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
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