1. § 2.2 - 2.3 The Banzhaf Power Index

2. 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 players can join together to pass a motion?
P1 and P2 (This group controls 94 votes).
P1 and P3 (This group controls 55 votes).
P2 and P3 (This group controls 51 votes).
P1 , P2 and P3 (This group controls all of the votes).

3. 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.
The grand coalition is the coalition consisting of all the players.

4. 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.

5. Example: Let us return to our [51 : 49, 45, 6] example and examine all of the possible coalitions that could be formed.
Coalition
Weight
Win/Lose
Critical Players 1
{P1} 49
Lose
N/A 2
{P2} 45 3
{P3} 6 4
{P1 ,P2} 94
Win
P1 ,P2 5
{P1 ,P3} 54
P1 ,P3
{P2 ,P3} 51
P2 ,P3 7
{P1 ,P2 ,P3}
100
None
There are a total of 6 Critical Players.
Each Player is a Critical Player in 2 coalitions.
So we could say that each Player’s ‘power’ is: /6 = 1/3

6. The Banzhaf Power Index
The Idea: A player’s power is proportional to the number of coalitions for which the player is critical.

7. 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 fraction B/T.

8. 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?

9. Example: Step 1. We have the following seven coalitions: 1 {P1} 2 {P2}3
{P3} 4
{P1 ,P2} 5
{P1 ,P3} 6
{P2 ,P3} 7
{P1 ,P2 ,P3}

10. Example: Step 2. We have the following winning coalitions: 1 {P1}
6 votes 2
{P2} 5 3
{P3} 4
{P1 ,P2} 11
{P1 ,P3} 10 6
{P2 ,P3} 9 7
{P1 ,P2 ,P3} 15

11. 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

12. Example: Step 4. P1 is critical three times. P2 is critical one time.
Winning Coalitions
Critical Players
{P1 ,P2}
P1 ,P2
{P1 ,P3}
P1 ,P3
{P1 ,P2 ,P3} P1

13. Example: Step 5. There are a total of 3 + 1 + 1 = 5 critical players.
The Banzhaf Power Index for each player is
P1 : 3/5 P2 : 1/5 P3 : 1/5
Winning Coalitions
Critical Players
{P1 ,P2}
P1 ,P2
{P1 ,P3}
P1 ,P3
{P1 ,P2 ,P3} P1

14. 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.

15. The Banzhaf Power Index
Set
{P1 ,P2}
{P1 ,P2 ,P3}
{P1 ,P2 ,P3,P4}
{P1 ,P2 ,P3,. . .,PN }
# of subsets 4 8 16 2N
Subsets
{ } {P1} {P2} {P1 ,P2}
{ } {P3} {P1} {P1 ,P3} {P2} {P2 ,P3} {P1 ,P2} {P1 ,P2,P3}
. . .
# of coalitions 3 7 15
2N - 1
One question we might care about is, “How many coalitions are there given a certain number of players?”

16. 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}

17. 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].

18. 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].

19. 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}

20. Example: Consider the weighted voting system described by [9 : 6, 4, 2, 1].
Here our Banzhaf power distribution looks like: P1 : 5/9 P2 : 4/9 P3 : 1/9 P4 : 0
(Notice that P4 has no power--this means that P4 is a dummy.)
Winning Coalitions
{P1 ,P2}
{P1 ,P2,P3}
{P1 ,P2,P4}
{P1 ,P3 ,P4}
{P1 ,P2,P3 ,P4}

21. Example: The system is now written as [9 : 5, 5, 2, 1] and our winning coalitions are:
Now suppose that P3--indignant over P1 ’s level of power demands that P1 give a vote up to P2.
Winning Coalitions
{P1 ,P2}
{P1 ,P2,P3}
{P1 ,P2,P4}
{P1 ,P2,P3 ,P4}

22. Example: The system is now written as [9 : 5, 5, 2, 1] and our winning coalitions are:
Now suppose that P4--indignant over P1 ’s level of power demands that P1 give a vote up to P3. In this case, our distribution becomes:
P1 : 1/2 P2 : 1/2 P3 : 0 P4 : 0
While P1 ‘s power has decreased so has P3 ‘s!
Winning Coalitions
{P1 ,P2}
{P1 ,P2,P3}
{P1 ,P2,P4}
{P1 ,P2,P3 ,P4}