# ML Wiki

## 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