Math 1340: Maths and Politics, Spring 2012 Homework Solutions Homework 2
Problem 1 Paris would clearly lose in a winner-by-majority or plurality contest since it is not the first choice of most of Kate's friends. Nor is it the Condorcet winner. Kate could
1/22/13
Math and Politics Lecture 1:
Email of Professor: [email protected]
TA- Katie Sulivan [email protected]
The winner of an election should be the candidate of the first choice of more votes than
another.
Makes a parallel to Al Gore and Bush el
Math 1340: Maths and Politics, Spring 2012
Homework Solutions
Homework 12
Problem 1
Step 1:
Annie gets: apartment lease (35), entertainment system (20), washer-dryer (15). Total: 70.
Ben gets: pool table (25), antique table (20). Total: 45.
Step 2:
Annies
Math 1340: Maths and Politics, Spring 2012 Homework Solutions Homework 3 Problem 1 Let's reproduce the preference lists of the 7 voters for ease of reference: a d b c a c d b b d c a b a d c c b d a a b d c d b c a
(a) Condorcet method: For each pair of c
3/26/13
SequentialGames
Prelim on Thursday PLS 233 Starts at 7:30PM
HW for 2nd AprilIndividual: Q21, 27
Group: Q24, 28
Sequential Games:
e.g. Tic-tac-toe, chess
Theory of Moves
Recall:
1. An initial state is chosen ie a choice of strategy for each player
1/29/13
MathandPoliticsLecture3
Individual HW: Q 23, 41
Group HW: Q 29, 33, 35, 37
Recap: Social Choice Procedure
Input [Voter Profile] Output
Voter profile Social Choice Set cfw_b
What is wrong with plurality?
-Minority leadership because more people can
Social Choice:
Recall: A social choice procedure takes input: voter profile and gives an output: social choice set.
Example: Majority rule n voters. The social choice set will consist of any candidate who is at the top of at least
n/2 preference lists.
Ma
MATH 1340 Test 1 (Voting and Social Choice)
This test has 4 problems (2 on the front, 2 on the back) with several parts to each. Write down
all solutions in a Cornell University Official Examination Book, with your name and signature on
the front cover. M
02/02/2017
Properties of Voting Systems:
Q. What are some properties that a good voting system should have?
1. Produce a winner (in every situation) (can be formalized by social choice)
2. Effective way to resolve a tie (sometimes , depends on what good m
M ATH 1340: Mathematics and Politics
Summer 2010
Properties of Yes-No Voting Systems
Introduction
We have studied several properties that yes-no voting systems can satisfy or fail to satisfy.
Here are the most important ones.
A yes-no voting system is we
M ATH 1340: Mathematics and Politics
Summer 2010
Symmetry in 2 2 games
Introduction: What is symmetry?
Consider a collection of objects, together with a bunch of ways of transforming one object into another. Generally speaking, an object is called symmetr
M ATH 1340: Mathematics and Politics
Summer 2010
Dominant strategies and Nash equilibria
A 2 2 ordinal game consists of two players, which we call R OW and C OLUMN,
each of whom can choose between two strategies, which we call C (cooperate)
and N (noncoop
MATH 1340, HOMEWORK #4
DUE THURSDAY, MARCH 2
Please show your work in the problems that require calculations. For the short answer questions, write in
complete sentences. All assertions must be justified to get full credit.
0. (For participation credit.)
Lecture22DollarAuction
Conservative Convention: If for other player at their turn they have to decide between two bids
which lead to the same pay-off, then they will choose the smallest bid.
Punishing convention:
This is a sequential game.
1. There are tw
L7MathematicsandPolitics
Prop: S. ONeil is the tallest man in the world.
Pf: ONeil is the tallest basketball player This is not a proof
Prop: S. ONeil is NOT the tallest man in the world.
Pf: Find someone taller! Counter example!
-More Yes-No voting time:
1/24/13
Math and Politics Lecture 2:
Reminder: Homework due in on Tuesday Groups of 3.
