tudents think that they are
familiar with the concept of voting.
After all, they have heard about governmental elections, Academy Award voting, and the ranking of the football teams. They have probably participated in club and school elections. Yet if you
ask students about voting methods, most can describe only one voting method, namely, plurality. Not only have they never questioned its fairness, but considering new methods is new to them.
"The plurality method can produce a winner who is liked least."
The following activities allow students to explore alternative voting methods. They discover what advantages and disadvantages each method offers and also see that each fails, in some way, to satisfy some desirable properties.
They are particularly surprised to discover that the plurality method can produce a winner who is liked least by a majority of the voters.
In addition, students look at how elections can be manipulated. One extension involves discussing Arrow's impossibility theorem, which states that it is impossible for a voting system to satisfy all the desirable features. Although these activities, including the statement and proof of Arrow's theorem, require
only basic arithmetic, they allow students to engage in high-level mathematical thinking.
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Note: This activity lends itself easily to interdisciplinary instruction. Current events on the national, local, or school level can be incorporated into the project. If students are involved in making such group
decisions as choosing a class gift or service project, selecting a time to hold an event, or arranging for refreshments, relevant activities can be substituted
for the activities on the worksheets. |
Directions
To introduce the topic and to familiarize students with the table format, I recommend the following large group activity. Have the students suggest activities or destinations for a hypothetical class trip, and write the first three suggestions on the chalkboard. Ask each student to list her or his first, second, and third choices on a piece of paper, permitting no tied rankings.
Suppose that your students suggested archery (A), biking (B), and canoeing (C). You would then create a table, similar to table 1, where the columns represent all the possible preference lists.
| Table 1 |
| Example Preference Table |
| First Choice |
A |
A |
B |
B |
C |
C |
| Second Choice |
B |
C |
A |
C |
A |
B |
| Third Choice |
C |
B |
C |
A |
B |
A | |
At the top of each column, write the number of students who ranked the options in the order given. Ask students which alternative wins and how they determined the winner. If students generate only one method of tallying the winner, ask them to think of a different way.
Use the activity sheets below to help students explore different voting methods. Each activity sheet is designed for groups of three to four students.
Student Activity Sheets
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Sheet 1: The Plurality Method and Other Voting Systems |
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Sheet 2: Strategic Voting |
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Sheet 3: Tournament Digraphs and Condorcet Winners |
Download
Activity
Sheets | |
Using Sheet 1:
The Plurality Method and Other Voting Systems
Part I
(Questions 1-5)
Plurality voting is the method most familiar to students. In this method, each member is given one vote and the option that receives the most votes wins. Variations of the plurality method are used in choosing state representatives and senators, ratifying proposals, and selecting Academy Award winners. When a candidate receives more than 50 percent of the vote, including
situations in which only two candidates are being considered, then plurality does produce a preferred candidate. However, in many situations, plurality may
not produce a clear preference.
Since students are comfortable with the plurality method, they can usually complete this activity sheet in small groups. They are often surprised to find that skiing comes in both first and last and often mention
their surprise in their responses to question 3. A reasonable answer for question 4 would be the speed and ease of the plurality method. If students get stuck on question 5, ask them to think how winners are
determined in sports tournaments or when many alternatives are available.
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Note: A nice follow-up to this activity sheet is a discussion about variations of the plurality method including runoff elections,
the electoral college, and two-thirds majority. Another variation of plurality, the Hare system of voting, involves a series of elections. At each stage the option, or options, with the least number of votes are eliminated from future ballots. The voters who originally voted for the eliminated option vote for the remaining option that they have ranked highest. I find that this point is worth
emphasizing because students often tend to overlook these voters.
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Part II
(Questions
6-10)
The second half of this activity sheet introduces students to three voting methods: the Hare system, Borda count, and sequential pairwise voting. Although the sheet gives instructions for each voting method, some student groups may need help following the directions.
Tip: With the Borda count, I show students how they can put a 0 next to third place, a 1 next to second place, and a 2 next to first place on the chart as an aid to totaling the points. A quick way for students to verify that their totals are reasonable is to determine the total number of votes, that is, three points per voter times forty voters, and verify that this result matches their total number of points.
Comparing the technique of sequential pairwise voting with a single elimination tournament, with byes, may help students understand the method, but care should be taken to distinguish between the two. In sequential pairwise voting, the pairings are sequential and no simultaneous pairings take place. Sketching the corresponding "tournament bracket" diagrams as a
demonstration may help clarify the distinction.
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Note: Ties do not occur in the problems given; however, the instructor should know how they are handled in the multistep voting systems.
With the Hare system, whenever two or more options share the least number of votes, they are eliminated at the same time. If all remaining candidates have the same number of votes, none are eliminated; they are all considered tied for the win. Whenever two candidates tie during a head-to-head contest in sequential pairwise voting, neither is eliminated; they both continue and compete in a three-way contest with the next candidate. |
Have students discuss the advantages and disadvantages of each of these voting methods, first in their small groups and then with the entire class.
