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Math 103, Section 11, Friday, March 11, 2011
Chapter 3 Handout 5
Have a wonderful spring break!!!!!!!
The Mathematics of Sharing
Assignment Chapter 3A
Please do Exercises 12,18,30, and 32, pages 105109 and upload to
Sakai.
Homework is due by Friday, March 11, 5 minutes before midnight
Assignment Chapter 3B
Please do Exercises 42, 56, and 70, pages
112117.
Homework is due by Friday, March 25, 5 minutes before midnight
Homework to be discussed in class. Page 114, Problem 53 Sealed
bids
To be discussed in class homework: Page 117, Problem 69 Method of
Markers
Chapter 3 Quiz will be a takehome quiz in which you can use your
notes.
However, try doing the quiz without your notes.
Can you predict under what circumstances there will be no surplus in
the method of sealed bids?
Lonechooser example:
Jared, Karla, and Lori are dividing the 18 inch long sub which has a 12 inch vegetarian piece
and a 6 inch meat piece
The sub is shown in the above picture. The three friends are dividing
the sub using the lonechooser method.
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View Full Document Jared likes the vegetarian and meatball parts equally well.
Karla is a strict vegetarian and does not eat meat at all, and
Lori likes the meatball part twice as much as she likes the vegetarian part.
Value table
Vegetarian
Meatball
Karla cuts
1 point
0 points
Jared
1 point
1 point
Lori
1 point
2 points
Karla and Jared are the dividers and Lori is the chooser.
In the first division, Karla divides the
sub into two shares (a left share s1 and a right share s2) and Jared picks the share he likes
better.
Part a.
Describe s1 and s2 using the ruler under the sandwich and using interval notation.
For example, [0,1] is interval notation.
Put your answers in the table below. Then complete the table
.
Fill in the following table:
Shares
s1=[0,6]
s2=[6,18]
value (% or fraction) to Karla
50%
50%
Value (% or fraction) to Jared
6/18=1/3
12/18=2/3
b. Which share does Jared pick? s1 or s2?
s2
Why
?
Larger value
c. Describe how Jared would then subdivide his share into three pieces. Use interval notation.
[6,10]
[10,14]
[14,18]
d. Describe how Karla would then subdivide her share into three pieces. Use interval notation.
[0,2]
[2,4]
[4,6]
e. Describe the final fair division of the sub and give the value of each player’s share (as a fraction of the
total value of the sub in the eyes of the player receiving it) by filling in the following table:
Player
Shares received
at end
Value as a fraction
Jared
[6,10] and
[10,14] which is
[6,14]
8/18=4/9
Karla
[0,2] and [2,4]
which is [0,4]
4/12=1/3
Lori
[14,18] and
[4,6]
10/24=5/12
To Lori the whole sandwich is worth 1*12+2*6=24 points
[14,18] is worth 8 points
[4,6] is worth 2 points.
For Solving Discrete Fair Division Problems:
3.6 The Method of Sealed Bids
Involves
Step 1 Bidding
Step 2 Allocation
Step 3 First settlement
Step 4 Division of the surplus
Step 5 Final Cash Settlement=first settlement + surplus
Here is a clarification of the Method of Sealed Bids.
This method involves a money and item exchange among the players.
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This note was uploaded on 01/20/2012 for the course MATH 103 taught by Professor Berkowitz during the Spring '07 term at Rutgers.
 Spring '07
 Berkowitz
 Math

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