8: Fair Division

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Maxie Inigo, Jennifer Jameson, Kathryn Kozak, Maya Lanzetta, & Kim Sonier
Coconino Community College

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Many students believe that mathematics hasn’t changed in several hundred years. However, the field of fair division is relatively young. Some of the methods discussed in this chapter were developed after the 1940s. This is an open field of study in mathematics. The methods we will look at do not always give the best possible answer but they are the best methods we have at this point in time.

Fair division tries to divide something in an equitable way. It can be used to divide up an estate, a jewelry collection, or a piece of land among heirs. Fair division can also be used to split up the assets of a business when a partnership is being dissolved. It can even be used by roommates to divide up the cleaning chores when the cleaning deposit is on the line.

8.1: Basic Concepts of Fair Division
8.2: Continuous Methods 1 - Divider/Chooser and Lone Divider Methods
The Divider/Chooser method and the Lone Divider method are two fairly simple methods for dividing a continuous set S. They can be used to split up a cake or to split up a piece of land. The Lone Divider method works for three or more players but works best with only three or four players. The Divider/Chooser method is a special case of the Lone Divider method for only two players.
8.3: Continuous Methods 2 - Lone Chooser and Last Diminisher Methods
8.4: Discrete Methods - Sealed Bids and Markers
There are two more fair division methods that deal with discrete objects. It two heirs have to split a house they cannot just cut the house in half. Instead we have to figure out a way to keep the house intact and still have both heirs feel like they received a fair share. The method of sealed bids is used for dividing up a small number of objects not necessarily similar in value. If there are many objects similar in value the method of markers can be used to find a fair division.
8.5: Exercises