Finite Math A Chapter 2, Weighted Voting Systems
Finite Mathematics Notes
Chapter 2: Weighted Voting Systems – The Power Game
PS.ED.2: Use election theory techniques to analyze election data. Use weighted voting
techniques to decide voting power within a group.
Academic Standards:
ONE PERSON – ONE VOTE
is an democratic idea of equality
But what if the voters are not PEOPLE but are governments? countries? states?
If the institutions are not equal, then the number of votes they control should not be equal.
The United Nations Security Council – 15 voting nations: 5
permanent members (Britain, China, France, Russia, United
States), 10 nonpermanent members appointed for a 2-year
rotation. Permanent members have more “votes” than non
permanent members.
Stock Holders/Shareholders : The more
stock you own, the more say you have in
decision making for the company.
The Electoral College—Each state gets a number of “electors” (votes) equal to the
number of Senators plus the number of Representatives in Congress. California
has 55 votes but North Dakota only has 3 votes. Each state is a voter but states
with heavy concentration of population receive a bigger “vote”.
This situation where each voter is not equal in the number of votes they control is called:
2.1 An Introduction to weighted voting
Important terms:
_________________________________________: A voting situation where voters are not necessarily equal in
the number of votes they control.
_________________: A vote with only two choices. (usually yes/no)
_________________: The voters (symbolized by P1, P2, P3, etc.)
_________________: The number of votes a player controls.
_________________: The smallest number of votes required to “pass” a motion.
Notation
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Finite Math A Chapter 2, Weighted Voting Systems
[q: w1, w2, w3, . . ., wn]
Example 1.
q=
2
w’s =
[14: 8, 6, 5, 1]
quota =
Player 1 (P1) = controls ___ votes /“has a weight of ____”
total votes =
Player 2 (P2) = controls ___ votes
Player 3 (P3) = controls ___ votes
Player 4 (P4) = controls ___ vote
Example 2.
Given the weighted voting system [16: 8, 6, 4, 4, 3, 1], state the following:
The number of players: ____
The total number of votes: ____
The weight of P5: ____
The minimum % of the quota to nearest whole %: ____
Common Types of Quotas:
U.S. Senate:
Simple Majority to pass an
ordinary law (51 votes)
Simple majority/Strict majority 
Two-thirds majority 
60 votes to stop a filibuster
Unanimity 
2/3 of the votes (67) to
override a presidential veto.
Weighted Voting Issues
Example 3:
Four partners decide to start a business. P1 buys 8 shares, P2 buys 7 shares, P3 buys 3 shares and P4 buys 2 shares.
One share = one vote.
a. The quota is set at two-thirds of the total number of votes. Describe as a weighted voting system.
b. The partnership above decides the quota is too high and changes the quota to 10 votes.
c. The partnership above decides to make the quota equal to 21 votes.
For a weighted voting system to be legal: the quota must be at least a __________________________________
and no more than ___________________________________________
Finite Math A Chapter 2, Weighted Voting Systems
Symbolically:
If V  w1  w2  w3  ...  wn , then
3
V
 q V
2
Example 3d.
What if our partnership changed the quota to 19?
4.
[q: 7, 2, 1, 1, 1]
What is the smallest legal quota? _____
What is the largest legal quota? _____
5.
What is the value of the quota if at least two-thirds of the votes are required to pass a motion?
______
What is the value of the quota if more than three-fourths of the votes are required to pass?
______
A committee has 4 members (P1, P2, P3, P4). P1 has twice as many votes as P2. P2 has twice as many votes
as P3. P3 and P4 have the same number of votes. The quota is 49. Describe the weighted voting system
using the notation [q: w1, w2, w3, w4] given the definitions of quota below. (Hint: write the weighted
voting system as [49: 4x, 2x, x, x] and then solve for x.
a) The quota is a simple majority
b) The quota is more than three-fourths
Dictators, Dummies, and Veto Power
Example 6:
[11: 12, 5, 4]
What do you notice about P1 ?
Note:
If any player is a
dictator, then
EVERY OTHER
PLAYER is a dummy.
Even if there is no
dictator, there may
Finite Math A Chapter 2, Weighted Voting Systems
4
P1 has all the power 
P2 and P3 have no power 
Example 7:
[30: 10, 10, 10, 9]
Example 8:
[12: 9, 5, 4, 2]
If a player is not a dictator, but
the other players cannot meet
the quota without his votes,
we say he has veto power.
