**Fair Division Activity
**

Prepared by:

Joseph Malkevitch

Department of Mathematics

York College (CUNY)

Jamaica, New York 11451

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Many situations arise where one or more items must be shared or divided up by a group of 2 or more people. What are fair procedures for doing this?

Here is an example designed to get you to think about the situation involved.

A rich uncle has left two identical twin sisters (as it happens both mathematics teachers) several items but has not specified who will get what. One way to deal with the situation is to sell the items and divide the money obtained equally but some of the items might fetch higher prices if they were not sold immediately but were perhaps sold in the future after a suitable buyer was found. Also, the items have considerable sentimental value to the twins who like the idea of keeping these things "in the family."

The items are: M = A collection of mathematics books, P = A large polyhedral puzzle made of of rare woods and E = an Escher print.

The twins have provided the executor of the estate with the value of each of the items to them in dollars, and have done this without taking into account the possible preferences of their twin. The results of this for M and P are shown in a valuation table, Table 1 below.

Question 1:

Develop an algorithm (method) that can be used to "fairly divide" the inheritance for the two items in Table 1 and which would apply to more than two items to divide, as well as a larger group of people trying to carry out a fair division of this type. Can you develop two different methods that seem appealing to you in this case?

Question 2:

Using the algorithm you developed for Question 1, extend what you did (if possible) to the case of the fair division problems in Tables 2 and 3 where there are more items and/or more people to divide the items using the valuations provided by the people involved in the fair division.

Table 1

Twin 1 | Twin 2 | |

M | 5400 | 1800 |

E | 3600 | 2700 |

Table 2

Twin 1 | Twin 2 | |

M | 5400 | 1800 |

E | 3600 | 2700 |

P | 1200 | 1500 |

Table 3

Twin 1 | Twin 2 | Twin 3 | |

M | 5400 | 1800 | 3000 |

E | 3600 | 2700 | 4200 |

P | 1200 | 1500 | 1800 |

Question 3:

In the tables above none of the items are given the same valuation by two different "players." What modification would be made in your procedure when there are ties in the evaluations by different players?