Question

Blocking Coalitions and the Banzhaf Power Index
The four members, A, B, C, and D, of an organization adopted the weighted voting system {6: 4, 3, 2, 1}. The table below shows the winning coalitions.

Winning coalition	Number of votes	Critical voters
{A, B}	7	A, B
{A, C}	6	A, C
{A, B, C}	9	A
{A, B, D}	8	A, B
{A, C, D}	7	A, C
{B, C, D}	6	B, C, D
{A, B, C, D}	10	None

Using the Banzhaf power index, we have

BPI(A) =

5

12

.

A blocking coalition is a group of voters who can prevent passage of a resolution. In this case, a critical voter is one who leaves a blocking coalition, thereby producing a coalition that is no longer capable of preventing the passage of a resolution. For the voting system from the table above, we have the following.

Blocking coalition	Number of votes	Number of remaining votes	Critical voters
{A, B}	7	3	A, B
{A, C}	6	4	A, C
{A, D}	5	5	A, D
{B, C}	5	5	B, C
{A, B, C}	9	1	None
{A, B, D}	8	2	A
{A, C, D}	7	3	A
{B, C, D}	6	4	B, C

If we count the number of times A is a critical voter in a winning or blocking coalition, we find what is called the Banzhaf index. In this case, the Banzhaf index is 10 and we write

BI(A) = 10.

Using both the winning coalition and the blocking coalition tables, we find that

BI(B) = 6,

BI(C) = 6,

and

BI(D) = 2.

This information can be used to create an alternative definition of the Banzhaf power index.

Applying this definition to the voting system given above, we have

BPI(A) =

BI(A)

BI(A) + BI(B) + BI(C) + BI(D)

=

10

10 + 6 + 6 + 2

=

10

24

=

5

12

.

Watch the video below then answer the question.

Blocking Coalitions and the Banzhaf Power Index

View Transcript

Using the data in the example, list all blocking coalitions. (Select all that apply.)

{A, B}{A, C}{A, D}{B, C}{B, D}{C, D}{A, B, C}{A, B, D}{A, C, D}{B, C, D}

Yexample Determine Winning Coalitions in a Weighted Vot ing System Suppose that the four owners ofa company, Ang, Bonhomme, Carmel, and Diaz, own, respectively, 500 shares, 375 shares, 225 shares, and 400 shares. There are total of this is 750, so the quota is 751. The weighted voting system for this 1500 votes; half company is (751: 500, 375, 225, 400}. a. Determine the winning coalitions. b. For each winning coalition, determine the critical voters. Solution a. A winning coalition must represent at least 751 votes. We will list these coalitions in the table below, in which we use A for Ang, B for Bonhomme, C for Carmel, and D for Diaz Winning coalition Number of votes A, B 875 900 A, D B, D 775 A, B, C 1100 A, B, D 1275 A, C, D 1125 {В. С, D} 1000 {А, В, С, D 1500 answer Yes. If any of the permanent members votes against a resolution, the resolution cannot pass b. A voter who leaves a winning coalition and thereby creates a losing coalition is a critical voter. For instance, for the winning coalition {A, B, C}, if A leaves, the number of remaining votes is 600, which is not enough to pass a resolution. If B leaves, the number of remaining votes is 725-again, not enough to pass a resolu- tion. If C leaves, the number of remaining votes is 875, which is greater than the quota. Therefore, A and B are critical voters for the coalition {A, B, C} and C is not a critical voter. The table below shows the critical voters for each winning coalition. Winning coalition Number of votes Critical voters A, B 875 A, B {A, D 900 A, D В. D B, D 775 {А, В, С} 1100 A, B A, B, D 1275 None {A, C, D 1125 A, D (В. С, D} 1000 B, D {А, В, С, D} 1500 None check your progress Many countries must govern by forming coali- tions from among many political parties. Suppose a country has five political parties named A, B, C, D, and E. The numbers of votes, respectively, for the five parties are 22, 18, 17, 10, and 5. a. Determine the winning coalitions if 37 votes are required to pass a resolution. b. For each winning coalition, determine the critical voters.

Answer 1

Blocking Coalih'ons Blocking Coals naas: Availalbe AB X A C A p G00 775 B D 625 CiD 110o A BIC (275 Ai BID 112 5 A B, C.D

Ci BPIt Cy KEI veter i is citical number o hmes givea Fudng c's Erom table 41 So. BPL CA) (2 BPI CB) (2 BPECC) 10 T2 BPE CDIE 11