Chapter 6.CR, Problem 1CR

To determine

a)

To find:

The total number of votes cast.

Answer to Problem 1CR

Solution:

The total number votes cast is $74$.

Explanation of Solution

Given:

The results are given as,

	Number of Ballots Cast
	$5$	$7$	$6$	$12$	$13$	$17$	$14$
1st choice	M	M	M	B	B	V	T
2nd choice	B	V	T	T	M	M	V
3rd choice	V	B	V	V	T	B	B
4th choice	T	T	B	M	V	T	M

Approach:

The number of votes cast can be found by the sum of rows representing the number of ballots cast for each ranking.

Calculation:

The number of votes cast is,

$\begin{array}{rll}\text{Total}& =& 5+7+6+12+13+17+14\\ & =& 74\end{array}$

Therefore, the total number votes cast is $74$.

Conclusion:

Hence, the total number votes cast is $74$.

To determine

b)

To find:

The winner of voting by plurality method.

Answer to Problem 1CR

Solution:

Beethoven wins based on plurality method.

Explanation of Solution

Approach:

The candidate that receives the most votes is selected for winning part according to the plurality method.

Calculation:

As shown in the given table, Beethoven received maximum total $\left(12+13=25\right)$ votes. So, Beethoven will be first choice among other three candidates.

Therefore, Beethoven wins based on plurality method.

Conclusion:

Hence, Beethoven wins based on plurality method.

To determine

c)

To find:

The percentage of votes that received by the winner in part (b).

Answer to Problem 1CR

Solution:

The percentage of votes that received by the winner is $34\%$.

Explanation of Solution

Approach:

The percentage of votes is given by,

$\text{Percentage}=\frac{\text{Votes}\text{received}}{\text{Total}\text{number}\text{of}\text{votes}}\times 100$

Calculation:

The percentage of votes that received by the winner is given by,

Substitute $25$ for votes received, and $74$ for total number of votes in the above formula.

$\begin{array}{rll}\text{Percentage}& =& \frac{\text{25}}{\text{74}}\times 100\\ & =& 0.3378\times 100\\ & \approx & 34\%\end{array}$

Therefore, the percentage of votes that received by the winner is $34\%$.

Conclusion:

Hence, the percentage of votes that received by the winner is $34\%$.

To determine

d)

To find:

The winner of voting by instant runoff method.

Answer to Problem 1CR

Solution:

Vivaldi wins based on instant runoff method.

Explanation of Solution

Approach:

If a candidate receives majority of the first choice votes, that candidate is declared as the winner according to the instant runoff method. If this is not happened then the event with the fewest first choice votes is eliminated and those votes are given to the next preferred candidate.

Calculation:

From the above calculation Beethoven received $34\%$ vote which is not majority.

Eliminate the candidate which has fewer majorities that is Tchaikovsky and modify voters preference table as shown below,

	Number of Ballots Cast
	$5$	$7$	$6$	$12$	$13$	$17$	$14$
1st choice	M	M	M	B	B	V	V
2nd choice	B	V	V	V	M	M	B
3rd choice	V	B	B	M	V	B	M

As shown in the above table, still any candidate does not receive majority of votes so eliminate Mozart.

	Number of Ballots Cast
	$5$	$7$	$6$	$12$	$13$	$17$	$14$
1st choice	B	V	B	B	B	V	V
2nd choice	V	B	V	V	V	B	B

As shown in the above table Vivaldi receives maximum votes.

Therefore, Vivaldi wins based on instant runoff method.

Conclusion:

Hence, Vivaldi wins based on instant runoff method.

Conclusion:

To determine

e)

To find:

The percentage of votes that received by the winner in part (d).

Answer to Problem 1CR

Solution:

The percentage of votes that received by the winner is $59\%$.

Explanation of Solution

Approach:

The percentage of votes is given by,

$\text{Percentage}=\frac{\text{Votes}\text{received}}{\text{Total}\text{number}\text{of}\text{votes}}\times 100$ … (1)

Calculation:

The percentage of votes that received by the winner is given by,

Substitute $44$ for votes received, and $74$ for total number of votes in the above formula.

$\begin{array}{rll}\text{Percentage}& =& \frac{\text{44}}{\text{74}}\times 100\\ & =& 0.5945\times 100\\ & =& 59.45\%\\ & \approx & 59\%\end{array}$

Therefore, the percentage of votes that received by the winner is $59\%$.

Conclusion:

Hence, the percentage of votes that received by the winner is $59\%$.

To determine

f)

To find:

The winner of voting by Borda count method.

Answer to Problem 1CR

Solution:

The winner is Beethoven.

Explanation of Solution

Calculation:

Each candidate receives $4$ points for each first choice vote, $3$ points for each second choice vote, $2$ points for each third choice vote and $1$ point for each fourth choice vote.

