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# Class March11 - Math 103 Section 11 Friday Chapter 3 Handout 5 Have a wonderful spring break The Mathematics of Sharing Assignment Chapter 3A

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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 105-109 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 112-117. 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 take-home 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? Lone-chooser 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 lone-chooser method.

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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.

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Class March11 - Math 103 Section 11 Friday Chapter 3 Handout 5 Have a wonderful spring break The Mathematics of Sharing Assignment Chapter 3A

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