Weighted Voting Systems
Brian Carrico
What is a weighted voting system?
A weighted voting system is a decision
making procedure in which the participants
have varying numbers of votes.
Examples:
Shareholder elections
Some legislative bodies
Electoral College
Key Terms and Notation
Weight
Quota
Shorthand notation:
[q: w1, w2, …, wn]
Coalition Building
Rarely will one voter have enough votes to
meet the quota so coalitions are
necessary to pass any measure
Types of coalitions
Winning Coalition
Losing Coalition
Blocking Coalition
Dummy voters
Coalition Illustration
On the right is a table of Shareholder # of shares
the weights of
shareholders of a Ruth Smith 9
company.
A simple majority (16 Ralph Smith 9
votes) is needed for any Albert Mansfield 7
measure.
Ide, Lambert, and Kathrine Ide 3
Edwards are all Dummy
Voters as any winning Gary Lambert 1
coalition including any
subset of those three Marjorie 1
would be a winning Edwards
coalition without them. Total 30
How do we Measure
an individual’s power?
Critical Voter
Banzhaf Power Index
Developed by John F Banzhaf III
1965- “Weighted Voting Doesn’t Work”
The number of winning or blocking coalitions
in which a participant is the critical voter
Critical Voter Illustration
Consider a committee of three members
The voting system follows this pattern:
[3: 2, 1, 1]
For ease, we’ll refer to the members as A, B, and C
A B C Votes Outcome A B C Votes Outcome
Y Y Y 4 Pass Y Y Y 4 Pass
Y NY 3 Pass NY Y 2 Fail
Extra Votes
A helpful concept in calculating Banzhaf
Power Index
A winning coalition with w votes has w-q
extra votes
Any voter with more votes than the extra
votes in the coalition is a critical voter
Calculating Banzhaf Index
Weight Winning Extra Weight Blocking Extra
Coalitions Votes Coalitions Votes
3 [A,B];[A,C] 0 2 [A];[B,C] 0
3 [A,B];[A,C] 1
4 [A,B,C] 1
4 [A,B,C] 2
In Winning Coalitions; A is a critical voter three
times, B and C are critical voters once
In Blocking Coaltions; A is a critical voter three
times, B and C are critical voters once
Banzhaf Index of this system: (6,2,2)
Notice a Pattern there?
Each voter is a critical voter in the same
number of winning coalitions as blocking
coalition
When a voter defects from a winning
coalition they become the critical voter in a
corresponding blocking coalition
[A, B, C]=>[A]
[A, B]=>[A, C]
[A, C]=>[A, B]
How does this help?
Because these numbers are identical, we
can calculate the Banzhaf Power Index by
finding the number of winning coalitions in
which a voter is the critical voter and
double it
Can make computations easier in systems
with many voters
[51: 40, 30, 20, 10]
Banzhaf Index
From the table above we can see that in
winning coalitions,
A is a critical vote 5 times
B and C are critical votes 3 times each
D is a critical vote once
So, their Banzhaf Index is twice that,
A=10, B=6, C=6, and D=2
Their voting power is
A=10/24 B=6/24 C=6/24 D=2/24
The Electoral College
Shapley-Shubik Power Index
For coalitions built one voter at a time
The voter whose vote turns a losing
coalition into a winning coalition is the
most important voter
Shapley-Shubik uses permutations to
calculate how often a voter serves as the
pivotal voter
This index takes into account commitment
to an issue
How do we find the pivotal voter?
The first voter in a permutation of voters
whose vote would make a the coalition a
winning coalition is the pivotal voter
The Shapley-Shubik power index is the
fraction of the permutations in which that
voter is pivotal
Formula:
(number times the voter is pivotal)
(number of permutations of voters)
What does this overlook?
Example
Permutations Weights
A B C 2 3 4
A C B 2 3 4
B A C 1 3 4
B C A 1 2 4
C A B 1 3 4
C B A 1 2 4
Shapley-Shubik indexes:
A=4/6 B=1/6 C=1/6
For a larger corporation
Larger Corporation (cont)
This is the same corporation we looked at
earlier distributed as [51: 40, 30, 20, 10]
The Shapley-Shubik Index for the four
people in the corporation is:
A=10/24 B=6/24 C=6/24 D=2/24
So here, the Banzhaf and Shapley Shubik
indexes agree, but is this always true?
Comparing the Indexes
The Banzhaf index assumes all votes are
cast with the same probability
Shapley-Shubik index allows for a wide
spectrum of opinions on an issue
Shapley-Shubik index takes commitment
to an issue into account
An illustration of the difference
Consider a corporation of 9001
shareholders
Such a large corporation can only be
analyzed if nearly all of the voters have the
same power
So, we will consider a corporation with 1
shareholder owning 1000 shares and 9000
shareholders each owning one share, and
assume a simple majority
Under Shapley-Shubik
The big voter will be the critical voter in
any permutation that positions at least
4001 of the small voters before him, but no
more than 5000
We can group the permutations into 9001
equal groups based on the location of the
big shareholder
Shapley-Shubik (cont)
We can see that the big shareholder is the
pivotal voter in all permutations in groups
4002 through 5001
So, the big shareholder has a Shapley-
Shubik index of 1000/9001
The remaining 8001/9001 power goes
equally to the 9000 small voters
Under Banzhaf
We can estimate the big shareholder’s
Banzhaf Power Index can be estimated
assuming a each small shareholder
decides his vote by a coin toss
The big shareholder will be a critical voter
unless his coalition is joined by fewer than
4001 small shareholders or at least 5001
small shareholders
Banzhaf (cont)
When the 9000 small shareholders toss their
coins, the expected number of heads is ½ *
9000 = 4500
The standard deviation is roughly 50
By the 68-95-99.7 rule we can see that there is a
68% chance of 4450-4550 heads
95% chance of 4400-4600 heads
99.7% chance of 4350-4650 heads
You can see that the big shareholder’s Banzhaf
Index is nearly 100%
Which seems fairer?
The Shapley-Shubik Index gave the big
shareholder roughly 11% of the power
while the Banzhaf Index gave him nearly
100% of the power
The big shareholder has roughly 11% of
the votes
Which index seems more realistic?
Why are the indexes so different when
earlier they came out the same?
Homework