Question

the employees at a company were asked to vote for the next coffee of the month. the choices were the iced coffee, the cappuccino, or the vanilla latte. (the coffee of the month is available for free in the employee lounge.) each employee ranked the coffees, and the votes are shown in the preference table.

\\(\

$$\begin{array}{lcccccc} \\text{number of votes} & 1 & 12 & 11 & 10 & 18 & 7 \\\\ \\text{first choice} & \\text{iced coffee} & \\text{vanilla latte} & \\text{vanilla latte} & \\text{cappuccino} & \\text{iced coffee} & \\text{cappuccino} \\\\ \\text{second choice} & \\text{cappuccino} & \\text{cappuccino} & \\text{iced coffee} & \\text{vanilla latte} & \\text{vanilla latte} & \\text{iced coffee} \\\\ \\text{third choice} & \\text{vanilla latte} & \\text{iced coffee} & \\text{cappuccino} & \\text{iced coffee} & \\text{cappuccino} & \\text{vanilla latte} \\end{array}$$

\\)

the plurality-with-elimination method was used to determine the winner.

construct a new preference table to show the results after one coffee was eliminated.

use the plurality-with-elimination method to determine which coffee was selected to be coffee of the month.

• the iced coffee. it had first-choice votes after one coffee was eliminated.
• the cappuccino. it had first-choice votes after one coffee was eliminated.
• the vanilla latte. it had first-choice votes after one coffee was eliminated.
• two or more coffees tied for the most votes. they each had first-choice votes after one coffee was eliminated.

Explanation:

Count first-choice votes

Using the Plurality-with-Elimination Method and Preference Schedules knowledge points
We sum the first-choice votes for each coffee candidate from the preference table:

• Iced coffee: \(1 + 18 = 19\) votes
• Vanilla latte: \(12 + 11 = 23\) votes
• Cappuccino: \(10 + 7 = 17\) votes

Total votes: \(1 + 12 + 11 + 10 + 18 + 7 = 59\) votes.
Majority threshold: \(\lfloor 59/2
floor + 1 = 30\) votes.
Since no candidate has at least 30 votes, we must eliminate the candidate with the fewest first-choice votes.

Eliminate the candidate with fewest votes

Using the Plurality-with-Elimination Method knowledge point
Cappuccino has the fewest first-choice votes (\(17\)) and is eliminated.
We remove Cappuccino from the preference schedule and shift the remaining choices up:

• Column 1 (1 vote): Iced coffee, Vanilla latte
• Column 2 (12 votes): Vanilla latte, Iced coffee
• Column 3 (11 votes): Vanilla latte, Iced coffee
• Column 4 (10 votes): Vanilla latte, Iced coffee
• Column 5 (18 votes): Iced coffee, Vanilla latte
• Column 6 (7 votes): Iced coffee, Vanilla latte

Construct the new preference table

Using the Preference Schedules knowledge point
We combine columns with identical remaining preferences:

• Preference order: Iced coffee, then Vanilla latte

Columns 1, 5, and 6 have this order.
Total votes: \(1 + 18 + 7 = 26\) votes.

• Preference order: Vanilla latte, then Iced coffee

Columns 2, 3, and 4 have this order.
Total votes: \(12 + 11 + 10 = 33\) votes.

The new preference table has two columns:

• Column 1: \(26\) votes (First choice: Iced coffee, Second choice: Vanilla latte)
• Column 2: \(33\) votes (First choice: Vanilla latte, Second choice: Iced coffee)

Determine the winner

Using the Plurality-with-Elimination Method knowledge point
After eliminating Cappuccino, the new first-choice vote counts are:

• Vanilla latte: \(33\) votes
• Iced coffee: \(26\) votes

Vanilla latte has \(33\) votes, which exceeds the majority threshold of \(30\). Thus, Vanilla latte is selected as the coffee of the month.

Answer:

Question 1

The new preference table after eliminating Cappuccino:

Number of votes	26	33
Second choice	Vanilla latte	Iced coffee

Question 2

• The iced coffee. It had 26 first-choice votes after one coffee was eliminated.
• The cappuccino. It had 17 first-choice votes after one coffee was eliminated.
• The vanilla latte. It had 33 first-choice votes after one coffee was eliminated. (Correct answer)
• Two or more coffees tied for the most votes. They each had [ ] first-choice votes after one coffee was eliminated.