Jan 18, 2018
Fair Division Algorithms
The Continuous Case Cake Division Problem Let's return to the
problem of fairly dividinga 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. 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. 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.
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. 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. 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. 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. 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. 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. 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 Anns three
pieces and one of Barts. 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. Anns 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. Barts Situation Barts case is similar to Anns
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 thanone-sixth of the cake and that his final
portion may be more than two-sixths or one-third. Carls 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. Carls
Situation (contd)
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. 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?
Practice Problems (contd)
In Exercises 3 and 4, suppose Carl feels that Anns initial division
is fair, that Anns subdivision is even, but that Barts 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? Practice Problems (contd)
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? Practice Problems
(contd)
5.In mathematics, a fundamental principle of counting is that if
there are m ways of performing one task and n ways of
performinganother, 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. Practice Problems (contd)
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? Practice Problems (contd)
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? Practice Problems (contd)
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? Practice Problems (contd):
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 Bettys piece and has the
option to do the same.The piece of cake is given to the last person
who cut from it. Practice Problems (contd)
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. Practice Problems
(contd)
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? Practice Problems (contd)
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?