think they are familiar with the concept of voting - after all, they
have heard about governmental elections, Academy Award voting, and the
ranking of the sports teams. They may have also participated in club
and school elections. Yet if you ask students about voting methods,
most can describe only one method: plurality. Most students have never
questioned the fairness of plurality, and have never considered
alternative voting methods.
The following activities allow students to explore alternative
voting methods. Students discover what advantages and disadvantages
each method offers and also see that each method fails, in some way, to
satisfy some desirable properties. Many students are particularly
surprised to discover that when the plurality method is used, the
winner could be the candidate who the majority of voters like the least.
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.
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
group decisions on such things as, for example, choosing a class gift
or service project, selecting a time to hold an event, or arranging for
refreshments, you may want to substitute relevant activities for those
on the worksheets.
To introduce the topic and to familiarize students with this table
format, you will probably want to start with the following whole-class
Have students suggest activities or destinations for a
hypothetical class trip, and write the first 3 suggestions on the
board. Ask each student to list her or his first, second, and third
choices on a piece of paper, permitting no tied rankings.
Understanding the Table
Ask 2 different students for their preferences. Ask the class to
help generate all 6 possible preference combinations. Suppose your
students suggested archery (A), biking (B), and canoeing (C). You would
then create a table, similar to the one below, where the columns
represent all the possible preference lists.
|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|
Tallying the Class's Preferences
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 1 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 3 or 4
Practice with the Plurality Method and Other Voting Systems
Plurality voting is the method most familiar to students. In
this method, each voter is given 1 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% of the vote, including situations in which there are only 2
candidates, then the plurality method does produce a preferred
candidate. However, in many situations, plurality may not produce a
Since students are comfortable with the plurality method, they can usually complete this activity sheet in small groups.
Question 3: Students are often surprised to find that skiing comes in both first and last.
Question 4: A reasonable answer is the speed and ease of the plurality method.
Question 5: If
students get stuck, ask them to think how winners are determined in
sports tournaments or when many alternatives are available.
The second half of this activity sheet introduces students to 3 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.
Question 7: With the Borda count, 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, 3 points per voter times 40
voters, and verify that this result matches their total number of
Question 8: 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 may help
clarify the distinction.
- 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. This point is worth emphasizing
because students tend to overlook these voters.
- Ties do not occur in the problems given; however, you should know
how they are handled in the multistep voting systems. With the Hare
system, whenever 2 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 2 candidates tie during a head-to-head
contest in sequential pairwise voting, neither is eliminated; they both
continue and compete in a 3-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
Solutions to sheet 1
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 problem on this sheet demonstrates that a
property known as independence of irrelevant alternatives (IIA)
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 voters
represented by column 1 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 2 preference lists, B was able to win.
Question 1 on this sheet is adapted from Introductory Graph Theory (Chartrand 1985, 168).
Solutions to sheet 2
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, or Hare system. Ask students to explain why it does hold for sequential pairwise voting.
- Question 1: Tournament digraphs are used to help students
visualize the results of pairwise voting. Most students expect
candidate B to win in every method.
- Question 2: The solution is the digraph in question
1. Students discover that although B might turn out to be the winner
under each of these voting methods, they can only be positive that B
wins by using sequential pairwise voting.
- Question 3: The questions 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
The Condorcet Winner Problem
Tournament Digraphs and Condorcet Winners Activity Sheet also introduces the term Condorcet winner.
Question 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
experiment later with different tables of preference lists to create
Solutions to sheet 3
- Written by Teresa D. Magnus. Mathematics Teacher, January 2000, page 18.
- Brams, Steven J., et al. "Social Choice: The Impossible Dream." In For All Practical Purposes, 4th ed., edited by COMAP, the Consortium for Mathematics and Its Applications, 411-42. New York: H. W. Freeman & Co., 1997.
- Chartrand, Gary. Introductory Graph Theory, New York: Dover publications, 1985.