test on social choice theory

(November 30, 2007)

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 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.
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.

For questions 1–3, consider the following two profiles, and suppose social welfare function F specifies the indicated social preference orderings.

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).
1. These profiles, along with the indicated social orderings, entail that F satisfies condition U.
2. These profiles, along with the indicated social orderings, entail that F violates condition U.
3. These profiles, along with the indicated social orderings, neither entail that F satisfies condition U nor entail that F violates condition U.

Follow the same instructions as for question 1, replacing all references to condition U with references to condition ND.

Follow the same instructions as for question 1, replacing all references to condition U with references to condition I.

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 four-alternative, three-person 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.

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 thus-specified 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?

Write two three-alternative, four-person 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.

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.
1. Arrow’s theorem implies Cook’s theorem.
2. Cook’s theorem implies Arrow’s theorem.
3. Each theorem implies the other.
4. Neither theorem implies the other.

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 quasi-decisive 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 quasi-decisive for some alternative over another:

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.

Many people believe the following, which we’ll call claim L-choice: 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 L-choice? 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 L-choice.)

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