two or more of the six conditions mentioned in answers a through f.
   8. R does not entail that MR violates any of the six conditions mentioned in answers a through f.
9. 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 multiple-choice question, not four true/false questions. Just answer a, b, c, or d.)
   1. the non-dictatorship condition, the Pareto condition, and the unrestricted-domain condition violates the independence condition
   2. the independence condition, the Pareto condition, and the unrestricted-domain condition violates the non-dictatorship condition
   3. the independence condition, the non-dictatorship condition, and the unrestricted-domain condition violates the Pareto condition
   4. the independence condition, the non-dictatorship condition, and the Pareto condition violates the unrestricted-domain condition
10. 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. 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.)

Instructions, revisited:

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