# UTPA STEM/CBI Courses/Contemporary Mathematics/Voting and Social Choice

Course Title: Contemporary Mathematics

Lecture Topic: Voting and Social Choice

Instructor: Roger Knobel

Institution: UTRGV

## Backwards Design

Course Objectives

• Primary Objectives- By the next class period students will be able to:
• Use four different methods for determining the winner of an election using voting by preference ballots.
• Apply fairness criteria to determine the fairness of a voting system.
• Conclude from Arrow’s Impossibility Theorem that there is no reasonable voting system that satisfies all measures of fairness.
• Sub Objectives- The objectives will require that students be able to:
• Find the winner of an election using the plurality method.
• Find the winner of an election using the plurality-with-elimination method.
• Find the winner of an election using the Borda count method.
• Find the winner of an election using the method of pairwise comparisons.
• Determine if a voting method satisfies the majority criterion for fairness.
• Determine if a voting method satisfies the Condorcet fairness criterion.
• Determine if a voting method satisfies the monotonicity criterion for fairness.
• Determine if a voting method is independent of irrelevant alternatives.
• Difficulties- Students may have difficulty:
• Distinguishing between plurality and majority.
• Using a preference schedule to determine which candidate is preferred by a group of voters in a head-to-head comparison between two candidates.
• Analyzing a voting example to determine if there was a violation of a fairness criterion.
• Distinguishing between the fairness of a voting system with the fairness of an individual election.
• Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
• Recognize voting systems used in businesses, government, and society.
• Awareness of potential unfairness in voting systems.

Model of Knowledge

• Concept Map
• Interpretation of data tables.
• Logic
• Content Priorities
• Enduring Understanding
• Identify the type of voting system being used in particular voting situations.
• Recognize the potential for violations of fairness in a voting system.
• Conclude from Arrow’s Impossibility Theorem that there is no reasonable voting system that satisfies all fairness criteria.
• Important to Do and Know
• Determine the winner of an election using different voting systems.
• Identify particular instances of violation of fairness in a voting system.
• Worth Being Familiar with
• Understand insincere voting and its potential to affect the outcome of an election.
• Examples of the implementation of different voting systems in practice.

Assessment of Learning

• Formative Assessment
• Student response system during class to check for understanding
• Quizzes to check for understanding on homework
• Homework based on in-class discussion and research
• Summative Assessment
• In-class exam over the unit on voting theory consisting of items which test the ability to
(a) determine the winner of an election using four voting methods (Plurality, Plurality-with-elimination, Borda count, and Pairwise comparisons),
(b) identify instances of violations of fairness criteria (majority, Condorcet, monotonicity, independence of irrelative alternatives), and
(c) conclude from Arrow’s Impossibility Theorem that there is no reasonable voting method that passes all fairness criteria.

## Legacy Cycle

OBJECTIVE

By the next class period, students will be able to:

• Use four different methods for determining the winner of an election using preference ballots.

Apply fairness criteria to determine the fairness of a voting system.

• Use Arrow’s Impossibility Theorem to conclude that there is no reasonable voting system that satisfies all measures of fairness.

The objectives will require that students be able to:

• Find the winner of an election using the plurality method.
• Find the winner of an election using the plurality-with-elimination method.
• Find the winner of an election using the Borda count method.
• Find the winner of an election using the method of pairwise comparisons.
• Determine if a voting method satisfies the majority criterion for fairness.
• Determine if a voting method satisfies the Condorcet fairness criterion.
• Determine if a voting method satisfies the monotonicity criterion for fairness.
• Determine if a voting method is independent of irrelevant alternatives

THE CHALLENGE

The Rio Grande Valley STEM Society is holding its annual election for president. There are four candidates running for president: Allen, Betty, Chris, and Debra. Each club member fills out a ballot ranking the candidates from first choice to forth choice. There are some common ballots – for example, 14 voters ranked the candidates in the order Betty, Debra, Chris, and Allen. The following table summarizes the ballots of the Society’s voters. Which candidate do you think best represents the choice for president of the entire voting membership?

Number of Voters 14 10 8 4 1
1st Choice Betty Chris Allen Debra Chris
2nd Choice Debra Debra Chris Allen Allen
3rd Choice Chris Allen Debra Chris Debra
4th Choice Allen Betty Betty Betty Betty

GENERATE IDEAS

1. Begin with a general discussion to check for understanding of the ballot table by asking questions to the class such as:
• How many voters were there?
• What does the “1” mean in the last column of the table?
• How would we rewrite the table if Allen were to remove himself from consideration before the winner was determined?
2. Form groups of three or four students. Give the groups 10-15 minutes to develop a method for using the table to determine a “winner” of the election.
3. Have one member from each group share their voting method and the candidate that “wins” using their method.
4. As a class, compare and contrast the methods; group together methods that are similar. Emphasize that a voting method should be an algorithm based on the data, not opinions. If a description of a voting method was given to different people, they should be able to follow the method and arrive at the same conclusion regarding the “winner”.

MULTIPLE PERSPECTIVES

1. Examples of voting using preference ballots, such as mayor elections in Burlington, VT and San Francisco, CA.
2. The video “Alternative Voting Explained” https://www.youtube.com/watch?v=3Y3jE3B8HsE
3. The Wall Street Journal article (February 6, 2009) “And the Oscar Goes to…Not its Voting System.”
4. Math in Society, a free open textbook with associated videos, available at http://www.opentextbookstore.com/mathinsociety/, with chapter Voting Theory.
5. In-class discussions.

RESEARCH & REVISE

1. On the first day, discuss the plurality method: the candidate with the most first-place votes is the winner of the election. Which candidate would win the election in “The Challenge” using plurality? (Betty) How well would the selection of Betty represent the choice of the entire voting group? (Not well; far more voters placed Betty last than first).
2. Continuing in class, discuss how to use a voting table to compare two candidates in a head-to-head comparison. Show that in the “Challenge” election, there is a candidate (Chris) that is preferred over each of the other three candidates in head-to-head comparisons, but would lose the election if using the plurality method.
3. For the next class period, students research alternative voting methods to plurality (Plurality with Elimination, Borda Count, and Pairwise Comparisons) and apply each method to the election in “The Challenge”. They should find each method produces a different “winner.” Complete homework practicing the different voting methods.
4. In class the second day, summarize the voting methods and the results when used on the “Challenge” election. Discuss fairness criteria (majority, Condorcet, monotonicity, independence of irrelevant alternatives) as a way to analyze the fairness of each voting method.
5. For the next class period, students research the fairness of the four voting methods, determining whether each voting method passes or fails each fairness criterion. Complete homework practicing the different fairness criteria.
6. In class the third day, summarize the fairness of the four voting methods with respect to the four fairness criteria. Notice that none of the four voting methods pass all four fairness criteria. Discuss Arrow’s Impossibility Theorem.

Complete the Test Your Mettle Quiz.

GO PUBLIC

On the last day of this unit, students will be divided into groups of three or four. Each group will be given a table summarizing the ballots cast for a certain election and a particular voting method. The group will then determine (a) the winner of the election, and (b) if the particular election demonstrates a violation of a fairness criterion. One member of the group will be randomly selected to present the election results and fairness analysis to the class.

## Pre-Lesson Quiz

Various material can be tested here, depending on time constraints.