Method in which relative control over the outcome of a motion within a
Weighted Voting System. Invented by
John F. Banzhaf III in order to win a court case. The situation went something like this (numbers improvised):
[ 60 : 33, 32, 29, 20, 3, 2 ]
Where the
Quota is 60, meaning 60 votes for the motion are needed for it to be passed, and there are 6
Players, who each have respective votes, or shares.
Coincidently the quota is 51% in this system, but keep in mind the quota is not always this. In the
Senate, for example, there are instances where the quota is 67%.
To find the
Banzhaf Power Index of this system, begin by listing the winning
Coalitions.
P1, P2, P3, P4, P5, P6
P1, P2, P3, P4, P5
P1, P2, P3, P4, P6
P1, P2, P3, P5, P6
P1, P2, P3, P4
P1, P2, P3, P5
P1, P2, P3, P6
P1, P2, P3
P1, P2, P4, P5, P6
P1, P2, P4, P5
P1, P2, P4, P6
P1, P2, P5, P6
P1, P2, P4
P1, P2, P5
P1, P2, P6
P1, P2
P1, P3, P4, P5, P6
P1, P3, P4, P5
P1, P3, P4, P6
P1, P3, P5, P6
P1, P3, P4
P1, P3, P5
P1, P3, P6
P1, P3
P2, P3, P4, P5, P6
P2, P3, P4, P5
P2, P3, P4, P6
P2, P3, P5, P6
P2, P3, P4
P2, P3, P5
P2, P3, P6
P2, P3
Every winning combination of each player is listed here. To continue, underline/highlight/write down the
Critical Players, where, if that player was removed, or were to vote against, the coalition could not pass the motion.
P1, P2, P3, P4, P5, P6
P1, P2, P3, P4, P5
P1, P2, P3, P4, P6
P1, P2, P3, P5, P6
P1, P2, P3, P4
P1, P2, P3, P5
P1, P2, P3, P6
P1, P2, P3
P1, P2, P4, P5, P6
P1, P2, P4, P5
P1, P2, P4, P6
P1, P2, P5, P6
P1, P2, P4
P1, P2, P5
P1, P2, P6
P1, P2
P1, P3, P4, P5, P6
P1, P3, P4, P5
P1, P3, P4, P6
P1, P3, P5, P6
P1, P3, P4
P1, P3, P5
P1, P3, P6
P1, P3
P2, P3, P4, P5, P6
P2, P3, P4, P5
P2, P3, P4, P6
P2, P3, P5, P6
P2, P3, P4
P2, P3, P5
P2, P3, P6
P2, P3
For each player, list the number of times he or she has been a critical player. Divide that by the total number of critical players. Behold the Banzhaf Power Index.
P1: 16/48 = 33%
P2: 16/48 = 33%
P3: 16/48 = 33%
P4: 0/48 = 0%
P5: 0/48 = 0%
P6: 0/48 = 0%
Players 1-3, dispite having different weighted votes, each have the same 'say' in the vote. Players 4-6 have no voice in the vote, and might as well stay home and watch
the simpsons. Not even P4, with a whopping 20 votes has enough weight to throw around. Players 4 thru 6 are labeled
Dummies. If a player were to have 100% power, he would be labeled a
Dictator.
And yes,
Banzhaf was arguing on behalf of the 3 smaller districts, and used this demonstration to prove his point. I've heard the
New York Justice System was so impressed by this, that apportionment of votes is now done by relative power, instead of
1 vote per X citizens.