# Math assignment: Fair Division

## Knaster Inheritance Procedure

The Knaster Inheritance Procedure allows for more than 2

parties and for assets which cannot be divided. Here's an

example if Carrie bids \$28,225 and Marti bids \$32,100 on

their aging parents car. The bigger bitter obviously should

get the car. But the winner has to pay out the loser a fair share.

The highest bidder places \$16,050 into a "kitty"; and the

loosing heir gets to withdraw 1/2 of the value he bid. This

leaves 1937.50 in the kitty and thats distributed to both of

them. In result at the end Marti gets the car plus \$968.75.

Carrie gets \$968.75 plus \$14,112.50.

## Lone Divider

the Lone Divider divides the goods into N pieces, N being the number of players. The Divider must be willing to accept any of the pieces. Other players indicate the pieces they are willing to accept. Let me give an example. Suppose Tiffany, Alice, Tim, and Lexi are dividing a piece of land. Lexi was chosen to be the divider through a coin toss. They had to 100% to give and they voted on 4 parts of the land. Tiffany voted p1=15% p2=30% p3=20% p4=35%. Alice put p1=30%, p2=35%, p3=10%, p4=25%. Tim voted p1=20%, p2=45%, p3=20%, p4=15%. Lexi put 25 on all. Its that way because the divider gets last unclaimed piece. The declarations are Tiffany gets p4, Alice gets p1, Tim gets P2, Lexi would get the last unclaimed peice which is p3.

## Last Diminisher

Definition of last-dimisher method-The last diminisher procedure is a procedure for

fair cake-cutting. It involves a certain heterogeneous and divisible resource,

such as a birthday cake, and n partners with different preferences over different

parts of the cake. It allows the n people to achieve a proportional division, i.e.,

divide the cake among them such that person receives a piece with a value of at

least 1/n of the total value according to his own subjective valuation.

Last-Diminisher method Lets say 4 salespeople are dividing

a state in four sales regions. They pull names from a hat to

decide play order. The first person picks a piece they like

then the next person states if he thinks its more or less than a

fair share. If they say its fair it stays the same if they say its

more than fair the amount of state is trimmed. If that person

trims it then that's their land now. The first person that

originally chose the first piece must chose another for

themselves. The last person gets the benefit because they

could just slightly trim the land and claim it four themselves.

So their first choices second choice has been taken. Now since

their are only two people left they did divider achievement.

So they flip a coin then the winner of that chooses the land they

want and the loser of the coin flip gets the left over land.

## Taking Turns

Taking turns is one of the simplest methods because it only involves two people and

distributing things equally. Say you have Cletus and Margret. They are getting a

divorce and have 4 large items they want to distribute. A mini van, their family

dog taco, a sports car and a freezer. They are given 100 points are they are asked

to rate the items with points on representing how important they are to them.

Cletus put 10 on the mini van, 20 on the family dog, 40 on the sports care and 30

for the freezer. Margaret put 40 on the mini van, 45 on the family dog, 5 on the

sports car and 10 on the freezer. In this problem you could tell who like what so

Cletus would end up getting the sports car and the freezer.

Margaret would get the mini van and the family dog.

## Bottom Up Strategy

Bottom Up strategy fair division scheme used for optimal strategy for rational players, each knowing the other's preferences. So basically if two people were splitting two items they would both rank them by preference and divide them evenly.