Question

17.  Consider the weighted voting system [36: 20, 18, 16, 2]

a) Find the Banzhaf power distribution of this system.

P1  P2  P3  P4                       P1  P2  P3                  P1  P2

P1  P2 P4                  P1  P3  

                                    P1  P3  P4

                                    P2  P3  P4

P1:                              P2:                              P3:                              P4:

b) Name the dictator in this system, if there is one.  If not, write "none".

c) Name the players with veto power in this system.  If there are none, write "none".

d) Name the dummies in this system.  If there are none, write "none".

Answer 1

The numbers [36: 20, 18, 16, 2] means that 36 votes are required to win, P1 has 20 votes, P2 has 18 votes, P3 has 16 votes, P4 has 2 votes, so the total number of votes is P1+P2+P3+P4 = 20+18+16+2 = 56.

We write all possible winning combinations. Of these, some are swing votes, meaning that in that particular vote, if only that group changes its vote, the proposal will not get 36 votes required to pass.

Examples:

P1 P2 is a winning combination. Both are swing votes as if one group changes its vote, the proposal will fail. We write this as: P1 P2

P1 P2 P4 is a winning combination. Of these, P1 and P2 are swing votes as if one group changes its vote, the proposal will fail. But, P4 is not a swing vote as the proposal will still pass. We write this as: P1 P2 P4

P1 P2 P3 P4 is a winning combination. Of these, no group is not a swing vote as the proposal will still pass by reaching 36 votes. We write this as: P1 P2 P3 P4

P3 P4 is not a winning combination, so it will not be listed.

So, writing for each combinatiion of winning votes and marking the swing votes, we get:

P1 P2 P3 P4
P1 P2 P3
P1 P2
P1 P2 P4
P1 P3
P1 P3 P4
P2 P3 P4

So, there are a total of 12 swing votes as marked above. Of this, P1 appears 5 times, P2 appears 3 times, P3 appears 3 times, and P4 appears 1 time. Dividing the number of swing votes for each player by the total number of swing votes gives us the Banzhaf power index.

Answer: So, the Banzhaf power distribution of this system is:

P1: 5/12
P2: 3/12
P3: 3/12
P4: 1/12

b) Answer: Dictator: none

A dictator is one who can alone reach 36 votes and others cannot reach 36 votes without the dictator's support. A dictator will be the only swing voter in the above list. The Banzhaf power index would be 1. Here, there is no dictator.

c) Answer: Players with veto power: none

A player with veto power is one whose support is required to reach 36 votes. Such a player will appear in all the swing votes. Here, there is no player with veto power.

d) Answer: Dummies: none

A player is a dummy whose vote does not influence the vote. Such a player will not appear in the swing votes. The Banzhaf power index would be 0. Here, there is no dummy.