# Banzhaf Power Index Consider 5

Banzhaf Power Index

Consider 5 voters, labeled A, B, C, D, and E, who areshareholders on a company board.

1) If there are 11 votes total and a 2/3 majority is required topass a motion, what is the quota? That is, howmany votes are required to pass a motion?

(Hint: the answer is a whole number between 0 and 11 thatrepresents at least a 2/3 majority).

2) Suppose A has 5 votes, B has 3 votes, and C, D, and E haveone vote each. Determine the Normalized BPI and the Absolute BPIfor each voter.

3) If A gives one vote to B, so that the new distribution ofvotes is: A has 4, B has 4, C has 1, D has 1, and E has 1, whathappens to voter A’s Normalized BPI (meaning, voter A’s share ofpower)–does it increase, decrease, or stay the same?

4) In a weighted voter scheme, a “dummy voter” is a voter whoeffectively has no power (NBPI = 0) even though they are allowedmore than zero votes.

Give an example of a voting situation (quota and distribution ofvotes among voters) where at least one of the voters is a dummyvoter.

Total votes = 11 multiplied by 2/3rd

= 7.33

= 8 (Whole number)

Hence the quota is 8

A critical player is one whose vote makes the difference betweenwinning or losing. Let T be the total number of criticalplayers.

The Banzhaf power index of a player P is the number of times Pis critical, divided by T.

 Winning coalitions Weight Critical players A, B 8 A, B A,B,C 9 A,B A,B,D 9 A,B A,B,E 9 A,B A,C,D,E 8 A,C,D,E A,B,C,D 10 A,B A,B,D,E 10 A,B A,B,C,E 10 A,B A,B,C,D,E 11 A

In the example, A is critical 8 times, B is critical 7 times,and C, D & E is only critical 1 time.

Normalised Banzhafindex: the number of swings as a proportion of the totalnumber of swings for all members. The indices sum to 1 over allmembers.

 T (Critical times) Power A 9 47.37% B 7 36.84% C 1 5.26% D 1 5.26% E 1 5.26% 19

Absolute Banzhafindex: the number of swings divided by the number ofpossible voting outcomes among the other members.

 T (Critical times) Power A 9 100.00% B 7 77.78% C 1 11.11% D 1 11.11% E 1 11.11% 19

 Winning coalitions Weight Critical players A, B 8 A, B A,B,C 9 A,B A,B,D 9 A,B A,B,E 9 A,B A,B,C,D 10 A,B A,B,D,E 10 A,B A,B,C,E 10 A,B A,B,C,D,E 11
 T (Critical times) Power A 7 50.00% B 7 50.00% C 0 0.00% D 0 0.00% E 0 0.00% 14 1

So the revise power of A reduces to 50% from 100%

In the answer 3 it can be observed that A & B are powervoter’s whereas C, D & E are dummy voters. Dummy voters arethose who do not have any weightage i.e. NBPI of 0%

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