Lesson 5: More than Two Choices

Students in a sixth-grade class were asked, “What activity would you most like to do for field day?” The results are shown in the table.

1. What percentage of the class voted for softball?

2. What percentage did not vote for softball as their first choice?

Suppose students at our school are voting for the lunch menu over the course of one week. The following is a list of options provided by the caterer.

To vote, draw one of the following symbols next to each menu option to show your first, second, third, and last choices. If you use the slips of paper from your teacher, use only the column that says “symbol.”

Here are two voting systems that can be used to determine the winner.

Voting System #1. Plurality: The option with the most first-choice votes (stars) wins.

1. People tend to be more satisfied with election results if their top choices win. For how many, and what percentage, of people was the winning option:

   1. their first choice? 
   2. their second choice?

   3. their third choice? 
   4. their last choice?

Voting System #2. Runoff: If no choice received a majority of the votes, leave out the choice that received the fewest first-choice votes (stars). Then have another vote.

If your first vote is still a choice, vote for that. If not, vote for your second choice that you wrote down.

If there is still no majority, leave out the choice that got the fewest votes, and then vote again. Vote for your first choice if it’s still in, and if not, vote for your second choice. If your second choice is also out, vote for your third choice.

4. After the second round of voting, did any choice get a majority? If so, is it the same choice that got a plurality in Voting System #1?
5. Which choice won?

6. How satisfied were the voters by the election results? For how many, and what percentage, of people was the winning option:

   1. their first choice? 
   2. their second choice?

   3. their third choice? 
   4. their last choice?
7. Compare the satisfaction results for the plurality voting rule and the runoff rule. Did one produce satisfactory results for more people than the other?

Let’s analyze a different election.

In another class, there are four clubs. Everyone in each club agrees to vote for the lunch menu exactly the same way, as shown in this table.

1. Figure out which option won the election by answering these questions.

   1. On the first vote, when everyone voted for their first choice, how many votes did each option get? Did any choice get a majority?
   2. Which option is removed from the next vote?
   3. On the second vote, how many votes did each of the remaining three menu options get? Did any option get a majority?
   4. Which menu option is removed from the next vote?
   5. On the third vote, how many votes did each of the remaining two options get? Which option won?

2. Estimate how satisfied all the voters were.

   1. For how many people was the winner their first choice?
   2. For how many people was the winner their second choice?
   3. For how many people was the winner their third choice?
   4. For how many people was the winner their last choice?
3. Compare the satisfaction results for the plurality voting rule and the runoff rule. Did one produce satisfactory results for more people than the other?

Your class just voted using the instant runoff system. Use the class data for following questions. 

1. For our class, which choice received the most points?
2. Does this result agree with that from the runoff election in an earlier activity?
3. For the other class, which choice received the most points?
4. Does this result agree with that from the runoff election in an earlier activity?
5. The runoff method uses information about people’s first, second, third, and last choices when it is not clear that there is a winner from everyone’s first choices. How does the instant runoff method include the same information?
6. After comparing the results for the three voting rules (plurality, runoff, instant runoff) and the satisfaction surveys, which method do you think is fairest? Explain.

Clare, Han, Mai, Tyler, and Noah are deciding what to do on the weekend. Their options are cooking, hiking, and bowling. Here are the points for their instant runoff vote. Each first choice gets 2 points, the second choice gets 1 point, and the last choice gets 0 points.

1. Which activity won using the instant runoff method? Show your calculations and use expressions or equations.

2. Which activity would have won if there was just a vote for their top choice, with a majority or plurality winning?

3. Which activity would have won if there was a runoff election?

4. Explain why this happened.