Question

1. Consider the weighted voting system [q : 8, 3, 3, 2], where q is an integer and 9 ≤ q ≤ 16.

(a) For what values of q is there a dummy?

(b) For what values of q do all the voters have the same power?

(c) If a voter is a dummy for a given quota, must the voter be a dummy for all larger quotas? Explain

Answer 1

Solution:

(a):

Given, weighted voting system =$\{q: 8,3,3,2\}$

For q=11,14, we have a dummy.

for q=11.

As we can observe in this case A have the veto and D is a dummy.

for q=14.

As we can observe in this case A, B and C have the veto and D is a dummy.

(b):

for q=16 all voters have the same power.

for q=16

The Banzhaf power index of a voter v, symbolized by $\mathrm{BPI}(v)$ , is given by:

$\begin{array}{l} \operatorname{BPI}(v)=\frac{\text { number of times voter } v \text { is a critical voter }}{\text { number of times any voter is a critical voter }} \\ \qquad \begin{aligned} B P I(A) &=\frac{1}{4} \\ &=0.25 \\ B P I(B) &=\frac{1}{4} \\ &=0.25 \\ B P I(C) &=\frac{1}{4} \\ &=0.25 \\ B P I(D) &=\frac{1}{4} \\ &=0.25 \end{aligned} \end{array}$

(c):

It is not necessary that if a voter is a dummy for a given quota, must the voter be a dummy for all larger quotas as in this can we can observe that D is a dummy for q=11 but not for q=13 as shown below:

We can observe in this case A have the veto and none of the voters is a dummy.