Number is not Power
WEIGHTED VOTING SYSTEMS

&
The Power Game


Examples of Weighted Voting Systems: We are going to take a look at voting situations in which voters are not necessarily equal in terms of the number of votes they control




To make things simple we are only going to look at a vote which only involves two choices. Referred to in your books as a motion.



Some Basic Definitions:



NOTATION & EXAMPLES

Standard notation: [ q; w1 , w2 , w3 , . . . . , wn ]
The quota is given first, followed by the respective weights of the individual players


EXAMPLE

Consider: [ 25; 8, 6, 5, 3, 3, 3, 2, 2, 1, 1, 1, 1 ]



BANZHAF POWER INDEX

Some Basic Definitions:




Finding the Power Index of Player P

The Banzhaf Power Index of player P is given by the fraction B/T






Here is an example:

We have a weighted voting system [ 6; 4, 3, 2 ]



Step 1.


Coalition
Coalition Weight
Win or Lose
{P1}
4
Lose
{P2}
3
Lose
{P3}
2
Lose
{P1, P2}
7
Win
{P1, P3}
6
Win
{P2, P3}
5
Lose
{P1, P2, P3}
9
Win



Step 2.

The winning coalitions are {P1, P2}, {P1, P3}, {P1, P2, P3}



Step 3.

Winning Coalitions
Critical Players
{P1, P2}
P1 and P2
{P1, P3}
P1 and P3
{P1, P2, P3}
P1 only




Step 4.

Step 5.
T = 5



Step 6. Power index for each player is given by B/T

We refer to the complete listing of the Banxhaf power indexes as the Banzhaf power distribution. (It is common practice to write power indexes as percentages, rather than fractions. )

For our example than:

P1; 60%

P2; 20%

P3; 20%





Journal Log