Chapter 17.CR, Problem 1CR

To determine

The winner using majority if any, if not, use plurality vote if a taste test is conducted on the Atlantic City Boardwalk. People are given samples of Coke, Pepsi and Safeway brands of cola in unmarked cups, and are then asked to rank them in order of preferences.

Answer to Problem 1CR

Solution:

There is no majority.

The item $\text{C}$ wins the plurality vote.

Explanation of Solution

For this election, the total number of voters is $n=18+15+12=45$

In Majority rule, if the number of voters is $n$ and $n$ is even, then the candidate with $\frac{n}{2}+1$ or more votes win.

If the number of voters is $n$ and $n$ is odd, then the candidate with $\frac{n+1}{2}$ or more votes win.

For this election, $n=45$ is odd.

Hence the majority is, $\frac{45+1}{2}=23$. That is the item having $23$ or more votes win.

From the results of the outcomes, look only at first choices and write the total votes for each item as below:

The total number of first-choice votes for the item $\text{A}$ is $15$.

The total number of first-choice votes for the item $\text{B}$ is $12$.

The total number of first-choice votes for the item $\text{C}$ is $18$.

Hence, according to Majority rule, no item receives a majority.

According to Plurality method, the item receiving the most votes is declared the winner.

Here item $\text{C}$ receives most votes as $18$.

Therefore the item $\text{C}$ is the winner using Plurality vote.