Majority Rule Equilibrium (Arrow"s Theorem)

Majority Rule Equilibrium (by Gary Miller, Lecture on Arrow)
I. Suppose our objective is to satisfy majority rule in deciding among three or more alternatives. Consider the example below where three groups vote on three alternatives of raising contributions for a public good:
Group I (45%) Group II (25%) Group III (30%)
1st Choice income tax property tax sales tax
2nd Choice property tax sales tax property tax
3rd Choice sales tax income tax income tax

A.
1. Now, if all three alternatives are voted on at the same time, selection of the alternative with the most votes (plurality) would result in a social choice of (select one):
a. Income Tax
b. Property Tax
c. Sales Tax

2. Would this social choice satisfy majority rule? Why so, or why not?

B.
1. Now, consider using plurality rule with a runoff (in which all alternatives are voted on at the same time, and the top two alternatives face each other in the second round of the election). Selection of an alternative using plurality rule with a runoff would result in a social choice of (select one): a. Income Tax
b. Property Tax
c. Sales Tax

2. Would this social choice satisfy majority rule? Why so, or why not?

C. Now, consider using a binary agenda; typically, majority rule decision-making operates by means of a binary agenda. Binary agenda is a means by which alternatives are voted on two at a time. In the example above there is an alternative that would beat every other alternative in a two-way vote. (Hint: if property tax is run against income tax, and the winner of that two-way race runs against the sales tax, which alternative wins both races? Now if sales tax is run against income tax and the winner of that two-way race runs against property tax, which alternative wins?) The alternative that beats every other alternative in a two-way vote is called the “majority rule winner” or “majority rule equilibrium.” What is the alternative that is a winner no matter what binary agenda is used? a. Income Tax
b. Property Tax
c. Sales Tax

D. What do we learn from observing the outcomes in A, B, and C above? (Explain)

II. When a majority equilibrium exists, the majority rule is a fair and meaningful way for groups to make decisions. It weighs every vote equally, and the outcome chosen is a meaningful expression of the collective will. The particular procedure (binary agenda) makes no difference in the outcome.

The problem is that there is not always a majority rule equilibrium. Consider the following example:

Group I (45%) Group II (25%) Group III(30%)
1st Choice income tax property tax sales tax
2nd Choice property tax sales tax income tax
3rd Choice sales tax income tax property tax

These preferences are just like the earlier preferences except now the sales tax supporters have income tax as a second choice instead of property tax.

E. Now, is there an alternative that beats every other alternative in a two-way race (that is, using binary agenda)?
a. Yes
b. No
F. How would you know? (Show the result through the use of binary agenda)
G. How is the final outcome determined
How often are the preferences such, that a “real” majority rule equilibrium exists? The answer to this question is “almost never.”