§ 2.2 - 2.3 The Banzhaf Power
Index
Example: Now we will continue with our
“Consensus Party” example from last time. We
saw yesterday that this hypothetical situation
could be written as [51 : 49, 45, 6]. We also
noticed that the number of votes each player
controls is not a good measure of the actual
power they possess in the system.
One question we might ask is, which sets of
• Pcan
players
joinPtogether
to passcontrols
a motion?94
1 and
2 (This group
votes).
• P1 and P3 (This group controls 55
votes).
• P2 and P3 (This group controls 51
Terminology
 A set of players that join forces to vote
together will be referred to as a
coalition.
 The total number of votes controlled by
the coalition is the weight of the
coalition.
 Coalitions that can pass a motion are
winning coalitions, those that cannot are
losing coalitions.
Terminology
 A critical player for a coalition is a
player whose absence would cause a
winning coalition to become a losing
coalition.
 We will use this concept to define the
Banzhaf Power Index.
Example: Let us return to our [51 : 49, 45,
6] example and examine all of the possible
coalitions that could be formed.
1
Coalitio Weight Win/Los Critical Players
n
There areea total of 6
{P1} Critical
49
Lose
N/A
Players.
2
{P2}
45
Lose
N/A
3
Each Player is a Critical
{P3} Player
6
N/A
in 2Lose
coalitions.
4
{P1 ,P2} 94
Win
P1 ,P2
{P1 ,P2
,P3}
Win
None
5
6
7
So we could say that
{P1 ,P3each
} 54 Player’s
Win ‘power’
P1 ,Pis:
3
{P2 ,P3} 51
Win= 1/3 P2 ,P3
2/6
100
The Banzhaf Power Index
 The Idea: A player’s power is
proportional to the number of coalitions
for which the player is critical.
The Banzhaf Power Index
Finding the Banzhaf Power Index of Player P :
Step 1. Make a list of all possible coalitions.
Step 2. Determine which coalitions are winning
coalitions.
Step 3. Determine which players are critical for
each winning coalition.
Step 4. Count the total number of times player P is
critical--call this number B.
Step 5. Count the total number of times all players
are critical--call this number T.
The Banzhaf Power Index for the player P is the
Example: The countries of Pottsylvania,
Moosylvania and Upper-Lower Watchikowistan
have decided to form an economic union.
Pottsylvania will have 6 votes, Moosylvania will
have 5 and Upper-Lower Watchikowistan will
have 4. For a motion to be accepted by the
union as a whole it must have the support of 10
votes. How is the power divided amongst the
three countries?
Example:
Step 1. We have the following seven
coalitions:
1
2
3
{P1}
{P2}
{P3}
4
5
6
7
{P1 ,P2}
{P1 ,P3}
{P2 ,P3}
{P1 ,P2
,P3}
Example:
Step 2. We have the following winning
coalitions:
1
2
3
{P1}
{P2}
{P3}
6 votes
5
4
4
5
6
7
{P1 ,P2}
{P1 ,P3}
{P2 ,P3}
{P1 ,P2
,P3}
11
10
9
15
Example:
Step 3. We have the following critical
players:
Winning
Coalitions
Critical
Players
{P1 ,P2}
P1 ,P2
{P1 ,P3}
P1 ,P3
{P1 ,P2 ,P3}
P1
Example:
Step 4.
P1 is critical three times.
Winning
P2 is critical
one time.
Coalitions
P3 is critical one time.
{P1 ,P2}
Critical
Players
P1 ,P2
{P1 ,P3}
P1 ,P3
{P1 ,P2 ,P3}
P1
Example:
Step 5.
There are a total of 3 + 1 + 1 = 5 critical
players.Winning
Critical
Coalitions
Players
{P1 ,P2}Power Index
P1 ,P
The Banzhaf
for2 each player is
{P
3}
P11 :,P3/5
P3 :,P1/5,P }
{P
1
2
3
P1 ,PP3 : 1/5
2
P1
The Banzhaf Power Index
 The complete list of every player’s
power indices is called the Banzhaf
power distribution.
