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.
5) Give your best explanation for the apparently paradoxicalanswer to #3 above.
Answer:
Answer to1.
Votes required topass a motion
Total votes = 11 multiplied by 2/3^{rd}
= 7.33
= 8 (Whole number)
Hence the quota is 8
Answer to2.
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 
Answer to3.
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 to4.
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%