Chapter 3
The Mathematics of Sharing

There are 20 pieces of candy and 4 children.
How do you divide this fairly for the children?

The truth is you cannot divide it evenly with
the information given.

Fairness can be expanded to everything.
◦ Land: The break up of Yugoslavia.
◦ Ocean Resources: 1982 Convention of the Law of
the Sea
◦ Global Responsibilities: 1997 Kyoto Treaty

The Goods- Informal name for the items that
are being divided. This can also be called “the
booty.”

The Players- Everyone that is participating in
the of S. Players will be called 𝑃1 , 𝑃2 , … , 𝑃𝑛 .

The Value Systems- Every player has an
internal value system that they judge the
value of the goods. This can be a dollar
amount or a percent.
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Rationality- Everyone will act rational. No one
will be different. Everyone wants the biggest
piece of S as possible.
Cooperation- Everyone will play. NO
QUITTERS
Privacy- No one knows each other.
Symmetry- We are all equal. No one deserves
more than anyone else.

You have a fair share if at least 𝑠 = 1
opinion.
𝑁
in P’s


S is divided into 𝑠1, 𝑠2 , 𝑠3 , 𝑠4 . 𝑃1 see the value of
the shares as 𝑠1 = 10%, 𝑠2 = 28%, 𝑠3 = 20%, and
𝑠4 = 42%.
The fair share that 𝑃1 sees is with 𝑠2 and 𝑠4
because they are both over the 25% that
would be considered fair.

Continuous- S can be divided infintly as many
ways as possible. It can be sliced and resliced. This can be dividing a cake, pizza, or
land.

Discrete- This is when S can not be divided in
anyway. This would be like a painting, couch,
house, or candy.


Mixed- When a fair-division game uses both
continuous and discrete objects.
To solve these you separate the objects into
two groups (continuous and discrete) then
use the correct method for each.

#2-14 even
Two Players

“I cut you choose”

Continuous division game

How do they divide the cake?

If done correctly:
◦ 𝑃1 will always get 50%
◦ 𝑃2 will get 50% or more
#16, 18, 20
More than two people



Division: Player D divides the S into three
equal shares.
Bidding: The two choosers will secretly bid on
the parts that they consider is fair shares.
(any part that is more than 1/n)
Distribute: Looking at the bids, give the parts
to those.
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D divides the project into three parts.
𝐶1 bids on section 1
𝐶2 bids on section 1 and 2
𝐶1 gets section 1
𝐶2 gets section 2
D gets section 3
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D divides the work table into three parts
𝐶1 bids on 1
𝐶2 bids on 1
D chooses between 2 and 3
Which ever one that D does not choose then
combine 1 and that number
𝐶1 𝑎𝑛𝑑 𝐶2 use the divider and chooser method
S1
S2
S3
Dale
33.3%
33.3%
33.3%
Cindy
35%
10%
55%
Cher
40%
25%
35%
Which pieces will Cindy and Cher bid on?
Cindy S1 S3, Cher S1 S3
How will the product be divided and what percent of share did each
person get?
Dale: S2
33.3%
Cindy: S3
55%
Cher: S1
40%
S1
S2
S3
Dale
33.3%
33.3%
33.3%
Cindy
20%
30%
50%
Cher
10%
20%
70%

Use the same process
S1
S2
S3
S4
Demi
25%
25%
25%
25%
Chan
20%
20%
20%
40%
Chloe
15%
35%
30%
20%
Chris
22%
23%
20%
35%

#22, 26, 28, 30



The two dividers use the divide and choose
method
Each divider then divides their section into 3
pieces
The chooser then picks one from each
dividers group

#34, 36, 38, 42

#44, 46, 48, 52, 71, 72
Ana
Belle
Chloe
Dresser
$150
$300
$275
Desk
$180
$150
$165
Vanity
$170
$200
$260
Tapestry
$400
$250
$500
Ana
Belle
Chloe
Dresser
$150
$300
$275
Desk
$180
$150
$165
Vanity
$170
$200
$260
Tapestry
$400
$250
$500
Total
$900
$900
$1200
Fair-Share
$300
$300
$400
Ana
Belle
Chloe
Dresser
$150
$300
$275
Desk
$180
$150
$165
Vanity
$170
$200
$260
Tapestry
$400
$250
$500
Total
$900
$900
$1200
Fair-Share
$300
$300
$400
Total Amount Earned
$180
$300
$760
Difference
+$120
$0
-$360



Ana has $120 dollars coming to her
Chloe has to give $360
Belle does not have any money coming in or
out

Once Ana has her difference covered there is
$240 left over.

This is the surplus
Divide this among everyone evenly

Final:

◦ Ana: Desk, and $200
◦ Belle: Dresser and $80
◦ Chloe: Vanity and Tapestry and loses $280

#54, 56, 76, 77
Clarke
Logic
Boring
Little
Sudoku
Set
Game
Total
Fair share
Morgan
James
Kyle

#62, 66, 68

#2-22 even, 26-30 even, 34, 36, 38, 42-48
even, 52, 54, 56, 62, 66, 68, 71, 72, 76, 77