Question

Four people are using a voting weight system to make decisions. They have the weights of 7,5,4,and 2. They use a quota of 13. Compute the Banzhaf power index for the voter of weight 5.

Answer 1

Let persons with weights be

A- 7

B- 5

C- 4

D- 2

There are total 15 ways in which they can combine i.e.

A- not critical as A cannot get the quota

B- not critical as B cannot get the quota

C- not critical as C cannot get the quota

D- not critical as D cannot get the quota

AB- not critical as AB cannot get the quota

AC- not critical as AC cannot get the quota

AD- not critical as AD cannot get the quota

BC- not critical as BC cannot get the quota

BD- not critical as BD cannot get the quota

CD- not critical as CD cannot get the quota

ABC- A, B, and C, all are critical as each of them are needed to get quota

ABD- A, B, and D, all are critical as each of them are needed to get quota

ACD- A, C, and D, all are critical as each of them are needed to get quota

BCD- not critical as BCD cannot get the quota

ABCD- A is critical as each of them are needed to get quota

So, A is critical 4 times and B, C & D are critical 2 times

So, Banzhaf power index of B (i.e the voter of weight 5) is 2/ (4+2+2+2) = 2/10= 1/5 = 0.2 (or 20%)

Please upvote. Thanks.