Question

Banzhaf Power Index

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

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

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

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

3) If A gives one vote to B, so that the new distribution of votes is: A has 4, B has 4, C has 1, D has 1, and E has 1, what happens to voter A's Normalized BPI (meaning, voter A's share of power)--does it increase, decrease, or stay the same?

4) In a weighted voter scheme, a "dummy voter" is a voter who effectively has no power (NBPI = 0) even though they are allowed more than zero votes.

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

5) Give your best explanation for the apparently paradoxical answer to #3 above.

Answer 1

Answer to 1.

Votes required to pass a motion

Total votes = 11 multiplied by 2/3^rd

= 7.33

= 8 (Whole number)

Hence the quota is 8

Answer to 2.

A critical player is one whose vote makes the difference between winning or losing. Let T be the total number of critical players.

The Banzhaf power index of a player P is the number of times P is 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 Banzhaf index: the number of swings as a proportion of the total number of swings for all members. The indices sum to 1 over all members.

	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 Banzhaf index: the number of swings divided by the number of possible 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

Answer to 3.

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%

Answer to 4.

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