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Will the Best Candidate Win?

  • Lesson
9-12
1
Number and OperationsData Analysis and Probability
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Location: Unknown

This lesson plan for grades 9‑12 is adapted from an article in the January 2000 edition of Mathematics Teacher. The following activities allow students to explore alternative voting methods. Students discover what advantages and disadvantages each method offers and also see that each fails, in some way, to satisfy some desirable properties.

Students 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 activity.

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 AABBCC
Second Choice BCACAB
Third Choice CBCABA

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 students.

Practice with the Plurality Method and Other Voting Systems 

pdficon Activity Sheet - 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 clear preference.

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 points.

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.

Notes:  

  • 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 entire class.

Strategic Voting 

pdficon Activity Sheet - 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 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 ABC
Second choice BAA
Third choice CCB

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).

Tournament Digraphs and Condorcet Winners 

pdficon Activity Sheet - 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,

    n(n - 1) 
    2

    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

    n(n -1) 
    2

    If students used patterns to discover the formula 1+2+3+...+ (n -1), you can show them that

     2539 candidate-TriangularFormula 

The Condorcet Winner Problem 

The 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 this situation.

Solutions to all sheets

References 

  • 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.
 

Assessments 

  1. Have students devise a question that can be answered by an election. They can then gather data and tally the results. Ask students to determine which choice is the winner from each voting method, and decide which methods are most effective, representative, or fair. 

Extensions 

  1. 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 these 4 voting methods do not satisfy the Condorcet-winner criterion (CWC) or the IIA. He also proved that it is impossible to create any voting system that does. To be more precise, any voting system that always produces at least 1 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 be no conflict.

  2. Presidential Primaries: Upcoming elections offer another opportunity for extending this activity and linking it with statistics and social studies. If more than 3 candidates are running for each party, have students survey a sample of adults, asking them to rank the leading Republican or Democratic 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, and findings in an essay or news story.
 

Questions for Students
1. Which voting systems are preferable if you are interested in both first- and second-place winners?
2. Which voting system is used in organized sports tournaments?
3. With which methods could "strategic voting" be effective?

Teacher Reflection 

  • How can you connect this lesson to other disciplines?
  • Describe the level of understanding students had of each voting method based on the given definition and after the use of an example.
  • Order the voting methods by amount of difficulty students had in developing an understanding of them.
  • In what order will you present the voting methods when you teach this lesson in the future?
  • What changes will you make to the activities to help students develop an understanding of each voting method?
 

Learning Objectives

By the end of this lesson, students will:

  • See connections between mathematics and other disciplines, including government, history, ethics, and sports
  • Develop skills in mathematical reasoning and apply those skills to everyday situations
  • Learn about various voting methods, ways in which these methods can be manipulated to achieve certain outcomes, and the impossibility of fair elections when more than two alternatives are available

NCTM Standards and Expectations

  • Develop a deeper understanding of very large and very small numbers and of various representations of them.
  • Compare and contrast the properties of numbers and number systems, including the rational and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions.
  • Use number-theory arguments to justify relationships involving whole numbers.