# 1 5 Weighted voting and power

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```					      1.5 Weightd voting and voting power

Goal: For given holdings, find the power index
of each stockholder.

Is a person’s influence or power proportional to
the fraction of votes he casts?
Weighted Voting Systems
Outline/learning Objectives
• Represent a weighted voting system using a
mathematical model.
• Use the Banzhaf index to calculate the
distribution of power in a weighted voting
system.
Weighted Voting Systems
• The Players
The voters in a weighted voting system.
• The Weights
That each player controls a certain number
• The Quota
The minimum number of votes needed to
• Dictator
The player’s weight is bigger than or equal
to the quota.

Consider [11:12, 5, 4]
P1
owns enough votes to carry a motion
single handedly.
• Dummy
A player with no power.

Consider [30: 10, 10, 10, 9]
P4
turns out to be a dummy! There is never
P4
going to be a time when      is going to
make a difference in the outcome of the
voting.
• Veto Power
If a motion cannot pass unless player votes
in favor of the motion.

Consider [12: 9, 5, 4, 2]
P1
has the power to obstruct by preventing
any motion from passing.
• Coalitions
Any set of players that might join forces and vote
the same way. The coalition consisting of all the
players is called a grand coalition.
• Winning Coalitions
Some coalitions have enough votes to win
and some don’t. We call the former
winning coalitions and the latter losing
coalitions.
• Critical players
In a winning coalition, a player is said to be a
critical player for the coalition if the coalition
must have that player’s votes to win. If and
W-w<q
only if
– Computing a Banzhaf Power Distribution
• Step 1. Make a list of all possible winning
coalitions.
Step 2. Within each winning coalition
determine which are the critical players. (To
determine if a given player is critical or not
in a given winning coalition, we subtract the
player’s weight from the total number of
votes in the coalition- if the difference drops
below the quota q, then that player is
critical. Otherwise, that player is not
essential/critical.
• Step 3.Count the number of times that
P 1is critical. Call this numberB1 Repeat for
each of the other players to find B 2, B3, ...,BN
Step 4. Find the total number of times
all players are critical. This total is
given by

T =B + B
1     2   + ... + BN
– Computing a Banzhaf Power Distribution
• Step 5. Find the ratio 1       .
=B T
1
This gives the Banzhaf power index of        .
P1
Repeat for each of the other players to find
2, 3, …, N . The complete list of ’s gives
the Banzhaf power distribution of the
weighted voting system.
Applications of Banzhaf Power
• The Nassau County Board of
Supervisors
John Banzhaf first introduced the concept
• The United Nations Security Council
Classic example of a weighted voting
system
• The European Union (EU)
Relative Weight vs Banzhaf Power Index

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