2.2 The Banzhaf Power Index
 
Topics
Coalitions
How many coalitions for n players?
Winning coalitions and critical players
The Banzhaf power index
An example


Coalitions Coalition: Any group of players that agree to vote the same way on a measure.  A single player will also be referred to as a "coalition".

Example:  [101: 99, 98, 3]


How many coalitions for n players?

The number of coalitions:

for a system of n players
there are

coalitions

Quiz:  How many coalitions for [5: 1, 1, 1, 1, 1, 2]? ··


Winning coalitions and critical players

Winning coalition:  any coalition that can pass a measure

For [101: 99, 98, 3], what are the winning coalitions?

··

Critical player (for a coalition):

Let's do it for all 7 coalitions:
Coalition Critical Players
{P1} ··
{P2} ··
{P3} ··
{P1, P2} ··
{P1, P3} ··
{P2, P3} ··
{P1, P2, P3}


The Banzhaf power index (John Banzhaf, 1965)

The idea:

Computing the Banzhaf power index of player
     
    1. list all possible coalitions (2n - 1 of them).
    2. for each winning coalition, list those players that are critical
    3. for player P, count the number of times P is critical = CP
    4. count the total number of times any player is critical = Ctotal
    5. The Banzhaf power index of P is 

The Banzhaf power distribution is formed by computing the Banzhaf power index for each of the players.

Analysis of Banzhaf power index results:


An example

Let's do it for: [4: 3, 2, 1]
 

Coalition Winning? Critical players
{P1} ·· ··
{P2} ·· ··
{P3} ·· ··
{P1, P2} ·· ··
{P1, P3} ·· ··
{P2, P3} ·· ··
{P1, P2, P3} ·· ··
    Ctotal··

Banzhaf power distribution

Player P
CP:
number of times P is critical
power in decimal: % of power:
P1 ·· ·· ·· 60 %
P2 ·· ·· ·· 20 %
P3 ·· ·· ·· 20 %

Notice that the %'s add up 100%.