History
What is known today as the Banzhaf power index was originally introduced by Penrose (1946) and went largely forgotten. It was reinvented by Banzhaf (1965), but it had to be reinvented once more by Coleman (1971) before it became part of the mainstream literature.
Banzhaf wanted to prove objectively that the Nassau County Board's voting system was unfair. As given in Game Theory and Strategy, votes were allocated as follows:
- Hempstead #1: 9
- Hempstead #2: 9
- North Hempstead: 7
- Oyster Bay: 3
- Glen Cove: 1
- Long Beach: 1
This is 30 total votes, and a simple majority of 16 votes was required for a measure to pass.
In Banzhaf's notation, are A-F in
There are 32 winning coalitions, and 48 swing votes:
AB AC BC ABC ABD ABE ABF ACD ACE ACF BCD BCE BCF ABCD ABCE ABCF ABDE ABDF ABEF ACDE ACDF ACEF BCDE BCDF BCEF ABCDE ABCDF ABCEF ABDEF ACDEF BCDEF ABCDEF
The Banzhaf index gives these values:
- Hempstead #1 = 16/48
- Hempstead #2 = 16/48
- North Hempstead = 16/48
- Oyster Bay = 0/48
- Glen Cove = 0/48
- Long Beach = 0/48
Banzhaf argued that a voting arrangement that gives 0% of the power to 16% of the population is unfair, and sued the board.
Today, the Banzhaf power index is an accepted way to measure voting power, along with the alternative Shapley–Shubik power index.
However, Banzhaf's analysis has been critiqued as treating votes like coin-flips, and an empirical model of voting rather than a random voting model as used by Banzhaf brings different results (Gelman & Katz 2002).
Read more about this topic: Banzhaf Power Index
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