 Generally these distributions are given
in percentage form.
The Banzhaf Power Index
Set
{P1 ,P2}
{P1 ,P2 ,P3}
{P1 ,P2 ,P3,P4}
{P1 ,P2 ,P3,. .
 One question we might care about.,Pis,
N}
“How
many
# of
4
8 coalitions16are there given
2N a
subsetscertain number of players?”
Subsets { }
{ }
...
...
{P1}
{P3}
{P2}
{P1}
{P1 ,P2}
{P1 ,P3}
{P2}
{P2 ,P3}
{P1 ,P2} {P1
,P2,P3}
# of
3
7
15
2N - 1
Notation: If we look at the previous
example one more time, there is another way
that critical players can be denoted. Here we
have listed the winning coalitions--critical
players are underlined.
Winning
Coalitions
{P1 ,P2}
{P1 ,P3}
{P1 ,P2 ,P3}
Example: The European Union, prior to its
recent expansion, was an economic and
political confederation consisting of 15
countries. The nations at the time were France,
Germany, Italy and the UK (10 votes each);
Spain (8 votes); Belgium, Greece, Netherlands
and Portugal (5 votes each); Austria and
Sweden (4 votes each); Denmark, Finland and
Ireland (3 votes each); Luxembourg (2 votes).
In this system there are a total of 87 votes and
a quota of 62.
This means the system can be fully described
as
[62: 10, 10, 10, 10, 8, 5, 5, 5, 5, 4, 4, 3, 3,
3, 2].
Example: The European Union, prior to its
recent expansion, was an economic and
political confederation consisting of 15
countries. The nations at the time were France,
Germany, Italy and the UK (10 votes each);
Spain (8 votes); Belgium, Greece, Netherlands
and Portugal (5 votes each); Austria and
Sweden (4 votes each); Denmark, Finland and
Ireland (3 votes each); Luxembourg (2 votes).
In this system there are a total of 87 votes and
a quota of 62.
This means the system can be fully described
as
[62: 10, 10, 10, 10, 8, 5, 5, 5, 5, 4, 4, 3, 3,
3, 2].
Example:
Consider the weighted voting
system described by [9 : 6, 4, 2, 1].
If we were to check the winning coalitions in
this example we would find the following:
Winning
Coalitions
{P1 ,P2}
{P1 ,P2,P3}
{P1 ,P2,P4}
{P1 ,P3 ,P4}
{P1 ,P2,P3 ,P4}
Example:
Consider the weighted voting
system described by [9 : 6, 4, 2, 1].
Here our Banzhaf power distribution looks like:
Winning
Coalitions
P : 5/9
P2 : 4/9
1
{P
1/9
1 ,P2}
P3 :
P4 : 0
{P1 ,P2,P3}
(Notice
that P4 has no power--this means that
{P
1 ,P2,P4}
P4 is a dummy.)
{P1 ,P3 ,P4}
{P1 ,P2,P3 ,P4}
Example:
system over
is now
[9
Now suppose thatThe
P --indignant
P ’swritten
level ofas
power
3
1
thatand
P1 give
vote up to
P2.
:demands
5, 5, 2, 1]
our awinning
coalitions
are:
Winning
Coalitions
{P1 ,P2}
{P1 ,P2,P3}
{P1 ,P2,P4}
{P1 ,P2,P3 ,P4}
Example:
The system is now written as [9
thatour
P4--indignant
over P1 ’s level
:Now
5, 5,suppose
2, 1] and
winning coalitions
are:of power
demands that P1 give a vote up to P3. In this case, our
distribution becomes:
Winning
Coalitions
{P1 ,P2}
P4 : 0
P1,P
: 1/2
{P
1
2,P3}
P2 : 1/2
{P1 ,P2,P4}
{P1 ,P2,P3 ,P4}
While P1 ‘s power has decreased so has P3 ‘s!
P3 : 0