First, tally the votes for each event as shown below,

	Beethoven	Mozart	Tchaikovsky	Vivaldi
1st choice	$25$	$18$	$14$	$17$
2nd choice	$05$	$30$	$18$	$21$
3rd choice	$38$	$0$	$13$	$23$
4th choice	$6$	$26$	$29$	$13$

The score for each event can be determined by multiplying the number of votes times the appropriate number of points as shown in table below,

	Beethoven	Mozart	Tchaikovsky	Vivaldi
1st choice (4 points each)	$100$	$72$	$56$	$68$
2nd choice (3 points each)	$15$	$90$	$54$	$63$
3rd choice (2 points each)	$76$	$0$	$26$	$46$
4th choice (1 point each)	$6$	$26$	$29$	$13$

As shown in the above table, maximum vote get by Beethoven.

Therefore, the winner is Beethoven.

Conclusion:

Hence, the winner is Beethoven.

To determine

g)

To find:

The number of points received by winner in part (f).

Answer to Problem 1CR

Solution:

The number of points received by winner is $197$.

Explanation of Solution

Approach:

Sum all the points given in table.

Calculation:

The results is shown in table below,

	Beethoven	Mozart	Tchaikovsky	Vivaldi
1st choice (4 points each)	$100$	$72$	$56$	$68$
2nd choice (3 points each)	$15$	$90$	$54$	$63$
3rd choice (2 points each)	$76$	$0$	$26$	$46$
4th choice (1 point each)	$6$	$26$	$29$	$13$
Total points	$197$	$188$	$165$	$190$

Therefore, the number of points received by winner is $197$.

Conclusion:

Hence, the number of points received by winner is $197$.

To determine

h)

To find:

The winner of voting by pair-wise comparison method.

Answer to Problem 1CR

Solution:

It is a tie between Beethoven and Vivaldi.

Explanation of Solution

Calculation:

There are four candidates, so according to pair-wise comparison method of voting, there are following comparisons,

$\begin{array}{rll}{}^4{C}_2& =& \frac{4!}{2!\left(4-2\right)!}\\ & =& 6\end{array}$

Investigate B versus M, M versus T, B versus V, M versus T, M versus V, T versus V as follows,

	Number of Ballots Cast
	$5$	$7$	$6$	$12$	$13$	$17$	$14$
1st choice	M	M	M	B	B	V	T
2nd choice	B	V	T	T	M	M	V
3rd choice	V	B	V	V	T	B	B
4th choice	T	T	B	M	V	T	M

Investigate B versus M,

$\begin{array}{rll}\text{Vote}\text{for}\text{B}& =& 12+13+14\\ & =& 39\\ \text{Vote}\text{for}\text{M}& =& 5+7+6+17\\ & =& 35\end{array}$

In the above comparison voters prefer B over M. consequently B receives 1 point.

Investigate B versus T,

$\begin{array}{rll}\text{Vote}\text{for}\text{B}& =& 05+07+12+13+17\\ & =& 54\\ \text{Vote}\text{for}\text{T}& =& 6+14\\ & =& 20\end{array}$

In the above comparison voters prefer B over T. consequently B receives 1 point.

Investigate B versus V,

$\begin{array}{rll}\text{Vote}\text{for}\text{B}& =& 05+12+13\\ & =& 30\\ \text{Vote}\text{for}\text{V}& =& 7+6+14+17\\ & =& 44\end{array}$

In the above comparison voters prefer V over B. consequently V receives 1 point.

Investigate M versus T,

$\begin{array}{rll}\text{Vote}\text{for}\text{M}& =& 05+07+6+13+17\\ & =& 48\\ \text{Vote}\text{for}\text{T}& =& 12+14\\ & =& 26\end{array}$

In the above comparison voters prefer M over T. consequently M receives 1 point.

Investigate M versus V,

$\begin{array}{rll}\text{Vote}\text{for}\text{M}& =& 05+07+6+13\\ & =& 31\\ \text{Vote}\text{for}\text{V}& =& 12+17+14\\ & =& 43\end{array}$

In the above comparison voters prefer V over M. consequently V receives 1 point.

Investigate T versus V,

$\begin{array}{rll}\text{Vote}\text{for}\text{T}& =& 6++12+13+14\\ & =& 45\\ \text{Vote}\text{for}\text{V}& =& 5+7+17\\ & =& 29\end{array}$

In the above comparison voters prefer T over V. consequently T receives 1 point.

According to the pair-wise comparison method of voting, it is a tie between Beethoven and Vivaldi because both received 2 points out of 6 points.

Therefore, it is a tie between Beethoven and Vivaldi.

Conclusion:

Hence, it is a tie between Beethoven and Vivaldi.

To determine

i)

To find:

The number of points received by winner in point (h).

Answer to Problem 1CR

Solution:

They both received 2 points out of 6 points according to the pair-wise comparison method.

Explanation of Solution

Calculation:

As shown in part (h) Beethoven and Vivaldi both received 2 points out of 6 points according to the pair-wise comparison method.

Therefore, they both received 2 points out of 6 points according to the pair-wise comparison method.

Conclusion:

Therefore, they both received 2 points out of 6 points according to the pair-wise comparison method.