Solutions to sheet 1
Using Sheet 2:
Strategic Voting
Here students investigate how a voter or block of voters can influence the results of an election by submitting a ballot that does not represent their true preferences. Although the terminology is avoided on these sheets, each of the problems on sheet 2 demonstrates that a property known as independence of irrelevant alternatives does not hold for these voting methods. In other
words, a losing candidate can win the election without any voters having moved the new winner ahead of the original winner in their preference lists. They may have moved other, irrelevant, candidates above or below one of these two. As an example, consider the following chart of preference lists:
| |
Number |
| Ranking |
2 |
3 |
4 |
| First Choice |
A |
B |
C |
| Second Choice |
B |
A |
A |
| Third Choice |
C |
C |
B | |
Candidate C wins using the plurality method. However, if the two people represented by the first column switched the positions of A and B in their preference list, B would win by plurality. Note that the order of B and C was not reversed. By moving B ahead of an irrelevant alternative, A, in two preference lists, B was able to win.
Question 1 on sheet 2 is adapted from Introductory Graph Theory (Chartrand 1985, 168).
Solutions to sheet 2
Using Sheet 3:
Tournament Digraphs and Condorcet Winners
At first, one would expect that if a candidate, called a Condorcet winner, could beat each of the other candidates in head-to-head contests, that candidate should win the election in which all candidates
compete. Students are surprised to discover that this so-called Condorcet-winner criterion does not hold for the plurality method, Borda count, and Hare system. Students should be asked to explain why it does hold for sequential pairwise voting.
In sheet 3, tournament digraphs are used to help students visualize the results of pairwise voting. In question 1, most students expect candidate B to win in every method. They discover in problem 2,
which has as its solution the digraph in 1, that although B may turn out to be the winner under each of these voting methods, they can be positive that B wins only in sequential pairwise voting.
Problem 3 is tangential to the main topic. You may wish to omit it or use it as a take-home bonus question.
With n candidates,
arrows are involved. Some students will write this expression in the form 1+2+3+...+(n - 1). One way of deriving the first formula is by noting that n points exist and that each point has an arrow to or from each of the other n - 1 points. Since each arrow touches two points, the number of arrows is
If students used patterns to discover the formula 1+2+3+...+ (n -1), you can show them that
| 1 + 2 + 3 +... + (n -1) |
| = |
(1 + 2 + 3 +... + (n -1)) |
+ |
((n -1) + ... + 3 + 2 + 1) |
| 2 |
2 | |
| = |
1 |
+ ((1 + (n - 1)) + (2 + (n - 2)) + (3 +
(n - 3)) + ... + ((n -1) + 1)) |
| 2 | |
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The Condorcet Winner Problem
Sheet 3 also introduces the term Condorcet winner. Problem 6 challenges students to be more creative and develop their own examples of tables of preference lists that show that the Hare system does not satisfy the
Condorcet winner criterion. Make sure that students understand that their tables of preference lists for this problem must produce a Condorcet winner. Suggest that students think about how a Condorcet winner might lose an election under the Hare system of voting. Note that the Condorcet winner must be eliminated at an early stage. It cannot have a lot of first place votes. Students can later
experiment with different tables of preference lists to create this situation.
Solutions to sheet 3
Follow-Up Activities
1. Discussing Arrow's Impossibility Theorem
Now that students have discovered that each of these voting methods may not produce the expected result, a new question arises. Does a voting system exist that satisfies all desirable criteria?
Kenneth Arrow proved that not only do these four voting methods not satisfy both the Condorcet-winner criterion (CWC) and independence of irrelevant alternatives (IIA) but that it is impossible to create any voting system that does. To be more precise, any voting system that always produces at least one winner cannot satisfy both CWC and IIA.
The proof of this theorem, which can be found in Brams et al. (1996, 426-30), requires an understanding of the topics on these worksheets along with high-level mathematical reasoning. Students have difficulty understanding that if a criterion does not hold for any one table of preference lists under a
particular voting method, then the voting method fails to satisfy the criterion. Other preference lists may exist for which there appears to appear no conflict.
2. Presidential Primaries
The upcoming presidential primaries offer another opportunity for extending this activity and linking it with statistics and social studies. Have students survey a sample of adults, asking them to rank the leading Republican--or Democratic, if more than three candidates run--candidates in order of preference. Ask students to create a table of preference lists illustrating their data, using each voting system studied. Students can report their sampling methods, calculations, findings, in an essay or news story.
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Student Activity Sheets
The Student Activity Sheets are available in PDF format. You can then print out the pages you desire. Click here to download the Student Activity Sheets. A preview of the four-page handout is below. (Click on a page to view a larger image.)
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