Weighted Voting
Systems
Chapter 2
Objective: Calculate the Banzhaf power Index for a weighted
voting system. Learn additional notation and terminology for
coalitions.

List the winning coalitions of a
weighted voting system. Identify
the critical players in each winning
coalition. Calculate the Banzhaf
Power distribution and determine
if the weighted voting system is
fair.
Closing Product
Exit Ticket

Any set of players that might join forces
and vote the same way. Can be as little as
one player to all players
Coalitions

Consists of all players in the set.
(unanimous vote)
Grand Coalition

A coalition that has enough votes to pass
the motion.
Winning Coalition

A coalition that has enough votes to
prevent a motion from passing.
Blocking Coalition

A player in a winning coalition, whose
votes are necessary in order to pass the
motion. In other words, the one(s) that
has veto power.
Critical Players
Consider the weighted voting system
[95: 65, 35, 30, 25]. Find the following:
a)
The total number of coalitions
b)
List the winning coalitions
Example #1
2n -1
Where n=total number of players
Total Number of Coalitions
Consider the weighted voting system [22: 10, 8, 7, 2, 1]
a)
What is the total number of coalitions?
b)
List all of the coalitions
Example #2

A mathematical measurement of a
player’s power in a weighted voting
system based on how many times that
player is a critical player.

β1 = # of time Player 1 is a critical player
sum of all critical players
Pronounced ‘beta-one’
Banzhaf Power Index

A complete list of the power indexes

β1 , β2 ,β3, …… βn

The sum of all the β’s is equal to 1. (or
100% if using percentages.
Banzhaf Power Distribution
i.
ii.
iii.
iv.
v.
Make a list of all winning coalitions.
Determine the critical player of each
coalition.
Count the number of times each player
is critical. (B1 for P1, B2 for P2, etc.)
Find the total number of times all
players are critical. (T = B1+ B2+ ..Bn)
Find the ratio for each player (β1 = B1/T)
Calculation
[49: 48, 24, 12, 12]

Winning Coalitions.
1 Player
2 Players
3 Player
4 Player
None
P1, P2
P1, P2, P3
P1, P2, P3, P4
P1, P3
P1, P2, P4
P1, P4
P1, P3, P4
B1= 7
B2= 1
B3= 1
B4= 1
T = 10
β1 = 7/10, β2 = 7/10 ,β3 = 7/10, β4 = 7/10
Consider the following …..
Do the next one on your own.
[14: 8, 4, 2, 1]

Winning Coalitions.
1 Player
2 Players
3 Player
4 Player
None
None
P1, P2, P3
P1, P2, P3, P4
B1= 2
B2= 2
B3= 2
B4= 0
T=6
β1 = 1/3, β2 = 1/3 ,β3 = 1/3, β4 = 0
What would happen if the weighted
voting system was [15: 16, 8, 4, 1] ?
Write down a sentence that
summarizes what the Banzhaf Power
Index would be for dictators and
dummies.
Consider the following ….

Dictator – Banzhaf
Power Index always
1, or 100%

Dummy – Banzhaf
Power Index always
0, or 0%
Consider the following …..

Used to determine if a weighted voted
system is set up in a fair manner. If it is
fair, the relative weight of the votes
should be comparable to the Banzhaf
power index.
Application of Banzhaf
Sued several times to change their
weighted voting system that gave more
power to bigger districts.
 Finally in 1991, NY Civil Liberties Union
sued and won (1993).
 It was shown that minority districts were
not receiving fair representation .

Nassau County Board of
Supervisors
Banzahf’s Power index showed for
example that Hempstead had 56% of
population but 70% of the power.
 Now districts are drawn equally, and they
no longer use a weighted system.

Nassau County Board of
Supervisors