3.3 Lonedivider method
fair division of a continuous asset among n players
inductive proof: start from the smallest cases(DividerChooser
works)
The LoneDivider method will always produce a fair division of
the assets among the n players
S1
S2
S3
D
33
Chapter 3: The math of sharing
Fair division
A number of players want to divide some assets fairly.
Continuous asset: money, water, cake (can be divided) (3.2
> 3.4)
Discrete asset: car, house (3.5 3.6)
A value function
Each player can define how val
Judgement and Decision Making
Math 11401 Syllabus
Fall 2016
Instructor : Ryan Pellico
Contact Information :
Office Hours: MWF 1:00  3:00 pm,
Tues 9:30  10:30 am,
or by appointment
Office:
MECC 273
Phone:
8602972297
Email:
ryan.pellico@trincoll.edu
Le
11/4/16
3.2 DividerChooser method
Fair division of a continuous asset bw 2 people
The 2 players randomly decide who is the divider and who is
the chooser
Divider devide into 2 shares, chooser chooses 1
> always fair share
Exam 2 Review:
(will be on
Hamiltons Method:
1.Give each its lower quota
o Ideal case: SQ1 +SQ2 +SQ3 = total # seats
o But usually LQ SQ
o > LQ1 + LQ2 + LQ3 total seats
2. If there is a surplus of seats, give them out 1 at a time
starting with the state with the largest residue
3.5
The Method of Sealed Bids
Procedure:
Each player makes a sealed bid, giving an estimate of the
dollar value of each item. Each players fair share is computed
by dividing the total of the players bids by the # of players.
Each item goes to the highes
A fair share is 1/N of the total value, where N is the number
of players
o If 2 people are dividing a cake, they should each get a
piece worth at least 50% to them
o If 3 people,., at least 33.33% to them
A fair division method is an algorithm for assign
11/16/16
How do we opposition seats to states based on population?
Apportion went
# seats you get /total # seats = population of your state/total
popu
Standard divisor (there is only one)
= total population/total #seats = #people reported by each
seat
DividerChooser
Dividing a sub that has turkey and ham
o
75%
Turkey
Deb
75
Connor
50
Ham
25
50
Deb divides so that s1 & s2 are each 50% to her
S1
50
33.3
25%
Deb
Connor
Where does she cut?
How does Connor value s1 & s2?
X is a fraction of turkey piece t
Math 11401
Exam 2 Review
1. A certain club consists of 6 freshmen, 4 sophomores, 3 seniors, and 2 faculty.
(a) How many coalitions can be formed with 2 freshmen and 2 sophomores?
(b) How many coalitions can be formed with 2 seniors and 2 faculty?
(c) How
10/7/16
cfw_S1,S2,S3,S4,S5,S6,F
There are
coalitions of cfw_S,S,F
There are
coaltions of cfw_S1,S,F
If there are n ways to do task 1 & if there are m ways to do task
2, there are m*n ways to do both.
A deck of cards: 4 suits & 13 ranks
10/14/16
Exam 2: Oc
9/30/16
2.2 Banzhaf power
H1,H2,OB,NH,LB,GC
[ 59: 31, 31, 28,21, 2, 2]
1. Find all winning coalitions
2. Find critical players for each winning coalition
3. Compute total critical count T = B1 + B2 +.+ Bn
4. Compute Banzhaf power index i = Bi/T , T =
10/3/16
[q: 10,9,5,2,1]
Find all q for which P1 and P2 have veto power (and no one else)
If P2 has veto power, so does P1
V = 27
14 q 27
P2 has veto power if the rest combined cant meet the quota
V w2 < q 18 < q (which is not not true)
If P2 has veto,
9/21/2016
Exam 1 on Monday, in class
The plurality with elimination methods can violate the monotonicity
criterion.
The plurality method satisfies the monotonicity criterion.
If youre the plurality winner and youre moved up in some
preference ballot(s),
Exam 1 Review:
If there is no Condorcet candidate, the Condorcet criteria
doesnt apply
Majority candidate Condorcet candidate (not the other way
around)
Chapter 2
[q:w1, w2, w3, wN]
V = w1 + w2 + wN total votes (Yes/No votes)
q: number of votes to pass a
Chap 16
What are the chances of something happening?
If roll a 6sided dice, chances to have an odd number :
Game: If odd # is rolled, you win $5; if even #, you lose
double the amount the # on the die
1
2
3
4
5
$5
$4
$5
$8
$5
What are my expected wi
Voting methods:
Plurality
Borda count
Plurality with elimination
Pairwaise comparisons
14
10
8
4
1
A
C
D
B
C
B
B
C
D
D
C
D
B
C
B
D
A
A
A
A
Plurality Election: A C D B
Borda count: B
Pluraity w/ elimination (if anyone has a majority of 1st, they win
 A coalition is a group of players
cfw_P1, P2 the coalition of P1 & P2
cfw_P1,P2,P3
A coalition is either winning or losing: win if its players have
enough votes to meet quota, lose otherwise
[40: 10, 10, 10, 10,9]
cfw_P1,P2
cfw_P3,P4 20 votes < q
Sections 2.1 & 2.2 Worksheet September 9, 2016 MATH 207
2.1 & 2.2: QUALITATIVE & QUANTITATIVE DATA
1.] Out of the 76 students enrolled in both MATH 207 sections, 39 are male and 37 are female. What
are the relative frequencies for the gender class?
.61 Vi

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