1. Fair Division Algorithms
The Continuous Case

2. Cake Division Problem
Let's return to the problem of fairly dividing a cake.
Because a cake is continuously divisible since it can be divided into any number of parts.
Remember that this is not true of discrete objects in an estate or seats in a legislature.

3. Notes About Fair Division
Remember that the resolution of a problem by an outside authority often does not result in a solution that is fair in the eyes of both individuals.
Additionally, the appeal to a random event such as a coin toss does not result in a division that is fair in the eyes of both parties.

4. Fairly Dividing the Cake
Remember that the division of a cake between two people is considered fair, only if each person feels that he or she has received at least half of the cake.
An acceptable solution to the problem of dividing an object such as a cake among several individuals is not possible without a definition of fairness.

5. Definition of Fairness
A division among n people is called fair if each person feels he or she has received at least 1/nth of the object.

6. Successful Division of the cake
Several assumptions must be made about the fair division problem:
Each individual is capable of dividing a portion of the cake into several portions that he or she feels are equal.
Each individual is capable of placing a value on any portion of the cake. The total of the values placed on all parts of a cake by an individual is 1, or 100%.
The value that each individual places on a portion of the cake may be based on more than just the size of the portion.

7. How to Divide the Cake
Cutting the cake into two pieces, required that one person cut the cake and that the other person choose the piece they desire.
This is the first assumption in the list.
The second assumption is that the person who doesn't cut the cake places the value on the cake that adds up to 1.

8. Is it Fair?
The individual that chooses may not feel that the pieces are equal.
As a result, they may choose the piece that he or she feels is more than half the cake.
The division is still fair because the definition requires only that each person feel that his or her piece is at least half of the cake.

9. Multiple Fair Divisions
How do you fairly divide cake among three, four, or more people?
No solution is adequate unless it adheres to the definition of a fair division.
A fair division among three people requires that each individual place a value of at least one-third on the received portion.

10. Unique Solutions
In math, there are often more than one way to solve a problem.
When this is the case, the solution is not unique.
Therefore, dividing a piece of cake between three or more people is not unique because it has more than one solution.

11. Three Person Division Problem
Three people: Ann, Bart and Carl.
Ann cuts the cake into two pieces that she feels are equal.
Bart chooses one of the pieces; Ann gets the other.
Ann cuts her piece into three pieces that she considers equal; Bart does the same with his.
Carl chooses one of Ann’s three pieces and one of Bart’s.

12. Fairly Divided?
To see if this division is fair, we must show that each person places a value of at least one-third on the portion that he or she received.
Consider Ann: In Step 1, she cuts the piece into two pieces which she considers equal.

13. Ann’s Situation
Since she cut the cake, she feels that she has half of the cake.
She feels that each piece she cut in Step 3 is one-third of half of the cake, or one-sixth of the cake.
She therefore feels that she received two-sixths or one-third of he cake in Step 4.

14. Bart’s Situation
Bart’s case is similar to Ann’s except that he may feel that the portion he chose in Step 2 is more than half of the cake.
Thus he may feel that the pieces which he cut in Step 3 is more than one-sixth of the cake and that his final portion may be more than two-sixths or one-third.

15. Carl’s Situation
Carl may feel that the two pieces that Ann cut in Step 1 are not equal.
His value could be, for example, 0.6 for one piece and 0.4 for the other.
Likewise, he may not feel that the cuts made in Step 3 are equal.
He could feel that the piece that he valued as 0.6 was divided into pieces that he values as 0.3, 0.2 and 0.1.

16. Carl’s Situation (cont’d)
Similarly, he could feel that the piece that was 0.4 was divided into pieces that he valued as 0.2, 0.1 and 0.1.
But, because he chooses first, he will pick the largest piece from each.
Thus, the portion he receives is = 0.5 which is more than 1/3.

17. Practice Problems
In the division among Ann, Bart and Carl, who will evaluate his or her share at exactly one-third? Who might feel that he or she received more than one-third?
Does the division among Ann, Bart and Carl result in three pieces or three portions?

18. Practice Problems (cont’d)
In Exercises 3 and 4, suppose Carl feels that Ann’s initial division is fair, that Ann’s subdivision is even, but that Bart’s subdivision is not. (Give your answers as fractions or decimals rounded to the nearest 0.01).
What value will Carl place on the piece he takes from Ann?

19. Practice Problems (cont’d)
Although Carl feels that the piece Bart divided is half of the cake, he does not feel that Bart subdivided it equally. He could, for example, place values of 0.3, 0.1 and 0.1, or values of 0.4, 0.06 and 0.04 on the three pieces. The largest value he could place on any of these three pieces is 0.5.
What is the smallest value he could place on the piece he takes from Bart?
What is the largest total value he could place on his two pieces?
What is the smallest value he could place on his two pieces?

20. Practice Problems (cont’d)
5. In mathematics, a fundamental principle of counting is that if there are m ways of performing one task and n ways of performing another, then there are m times n ways of performing both. For example, a tossed coin may land in two ways and a rolled die may land in six ways. Together they land in a total of 2 times 6 = 12 ways.

21. Practice Problems (cont’d)
If two people each have a piece of cake and each cuts his or her piece into three pieces, how many pieces will results?
If k people each have a piece of cake and each cuts his or her piece into k + 1 pieces, what are two equivalent expressions for the total number of pieces that result?

22. Practice Problems (cont’d)
If k + 1 boxes each contain k + 5 toothpicks, what are two equivalent expressions for the total number of toothpicks?
Two offices are to be filled in an election: mayor and governor. If there are three candidates for governor and four for mayor and conventional voting procedures are used, in how many ways may one vote?

23. Practice Problems (cont’d)
Consider the following division of a cake among three people: Arnold, Betty and Charlie. Arnold cuts the cake into three pieces he considers equal. Betty choose one of the pieces and Charlie chooses either of the remaining two. Arnold gets the third piece.
a. Will Arnold feel he has received at least one-third of the cake? Might he feel he has received more?
b. Will Betty feel she has received at lest one-third of the cake? Might she feel she has received more?
C. Will Charlie feel he received at least one-third of the cake? Might he feel he has received more?

24. Practice Problems (cont’d): Inspection Method
7. Arnold, Betty and Charlie decide to divide a cake in the following way: Arnold slices a piece he considers one-third of the cake. Betty inspects the piece. If she feels it is more than one-third of the cake, she will cut enough from the cake so that she feels it is one-third of the cake. The removed portion is returned to the cake. Charlie now inspects Betty’s piece and has the option to do the same. The piece of cake is given to the last person who cut from it.

25. Practice Problems (cont’d)
One of the remaining two people slices a piece that s/he feels is half of the remaining part of the cake. The other person inspects the piece with the option of removing some of the cake if s/he feels it is more than half of the remainder.

26. Practice Problems (cont’d)
Will the person who receives the first piece feels that it is at least one-third of the cake? Could s/he feel it is more than one-third?
Will the person who receives the second piece of cake feel that it is at least one-third of the cake? Could s/he feel it is more than one-third?

27. Practice Problems (cont’d)
c. Will the person who receives the third piece feel that it is at least one-third of the cake? Could s/he feel it is more than one-third?