Collaboration policy: Collaboration is permitted, but each individual must try each problem alone first, and ultimately each student must write down and turn in their own solution to each problem set. Each student must fully understand and be able to explain his/her own answers.
Part I: Individual Preferences and Rationality.
Suppose that a person has complete preferences on a set of 3 alternatives {x, y, z}. In class, we showed that if that person's preferences are transitive, then she has a best choice in that set-i.e. transitivity is sufficient to insure that there is an alternative in the set that qualifies as a "rational choice". However, it turns out that transitivity is not necessary for her to have a best choice. In particular, it is possible for a person to have preferences over the three alternatives that are complete, not transitive, but still admit a best choice. The first of the following two problems asks you to give an example of a preference that shows this to be true.
Problem 1. Write down a preference over three alternatives {x, y, z} that shows the following statement to be false.
"If a person's preferences on a set consisting of three alternatives {x, y, z} are complete but not transitive, then that person does not have a best choice in that set."
That is, give a preference Ri over {x, y, z} that is complete, not transitive, but has a best choice.
Problem 2. We say that a person's preferences on a set of alternatives are "proper", if that person is not indifferent between any pair of alternatives. I.e. for every alternative {x,y}, either xPiy, yPix, or x and y are incomparable, but not xIiy. Give a proof of the following statement.
"If a person has complete and proper preferences on a set of three alternatives {x, y, z}, but that person's preferences are not transitive, then the person does not have a best choice in that set."
Problem 3. Construct an example with 4 alternatives, in which Ri is complete but not transitive, preferences are proper, so there is no indifference, and agent i has a best choice.
Part II: Social Choice and Voting Rules.
A group of 13 persons must choose between three alternatives {x=Republican, y=Democratic, z=Other}. Their preferences are as follows:
4 persons have preferences xPiyPiz
4 persons have preferences zPiyPix
2 persons have preferences yPizPix
2 persons have preferences xPizPiy
1 person has preferences zPixPiy.
In each of the following questions, you are asked to select a social preference rule under which a particular outcome is the "best social choice" (as that term was defined in class).
You are free to use one of the social choice rules introduced in class, i.e. majority rule, top-two rule, plurality rule, or the Borda count. You are also free to use some other rule (as long as that rule is both non-dictatorial and satisfies unanimity (recall these terms were defined in class). In each case, make sure to give an argument that verifies that the rule will indeed select the alternative intended.
Hint: For each of these questions at least one of the rules introduced in class -i.e. majority rule, the top-two rule, plurality rule, or the Borda count "works" to give the desired outcome. That is, although designing your own rule is acceptable, you don't need to do so.
Problem 4. You are Karl Rove. Find a social choice rule under which the Republican alternative x is the "best" social choice. Show that the rule will indeed select your desired Republican alternative.
Problem 5. You are a clever Democrat political strategist (I could not think of real life examples, sorry). Find a social choice rule under which the Democratic alternative y is the "best" social choice. Show that the rule will indeed select your desired Democratic alternative.
Problem 6. You have become a maverick reformer of American politics! Find a social choice rule under which the third alternative z is the "best" social choice. Show that the rule will indeed select your desired alternative.
Part III: Arrow's Theorem.
Suppose individuals {1, 2, 3} have one of the following two preference profiles over the set of alternatives X={x, y, z, w}:
Profile I:
individual 1: x P1 y P1 z P1 w
individual 2: y P2 z P2 x P2 w
individual 3: z P3 y P3 x P3 w
Profile II:
individual 1: y P1 x P1 z P1 w
individual 2: z P2 y P2 x P2 w
individual 3: x P3 y P3 z P3 w
Problem 7. Show that the choice by majority rule is transitive on each of these profiles. That is, show that with each profile, and any a,b,c in X, if a is chosen over b by majority rule, and b is chosen over c by majority rule, then a is chosen over c by majority rule.
Problem 8. Show or argue that majority rule satisfies Pareto Optimality (Hint: All agents have the same preference ordering between each alternative and w in each profile).
Problem 9. Show or argue that majority rule satisfies IIA.
Problem 10. Majority rule is non-dictatorial. Thus by the results of (7) through (9), majority rule satisfies rationality, Pareto-Optimality, IIA and is non-dictatorial