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Weighted Voting and Voting Power
Section 1.5
WEIGHTED VOTING
Definition: method of voting when some people have
more votes than others.
i.e., The votes of some voters carry more weight than
the vote of others... Not all votes are considered
equals!
Examples: Electoral College, Corporate stockholder
meetings, Mom, etc.
the # of votes is based on population in each state
http://www.uselectionatlas.org/USPRESIDENT/national.php?f=0
EXAMPLE:
A student council has 1 representative per class, but
there are 500 sophomores, 300 juniors, and 300
seniors. How could votes be weighted to represent
everyone fairly?
Sophomore Rep gets 5 votes,
Junior & Senior Reps each get 3 votes
The passage of any issue requires
6
a majority of ____ votes.
Let’s pretend student council was voting
to change the exam exemption policy.
The proposal is to allow exemptions for
everyone…(not just seniors.)
COALITIONS:
Could any group win
Sophomores get 5 votes by themselves? NO!!
Juniors get 3 votes
So, join forces!
Seniors get 3 votes
Notation: {So;5}, {Jr;3}, {Sr;3}
{So,Jr; 8}
{So,Sr; 8}
Remember, You need 6
votes to WIN! {Jr,Sr; 6}
{So,Jr,Sr; 11}
WINNING COALITIONS:
Each collection is known as a Coalition.
{So;5}, {Jr;3}, {So,Jr;8}, {Jr,Sr;6} …
Those with enough votes to pass an issue are known as
Winning Coalitions. (A winning coalition is like an
alliance in “Survivor.”)
A voter is essential when their vote is NECESSARY to
win. (i.e. if you remove it, the winning coalition becomes
a losing coalition.)
So, for this example:
Winning
Coalitions:
So,Jr;8 So,Sr;8 Jr,Sr;6 So,Jr,Sr;11
2
Sophomores are essential for _____ coalitions
2
Juniors are essential for ______ coalitions
2
Seniors are essential for _____ coalitions
*Paradox: Although the votes were distributed to give
more power to sophomores, the outcome is that all
members have the same amount of power.
Banzhaf Power Index:
determines the power of a member of a voting body
Step 1. Make a list of all possible coalitions
Step 2. Determine which of them are winning coalitions
Step 3. In each winning coalition, determine which of
the players are essential players
Step 4. Count the total number of times player P is
essential (B)
Step 5. Count the total number of times all players are
essential (T)
The Banzhaf Power Index of player P is given by the
fraction B/T
Power Index Continued:
Consider the previous example:
Sophomores: 2/6
Juniors: 2/6
Seniors: 2/6
The Power Index should always add up to 100% or 1.
Dictators & Dummies
A dictator is a member of a voting body
who has all the power.
A dummy is a member who has no power.
#1
10 committee members vote by approval
voting on 4 candidates for a new
chairperson of the committee. Which
candidate wins? Candidate B
#2
Which candidate finishes last?
Candidate D
#3
If committee members # 5 and # 8 are
adamantly opposed to candidates B and D and
they have prior knowledge of the others’ votes,
how might they have voted differently when
using approval voting? Vote for A
#4
If all candidates earning over 50% of the votes will be
entered in a run-off, who would be in the run-off?
A,B, and C
#5
Consider the weighted voting situation:
Voter A – 15 votes; Voter B – 12 votes
Voter C – 6 votes; Voter D – 3 votes
24 votes are needed to pass…
Is coalition {B, C, D} a winning coalition?
No
#6
Consider the weighted voting situation:
Voter A – 15 votes; Voter B – 12 votes
Voter C – 6 votes; Voter D – 3 votes
24 votes needed to pass…
Which players are essential in the coalition
{A, B, C; 33}?
A and B
#7
Consider the weighted voting situation:
Voter A – 15 votes; Voter B – 12 votes
Voter C – 6 votes; Voter D – 3 votes
24 votes needed to pass…
Which players are essential in the coalition
{A, B, C, D; 36}?
A
#8
Consider the weighted voting situation:
Voter A – 15 votes; Voter B – 12 votes
Voter C – 6 votes; Voter D – 3 votes
24 votes needed to pass…
List 3 winning coalitions.
{A,B;27}, {A,B,C;33}, {A,B,D:30}…
#9
Consider the weighted voting situation:
Voter A – 15 votes; Voter B – 12 votes
Voter C – 6 votes; Voter D – 3 votes
24 votes needed to pass…
List 2 other winning coalitions.
#10
Consider the weighted voting situation:
Voter A – 15 votes; Voter B – 12 votes Voter C – 6 votes;
Voter D – 3 votes 24 votes needed to pass…
Using the winning coalitions {A, B: 27} {A, B, C: 33}
{A, B, D: 30} {A, C, D: 24} {A, B, C, D:36}
Find the power index for voter A. 5
#11
Consider the weighted voting situation:
Voter A – 15 votes; Voter B – 12 votes Voter C – 6 votes;
Voter D – 3 votes 24 votes needed to pass…
Using the winning coalitions {A, B: 27} {A, B, C: 33}
{A, B, D: 30} {A, C, D: 24} {A, B, C, D:36}
Find the power index for voter B.
4
#12
Consider the weighted voting situation:
Voter A – 15 votes; Voter B – 12 votes Voter C – 6 votes;
Voter D – 3 votes 24 votes needed to pass…
Using the winning coalitions {A, B: 27} {A, B, C: 33}
{A, B, D: 30} {A, C, D: 24} {A, B, C, D:36}
Find the power index for voter C.
1
#13
Consider the weighted voting situation:
Voter A – 15 votes; Voter B – 12 votes Voter C – 6 votes;
Voter D – 3 votes 24 votes needed to pass…
Using the winning coalitions {A, B: 27} {A, B, C: 33}
{A, B, D: 30} {A, C, D: 24} {A, B, C, D:36}
Find the power index for voter D.
1
#14
Which of Arrow’s Conditions is violated in
this situation:
Every student in the class voted for cake and ice
cream over pizza for a class party. However,
since we ranked them (along with Chick-fil-a and
Brownie sundaes) and used the Borda Count
Method, pizza won.
Condition 3
