Question

In: Accounting

Banzhaf Power Index Consider 5 voters, labeled A, B, C, D, and E, who are shareholders...

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.

Solutions

Expert Solution

Answer to 1.

Votes required to pass a motion

Total votes = 11 multiplied by 2/3rd

= 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%


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