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.