Office hours this week (511 Mallot) Mon 9-10 and 11:10-12:10
Homework Questions:
Individual- Q 4, 8
Group- 9, 12, 17, 18
Social Choice Procedure- Input Output it can le
L11DonationVotinginaYesorNo
VotingSystem
The Donation Paradox
F-4 L-1
F-3 L-2
Example:
A-5
B-3
C-1
D-1
E-1
Quota-8
Winning coalition:
AB
ABC
ABD
ABE
ABCD
ABCE
ABDE
ABCDE
ACDE
Total BPBI (A)- 9/19
BI (B)- 7/19
BI (C)- 1/19
A gives a vote to B
A-4
B-4
C-1
D
2/11/2013
MathematicsandPoliticsL5:TheEndof
Chapter1.TheBeginningofChapter
2
Individual HW: Q4, 8
Group: Ch1: Q39, 42
Ch2: Q1, 9
+ read The fairest vote of all
Condorcet
Dictatorship
AAW
N
Y
CWC
Y
X
Pareto
Y
Y
Mono
Y
Y
IIA
Y
Y
Consider the following SCP:
3/5/13
L13ElectoralCollege
Electoral College
According to 2000 dates:
BI of California -11.4%
BI of NY- 5.6%
BI of DC- 0.5%
Pop of these states- BI/10,000,000 ppl
Cali= 33.8 million . 3.4%
NY= 18.9 million .2.9%
DC= 570k8.9%
Game Theory Time:
This trend c
Adjusted Winner
Yet another divorce settlementApartment
Holiday Home
Custody
Piano
Rare Wine Collection
RV
Ann
30
15
35
5
10
5
Bob
25
6
20
20
9
20
Note: There are their honest valuations.
Initial Allocations:
Ann- 30+15+35+10= 90
Bob-20+20=40
Ann is the i
Chapter2:YesNoVotingConditionsL8
Office Hours: 511 Mallot. 9:30-12:00
Katie: 1-4pm
HW: Ind: Q13,26
Group: Q17, 18, 19, 21
Will not be graded, but may appear on prelim
Prelim: Tuesday, starting promptly at 7:30pm. PLS 233 (Plant Sciences, Ag Quad)
Trade ro
ApportionmentL19
Bias
Quota Property
Alabama Paradox
Population Paradox
Monotonicity
Hill-Huntingdown
HAMILTONS METHOD
Pop
State of Being
86
Washington DC
270
AFC Wimbledon
85
Mongolia
271
State of Emergency 33
New Los Florida
135
State of the Union
185
I
Calculate SSI for A, B, C:
A- 50
B- 49
C- 1
Quota- 51
List winning coalitionsA+C
A+B
A+B+C
AA B B C C
BCACAB
CBCABA
Therefore A is pivotal four times out of 6 possible options As SSI is 2/3
SSI (A)=2/3
SSI (B)=1/6
SSI (C)=1/6
Odd because even though B has
CalculatingthePowerofthePresident
L12
Shapley-Shubick Index
Reminder: Us federal system has one president and VP and 100 Senators and 435 members of
the House
Let x be a coalition. If x contains P, then it will be winning if it contains at least 51 senato
Mo Game Theory
Split or Steal!
Split
Steal
Split
(.5, .5)
(0, 1)
Steal
(1, 0)
(0, 0)+Shame
Split
Split
(3, 3)
Steal
(2, 4)
Steal
(4, 2)
(1, 1)
Ordinal Game
This game is called chicken.
Chicken
Swerve
Not Swerve
Swerve
both lose face, but survive
Column w
Why Did Lance Armstrong Cheat?
Rivalry between Armstrong and Jan Ullrick. In 2002, Ullrick was caught cheating and tested
positive for performance enhancing drugs.
Armstrong has the choice to:
Ullrich
Dope
Not Dope
Armstrong
Dope
Fair advantage
Lance Vict
1/31/13
Mathematics and Politics Lecture 4: Finishing the Chart of SP criterion
Reminders:
Prelims are evenings of Feb 19, March 28
Email him if you have a conflict
Practice-aaabbcddd
bccddbbbc
cbbaaaccb
dddccdaaa
Hare- A B D
Borda- B wins
a-12
b-17
c-12
MathematicsandPoliticsL6:Weighted
Yes/Nosystems
Definition: A minimal winning coalition is the smallest winning coalition that you can get. This
group exists if the condition satisfies the criterion that if any member of the coalition is removed
then this
MATH 1340, HOMEWORK #2 SOLUTIONS
DUE THURSDAY, FEBRUARY 9
1. [TP 1.13] Consider the following social choice procedure: if there is a Condorcet winner, then
that alternative is the social choice; otherwise use the Borda count to determine the social choice
MATH 1340, HOMEWORK #5 SOLUTIONS
DUE THURSDAY, MARCH 9
1. [TP 3.15] Consider a voting system for the six New England states where there are a total of
seventeen votes and twelve or more are required for passage. Votes are distributed as follows:
MA:4 ME:3
MATH 1340, HOMEWORK #3 SOLUTIONS
DUE THURSDAY, FEBRUARY 16
1. [TP 2.1] One way to have a case heard by the Supreme Court of the United States involves what
is called a grant of certiorari. The effect of this is that a case will be considered by the court
MATH 1340, HOMEWORK #1 SOLUTIONS
DUE THURSDAY, FEBRUARY 2
1. [TP 1.1] Let X and Y both be the set of non-negative integers 0, 1, 2, 3, . . . Determine if the
given procedures are functions from X to Y .
(a) The procedure corresponding to taking the square