Let's break down the problem step by step to describe the committee using the weighted voting system notation $[q: w_1, w_2, w_3, w_4]$.

Step 1: Set up the equation

We know the following relationships between the weights (number of votes) of the members:

• $w_1 = 4x$ (since $P_1$ has twice as many votes as $P_2$)
• $w_2 = 2x$ (since $P_2$ has twice as many votes as $P_3$)
• $w_3 = x$ (since $P_3$ and $P_4$ have the same number of votes)
• $w_4 = x$ (since $P_3$ and $P_4$ have the same number of votes)

Thus, the weighted voting system is of the form: $[93: 4x, 2x, x, x]$

Step 2: Solve for $x$

We now need to consider the quota $q = 93$ given in the problem to solve for $x$.

(a) When $q = 93$

In this scenario, the equation remains: $[93: 4x, 2x, x, x]$ But $x$ is arbitrary in this context since $q = 93$ is a fixed quota, and no specific total votes are provided. Thus, we don't solve for $x$ directly as we don't have the total number of votes specified.

So, the notation for the committee remains: $[93: 4x, 2x, x, x]$

(b) When $q = \frac{3}{4}$ of total votes

Here, we need to find the total number of votes and set the quota $q = \frac{3}{4}$ of the total votes. The total number of votes $V$ is: $V = 4x + 2x + x + x = 8x$

So, the quota $q$ is: $q = \frac{3}{4} \times 8x = 6x$

Thus, the weighted voting system becomes: $[6x: 4x, 2x, x, x]$

(c) When $q = \frac{1}{2}$ of total votes

Similarly, for this case, the quota $q$ is: $q = \frac{1}{2} \times 8x = 4x$

So, the weighted voting system is: $[4x: 4x, 2x, x, x]$

Summary of Answers:

• (a) $q = 93$: The committee is described as $[93: 4x, 2x, x, x]$.
• (b) $q = \frac{3}{4}$ of total votes: The committee is described as $[6x: 4x, 2x, x, x]$.
• (c) $q = \frac{1}{2}$ of total votes: The committee is described as $[4x: 4x, 2x, x, x]$.

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Would you like any further details or have any questions about these solutions?

Related Questions:

1. How does the change in the quota affect the voting power of each committee member?
2. What happens if $q$ is greater than the total number of votes?
3. How would the system change if $P_1$ had three times as many votes as $P_2$?
4. Can $P_3$ or $P_4$ ever have veto power in these setups?
5. How can we calculate