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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:

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:

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:

So we see that both $N$ and $E$ are equally important, even though they don't have the same number of shares.

Consider the following districts:

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).