Fairness and Equity: Notes 5

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

In light of the fact that fairness and equity problems are very complex, we will need to look at these issues from a variety of perspectives before we can actually "wade in" and try solving (resolving) specific equity problems. (Remember the parable of trying to describe an elephant to a blind person; an accurate description of the leg, of the ear, and the trunk of the elephant may not easily enable one to get a good idea of what an elephant really looks like.)

Consider, as an example, the issue of what to do as part of a divorce settlement between Mary and John to "divide" a house that they own jointly. Suppose we know no other information about Mary and John. How might we solve this question?

a. Sell the house and divide up what is gotten from the sale.

b. Both Mary and John move out and they rent the house and share the
income.

c. The house is divided into two apartments and each gets one of the two apartments.

d. A fair coin is flipped and one of them gets the house and the other gets nothing.

e. Mary uses the house from Jan. 1 to June 30 and the John uses the house from June 30 to Dec. 31. The following year they reverse the pattern of usage.

Note: If the house is not a "primary" dwelling but a "summer home," then a time sharing arrangement to use the house in alternating summers might work. Mary and John could share rent income if the summer home could be rented the rest of the time.
Note that the flipping the coin approach is a fair procedure in that each party has an equal chance of getting the house (if a fair coin is used) but the outcome is certainly not "equal." It is often important to distinguish between fair procedures and fair outcomes. Use of randomization is often done in sports as a fairness mechanism. A coin is flipped in football to see which team is to receive and which is to punt.

There is also the issue of what is called an information set. In this case, we pretended not to know much about the circumstances of John and Mary, but having more information will almost always alter the way we go about solving the equity problem.

For example, here is potentially useful additional information:

a. John and Mary earn the same income.

b. John comes from a wealthy family and shortly expects to inherit a lot of money.

c. Mary had more money in the bank at the time of the marriage and contributed more to the down payment of the house than John did.

d. John and Mary have no children.

(What additional complications arise if they do have children?)

A house is an example of something that can not be divided into two parts and retain its value. On the other hand, a chocolate bar could be so divided. It is convenient to have a taxonomy of how to classify fairness problems. We can distinguish between objects that are divisible and indivisible. Water and land are divisible. We can also distinguish between homogeneous objects and unhomogeneous ones. Water is homogeneous and divisible. A house is unhomogeneous and indivisible.

Comment: Money is not strictly speaking divisible because there is a smallest unit of currency. In America, it is the penny. Years ago in England the British pound was divided into 240 parts. Thus, the smallest price increase could be quite small. After the currency was decimalized the smallest part was 1/100 of a pound. The smallest price increase became much larger, contributing to inflation. The US is one of the few countries in the world that does not use the metric system. Recently we decimalized stock prices, so whereas in the past stocks went up or down 1/8, now they can go up or down a penny. This changes the nature of the market for stocks in a subtle way.

Here is another example, to illustrate some of the issues involved with how our view of solving equity questions changes with the information set available.

Situation:

Two communities I and II have been collectively allocated \$1,000,000 for public transportation?

How much should each community get?

Now suppose you are told the additional information shown below one fact at a time:

I; II

Population 230,000; Population 70,000

Per capita income \$40,000; Per capita income \$90,000

Area: 6 square miles; Area: 200 square miles

Public bus system; No public transportation

What other information might you want to collect in order to help you decide how to allocate the money?

Questions:

1. Suppose two people solve a fairness problem. How can one decide which of the two solutions is better?

2. Do people always assign the same value to the same things?

3. Do people always assign the same value to money?

Example: If your back aches, how large a bill on the ground does it take before you bend over to pick it up?

4. Is it possible that two people think they have been fairly treated because each has 1/2 of a "cake" but there is a different way of dividing the "cake" so that each person feels they have more than 1/2 the "cake?"