Chapter 2, Problem 1E

Five partners $\left(P_1,P_2,P_3,P_4\text{, and }{P}_5\right)$ jointly own the Gaussian Electric Company. $P_1$ owns 15 shares of the company, $P_2$ owns 12 shares, $P_3$ and $P_4$ each owns 10 shares and $P_5$ owns 3 shares, with the usual agreement that one share equals one vote. Describe the partnership as a weighted voting system using the standard notation $\left[q:w_1,w_{-}\left\{2\right\},\mathrm{...},w_N\right]$ if

a. decisions in the partnership are made by simple majority.

b. decisions in the partnership require two-thirds of the votes.

To determine

a)

To describe:

The partnership as a weighted voting system where decisions in the partnership are made by simple majority.

Answer to Problem 1E

Solution:

The partnership as a weighted voting system by simple majority is given as,

$\left[26:15,12,10,10,3\right]$

Explanation of Solution

Given:

In the Gaussian Electric Company, $P_1$ owns 15 shares of the company, $P_2$ owns 12 shares, $P_3$ and $P_4$ each owns 10 shares and $P_5$ owns 3 shares, with the usual agreement that one share equals one vote.

Procedure:

A given number of votes are controlled by each player in a formal voting arrangement, it is said to be weighted voting system.

Calculation:

The total share of the company is 100%. The distribution of shares of the company gives the number of votes owned by $P_1$, $P_2$, $P_3$, $P_4$ and $P_5$.15% shares are owned by $P_1$, and it represents 15 votes of the company. 12% shares are owned by $P_2$, and it represents 12 votes of the company. 10% shares are owned by $P_3$, which is 10 votes of the company. 10% shares are owned by $P_4$, which is 10 votes of the company and 3% shares are owned by $P_5$, which is 3 votes of the company.

Total number of votes is given by,

$15+12+10+10+3=50$

The simple majority is considered to be more than 50% of the total number of votes.

Majority is given by,

$\begin{array}{rll}\frac{50}{2}+1& =& 25+1\\ & =& 26\end{array}$

The standard notation of weighted voting system where decisions in the partnership are made by simple majority is given as,

$\left[26:15,12,10,10,3\right]$

Conclusion:

Thus, the partnership as a weighted voting system by simple majority is given as,

$\left[26:15,12,10,10,3\right]$

To determine

b)

To calculate:

The partnership as a weighted voting system where decisions in the partnership require two-thirds of the votes.

Answer to Problem 1E

Solution:

The partnership as a weighted voting system where decisions in the partnership require two-thirds of the votes is given as,

$\left[34:15,12,10,10,3\right]$

Explanation of Solution

Given:

In the Gaussian Electric Company, $P_1$ owns 15 shares of the company, $P_2$ owns 12 shares, $P_3$ and $P_4$ each owns 10 shares and $P_5$ owns 3 shares, with the usual agreement that one share equals one vote.

Procedure:

A given number of votes are controlled by each player in a formal voting arrangement, it is said to be weighted voting system.

Calculation:

The total share of the company is 100%. The distribution of shares of the company gives the number of votes owned by $P_1$, $P_2$, $P_3$, $P_4$ and $P_5$.15% shares are owned by $P_1$, and it represents 15 votes of the company. 12% shares are owned by $P_2$, and it represents 12 votes of the company. 10% shares are owned by $P_3$, which is 10 votes of the company. 10% shares are owned by $P_4$, which is 10 votes of the company and 3% shares are owned by $P_5$, which is 3 votes of the company.

The standard notation of weighted voting system where decisions in the partnership require two-thirds of the votes is given as,

$\begin{array}{rll}q& =& \frac{2}{3}\text{ of }50\\ & =& 34\end{array}$

$\left[34:15,12,10,10,3\right]$

Conclusion:

Thus, the partnership as a weighted voting system where decisions in the partnership require two-thirds of the votes is given as,

$\left[34:15,12,10,10,3\right]$