Sometimes, more than one
player will have veto power.
Is there a dictator ?
If P1 chooses to vote against the motion, can the other
players combine weight to meet the quota?
Example 9.
Determine which players, if any, are: dictators, veto power, dummies
a) [15: 16, 8, 4, 1]
Example 10.
b) [18: 16, 8, 4, 1]
c) [24: 16, 8, 4, 1]
Consider [q: 8, 4, 2]. Find the smallest value of q for which
a) all three players have veto power
c) P3 is the only dummy
2.2/2.3 The Banzhaf Power Index
Who is the most POWERFUL player?
b) P2 has veto power, but P3 does not
Finite Math A Chapter 2, Weighted Voting Systems
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_________________: A group of players who choose to vote together
_______________________________: The set of all voters. This represents a unanimous vote.
Weight of the coalition:
Winning coalitions—
Losing coalitions—
________________________ : Any player who MUST BE PRESENT in a winning
coalition in order for it to remain a winning coalition.
Note:
If you subtract the
critical player’s votes
from the coalition, the
number of votes drops
below the quota.
Example 1: Find the critical player or critical players in each of the following coalitions.
[15: 13, 9, 5, 2]
[51: 30, 25, 25, 20]
The Banzhaf Power Index:
a) {P1, P4}
b) {P2, P3, P4}
c) {P3, P4}
d) {P1, P2, P3}
a) {P1, P3}
b) {P1, P2, P3}
c) {P2, P3, P4}
d) {P2, P3}
A player’s power is proportional to the number of coalitions for which that player
is critical. The more often a player is critical, the more power he holds.
Finite Math A Chapter 2, Weighted Voting Systems
Calculate the Banzhaf Power
Index:
Idea:
Step 1:
Make a list of all WINNING coalitions.
Step 2:
Determine which players are critical in each coalition. (circle,
underline, highlight)
Step 3:
Count the total number of times each player is critical
Step 4:
Add the total number of times each player is critical to find the
grand total number of critical players.
The Banzhaf Power INDEX number for
each player = step 3 ÷ step 4
The Banzhaf Power DISTRIBUTION for
the weighted voting system is the % of
power each player holds.
Example 2:
Find the Banzhaf Power index for the weighted voting system:
[101: 99, 98, 3]
Example 3:
Find the Banzhaf Power Distribution for
[4: 3, 2, 1]
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Finite Math A Chapter 2, Weighted Voting Systems
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How do you know you have all the possible coalitions written down?
If n = number of players in a weighted voting system,
Be systematic or use the formula!
Then the number of possible coalitions is: 2n – 1
How many coalitions if 4 players?
How many coalitions if 5 players?
Example 4: Find the Banzhaf Power Distribution for
[6: 4, 3, 2, 1]
Banzhaf Coalitions: 4 Players
{P1}
{P2}
{P3}
{P4}
{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}
Example 5:
Consider the weighted voting system [q: 8, 4, 2, 1] . Find the Banzhaf Power Distribution of this weighted voting
system when:
a) q = 8
b) q = 10
c) q = 14
Finite Math A Chapter 2, Weighted Voting Systems
What I expect to see for “work” on your homework:
1. Write down all possible coalitions and cross off losers OR just the winning coalitions.
2. Critical Players should be circled or underlined.
3. Show fraction of BPI for each player AND calculate the % for BPD.
Banzhaf Coalitions: 4 Players
Possible Coalitions:
Use these to help you:
Banzhaf Coalitions: 3 Players
{P1}
{P1,P2}
{P1, P2, P3}
{P2}
{P1,P3}
{P3}
{P2,P3}
{P1}
{P2}
{P3}
{P4}
{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}
Where weighted voting systems/Banzhaf are used:
Banzhaf is used to QUANTIFY the amount of power each player holds.
1.
Nassau County Board of Supervisors (see p. 55):
Votes were given to districts according to
population and quota was simple majority. [58: 31,
31, 28, 21, 2, 2] Banzhaf showed that two of the six
counties actually had no voting power—that they
were actually dummy voters. Final result: 1993
court decision abolishing weighted voting in New
York States. “Districts” were created of roughly the
same population and each given one voted.
2.
United Nations Security Council: Banzhaf shows
that a permanent member of the council holds
more than 10 times the amount of power as one of
the non-permanent members. There are 5
permanent members (Britain, China, France,
Russia, US) and 10 non-permanent members. This
voting arrangement may change as others are
being considered for permanent membership.
