Banzhaf Power Index
'’Coalition’’ is a group of people/parties that need to achieve some quota when voting for a law. Otherwise this law will not pass.
The ‘‘Banzhaf Power Index’’ shows how strong a party is.
Suppose we have a company with 200 shares in total.
There are three shareholders:
- $D$ Doug: 101 shares,
- $N$ Nicolas: 97 shares,
- $E$ Elizabeth: 2 shares
For a decree to pass it should have 103 shares
Is $N$ 48 times more important then $E$? To assess the importance we use the Banzhaf index:
The Power Index
Critical Voter:
- A coalition is ‘‘winning’’ if it has enough power to pass a low/decree/whatever.
- A voter in a winning coalition is ‘‘critical’’ if his withdrawal causes the coalition to become a loosing coalition
Example:
- there are $2^3$ coalitions in total, and $3$ of them are winning
- ${D, E}$
- 103 votes - this is a winning coalition
- $E$ is a critical voter: if she withdraws, the coalition is no longer winning
- $D$ also is a critical voter
- ${D, N}$
- both $D$ and $N$ are critical
- ${D, N, E}$ (Unanimity)
- everybody agrees: 200 votes
- $D$ and $N$ are critical voters
- but now $E$ is not: if she withdraws, the coalition is still winning
The Power:
- The ‘‘Banzhaf Power’’ $BP(a)$ of a voter $a$ is the number of winning coalitions in which $a$ is critical.
- The ‘‘Total Banzhaf Power’’ of a voting game is the sum of all Bahnzaf powers of all voters: $TBP = \sum_{a} BP(a)$
- The ‘‘Banzhaf Index’’ of a voter $a$ is $\cfrac{BP(a)}{TBP}$
Example: | Voter | $BP$ | Index | $D: 101$ | 3 | 3/5 || $N: 97$ | 1 | 1/5 || $D: 101$ | 1 | 1/5 || | $TBP = 5$ | | So we see that both $N$ and $E$ are equally important, even though they don’t have the same number of shares.
Example: Nassau County
Consider the following districts: | | District | Weight | (1) | Hempstead 1 | 31 || (2) | Hempstead 2 | 31 || (3) | Oyster Bay | 28 || (4) | North Hempstead | 21 || (5) | Long Beach | 2 || (6) | Glen Cove | 2 | The threshold for a law to pass is $Q=58$
In this example all the power in equally distributed withing the 3 first districts (1), (2) and (3).
- any 2 of these 3 always form a winning coalition
- no other two districts can form such a winning coalition
Links
- http://en.wikipedia.org/wiki/Banzhaf_power_index