Math 103, Section 11, Tuesday, February 8, 2011
1. Submitted Homework Assignment #1 for Chapter 2 is at Sakai.
It is in the Assignments section and is called:
Assignment Chapter 2A
Please do Exercises 18b,18c,20 and 24 on page 68 and upload to
Sakai.
Homework is due by Friday, February 11 at midnight
The Exercises are at Sakai, Resources, Chapter 2 Exercises from
text.doc
Quiz on Friday, February 11
Submitted Homework Assignment #2 for Chapter 2 is at Sakai.
It is in the Assignments section and is called:
Assignment Chapter 2B
Please do Exercises 28d, 34, and 36 on page 69 and upload to Sakai.
Homework is due by Friday, February 18 at midnight
The Exercises are at Sakai, Resources, Chapter 2 Exercises from
text.doc
Homework to be discussed in class on Friday, February 11:
The following 6 problems:
Exercises 20, 27b, 33, and 35 and
1.
UN Problem
Consider the following simplified version of the UN Security Council: There are 4 players, each with one
vote,
and a majority of votes is needed to pass a resolution. Moreover, P
1
has veto power.
(a) Find the Banzhaf power distribution of this weighted voting system.
(b) In order to find the ShapleyShubik power distribution, how many sequential coalitions must you
examine?
Since there are 4 players, there are 4!= (4)(3)(2)(1) = 24 sequential coalitions.
(c)
Find the ShapleyShubik power distribution of this weighted voting system.
Start by listing all the sequential coalitions.
Then fill in the following
table:
Banzhaf power index
ShapleyShubik power index
P1
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P2
P3
P4
2. Given the weighted voting system
{4: 3, 2, 2}
Fill in the following table:
Banzhaf power index
ShapleyShubik power index
P1
P2
P3
Correction to the handout from our last class:
3. Consider the weighted voting system [q:7, 5, 3, 2].
3d. At least how much would the quota have to be so that all the players have veto power?
In class I said 17, but the correct quota is 16.
In fact,
For P1 to have veto power, the quota must be at least: 11
Why?
For P2 to have veto power, the quota must be at least: 13
Why?
For P3 to have veto power, the quota must be at least: 15
Why?
For P4 to have veto power, the quota must be at least: 16
Why?
Definition.
A player P has veto power
if and only if the sum of the weights of the other players is less than the
quota.
Consider [the weighted voting system:
[ q:P1,P2,P3,P4]
V/2<q<=V
where the weight of P1 is w1. The weight of P2 is w2. The weight of P3 is w3. The weight of P4 is
w4.
and V=w1+w2+…+w4
P1 has veto power means that w2+w3+w4<q
When listing winning coalitions, remember that
For 3 players, there are 2^313=4 coalitions to consider:
{P1,P2}
{P1,P3}
{P2,P3} and {P1,P2,P3}
For 4 players, there are 2^414=11 coalitions to consider.
{P1,P2} {P1,P3} {P1,P4}
{P2,P3}
{P2,P4}
{P3,P4}
{P1,P2,P3}
{P1,P2,P4}
{P1,P3,P4}
{P2,P3,P4}
{P1,P2,P3,P4}
Problem 3. For the weighted voting system [16:12,10,4], find the Banzhaf power distribution.
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 Spring '07
 Berkowitz
 Math, Game Theory, Voting system, Plurality voting system, UN Security Council, Banzhaf power index

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