3.
European Union Banzhaf quantifies the amount of power each nation has and shows that smaller nations such as
Luxembourg and Malta still hold some power.
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Finite Math A Chapter 2, Weighted Voting Systems
2.4/2.5 The Shapley Shubik Power Index:
The Shapley-Shubik Power Index:
A player’s power is proportional to the
number of sequential-coalitions for which
that player is pivotal. The more times a
player is pivotal, the more power he holds.
Sequential coalition:
Shapley-Shubik: ⟨𝑃1 , 𝑃3 , 𝑃2 ⟩
Banzhaf: { P1, P2, P3}
These 3 players decide to vote together.
They form a coalition.
Order listed in the { } doesn’t matter.
These 3 players decide to vote together.
P1 votes 1st, P3 votes 2nd , P2 votes 3rd.
They form a sequential coalition.
Order listed in the ⟨ ⟩ is important.
Pivotal player:
Example: Find the Pivotal Player
1. Given the weighted voting system [5: 3,2,1,1} find the pivotal player for the given sequential coalition.
a) [P1,P4,P3,P2]
b) [P3,P1,P2,P4]
c) [P4,P3,P2,P1]
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Finite Math A Chapter 2, Weighted Voting Systems
Counting Sequential Coalitions:
List the possible sequence for 3 players. How many are there?
How many sequential coalitions are there for 4 players? For 5 players?
Multiplication Rule:
If there are m ways to do task
1 and n ways to do task 2,
then there are mxn ways to do
both tasks together.
Factorials:
Shapley-Shubik Power Distribution
Step 1:
Make a list of all sequential coalitions
Step 2:
For each sequential coalition, determine the pivotal
player.
Step 3:
For each player, count the number of times they are
pivotal and divide by the number of sequential
coalitions.
Calculate the %.
Example 2: Find the Shapely
Shubik Power Distribution
for
[4: 3, 2, 1]
Sequential
Coalitions:
3 Players
[P1,P2,P3]
[P1,P3,P2]
[P2,P1,P3]
[P2,P3,P1]
[P3,P1,P2]
[P3,P2,P1]
If N= the number of players,
then the number of sequential
coalitions is N!
N! = N x (N-1) x . . . x 3 x 2 x 1
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Finite Math A Chapter 2, Weighted Voting Systems
Example 3:
Find the Shapley-Shubik Power
Distribution for [6: 4, 3, 2, 1]
Sequential Coalitions: 4 Players
[P1,P2,P3,P4]
[P2,P1,P3,P4]
[P3,P1,P2,P4]
[P4,P1,P2,P3]
[P1,P2,P4,P3]
[P2,P1,P4,P3]
[P3,P1,P4,P2]
[P4,P1,P3,P2]
[P1,P3,P2,P4]
[P2,P3,P1,P4]
[P3,P2,P1,P4]
[P4,P2,P1,P3]
[P1,P3,P4,P2]
[P2,P3,P4,P1]
[P3,P2,P4,P1]
[P4,P2,P3,P1]
[P1,P4,P2,P3]
[P2,P4,P1,P3]
[P3,P4,P1,P2]
[P4,P3,P1,P2]
[P1,P4,P3,P2]
[P2,P4,P3,P1]
[P3,P4,P2,P1]
[P4,P3,P2,P1]
Example 4:
Find the Shapley-Shubik Power
Distribution for [10: 8, 4, 2, 1]
Sequential Coalitions: 4 Players
[P1,P2,P3,P4]
[P2,P1,P3,P4]
[P3,P1,P2,P4]
[P4,P1,P2,P3]
[P1,P2,P4,P3]
[P2,P1,P4,P3]
[P3,P1,P4,P2]
[P4,P1,P3,P2]
[P1,P3,P2,P4]
[P2,P3,P1,P4]
[P3,P2,P1,P4]
[P4,P2,P1,P3]
[P1,P3,P4,P2]
[P2,P3,P4,P1]
[P3,P2,P4,P1]
[P4,P2,P3,P1]
[P1,P4,P2,P3]
[P2,P4,P1,P3]
[P3,P4,P1,P2]
[P4,P3,P1,P2]
[P1,P4,P3,P2]
[P2,P4,P3,P1]
[P3,P4,P2,P1]
[P4,P3,P2,P1]
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