Voting: The Importance of Method

Author: JoAnne Growney

When people put an issue to a vote, they generally suppose that they are using a fair and accurate way of discovering what the group prefers. Occasionally, they are concerned about possible dishonesty or voter apathy, but they seldom ask whether their voting method may be flawed. This is a serious omission, for a voting method may select a choice that is not what the voters want most. This Pull-Out Section points out the weaknesses in the sequential and plurality voting methods and proposes an alternative voting process call the Black method.

Sequential Pairwise Voting
The process called sequential pairvise voting is used when there are more than two choices. First, one pair of decision options is considered and a vote is taken; then, the winner is paired with another option for another vote. This method of pairing and voting continues until a final winner is selected. A flaw in this method is illustrated by the following example.

Each year, to honor the person who has contributed the most to the council, the officers of the Student Council at Central High School select the "Member of the Year." This year, there is a lot of concern about the selection, for it is rumored that last year's officers selected poorly. That rumor is true. Last year's selection was David Kovacs. David was not unworthy but, after the choice had been announced, it was learned that each of the three officers (president, vice president, and secretary-treasurer) preferred Barbara Ostrowski to David Kovacs. (See Figure 1.)

Note: The information below was created with the assistance of AI.

Level of Mathematics

This lesson is well-suited for:

High school students, especially those in grades 11–12, enrolled in Algebra II, Precalculus, Discrete Mathematics, or Civics with mathematical extensions.
College-level liberal arts math or mathematics for social science courses.
Some sections (like Condorcet and Black methods) assume moderate comfort with structured reasoning and analyzing outcomes—ideal for students who are already proficient with basic mathematical logic.

Application Areas

This lesson integrates mathematics with:

Civics and political science: Understanding electoral systems and fairness in group decisions.
Social choice theory: A major application of discrete mathematics.
Ethics and public policy: Investigating fairness and representation.
Decision science: Frameworks for group and personal decision-making.
Personal life skills: Application to job choice, organizational voting, and consensus methods.

It draws from real-world processes and historical frameworks, fostering interdisciplinary insight.

Prerequisites

Students should have:

A solid grasp of ordinal ranking and preference notation.
Comfort with interpreting data tables and digraphs.
An understanding of logic and basic set theory concepts (e.g., pairwise comparisons).
Ability to follow multi-step reasoning and apply it systematically.
No calculus is required, but reasoning and abstraction skills are important.

Subject Matter

The core mathematical topics include:

1. Voting Systems and Methods

Sequential pairwise voting (agenda-sensitive).
Plurality voting.
Condorcet method: Wins all head-to-head matchups.
Black method: Condorcet + vote totals as tie-breaker.

2. The Agenda Effect

Demonstrates how voting order affects outcomes.
Explores manipulability of decisions.
Introduces the concept of non-transitive preferences and voting cycles.

3. Graph Theory and Digraphs

Use of directed graphs (digraphs) to model voting outcomes.
Helps visualize cyclical preferences, vote counts, and consensus challenges.

4. Decision-Making Models

Extends voting models to personal decisions, such as job selection.
Weighting and combining preference criteria.

5. Fairness and Limitations

Introduces Arrow’s Impossibility Theorem: No perfect voting system exists that satisfies all fairness criteria.

Correlation to Mathematics Standards
Common Core State Standards – High School

Modeling with Mathematics

HSA-CED.A.3: Represent constraints by equations and use them to solve problems.
HSA-REI.D.10-12: Interpret and reason with systems and solutions (interpreting voting patterns).
HSM-MD.B.5: Analyze decisions and strategies using mathematical models.