Real-world examples of studying the mathematics of voting systems

Examples of studying the mathematics of voting systems using real elections

When teachers ask for examples of studying the mathematics of voting systems, they usually want something more than a made‑up classroom election. The strongest projects use real data, clear rules, and measurable outcomes. Here are several approaches that can turn into serious science fair projects.

One example of a strong project is to take ranked-choice election data from a city that already uses it and compare it to how the result would look under simple plurality (whoever gets the most first‑place votes wins). Cities like New York City and San Francisco publish detailed ballot data for some elections. By simulating both systems, you can show whether the same candidate wins, how many ballots get exhausted (no remaining ranked choices), and how many rounds of counting it takes. This is one of the best examples of using mathematics to highlight trade‑offs between simplicity and representation.

Another example of a project is to simulate strategic voting. You can design a survey where classmates rank their favorite school lunch options or student council candidates. Then you compare what happens when people vote honestly versus when they try to vote strategically (for example, ranking a strong opponent last to hurt their chances). These examples of simulated elections reveal how some systems invite strategic behavior more than others.

Classic mathematical examples of voting system paradoxes

Some of the best examples of studying the mathematics of voting systems focus on paradoxes—situations where a system behaves in a way that feels unfair or inconsistent.

One famous example of this is the Condorcet paradox. Imagine three candidates: A, B, and C, and three groups of voters with these preferences:

• Group 1 (35%): A > B > C
• Group 2 (33%): B > C > A
• Group 3 (32%): C > A > B

Pairwise comparisons show that A beats B, B beats C, and C beats A. There is no single candidate who beats all others head‑to‑head. This is a powerful example of how majority rule can “cycle,” and it is one of the classic examples of studying the mathematics of voting systems in textbooks and research.

Another classic example of a paradox is the spoiler effect under plurality voting. Suppose candidate X is popular with 45% of voters, candidate Y with 40%, and candidate Z with 15%, but Z’s supporters actually prefer Y over X. If Z drops out, Y might win with 55% against X’s 45%. But with all three on the ballot, X wins with 45%, even though 55% of voters prefer Y over X. This is an accessible example of how adding a candidate can change the winner.

These paradoxes are perfect for a science fair board: they are surprising, easy to explain with charts, and show why the mathematics of voting is not just about counting.

Examples of comparing different voting rules for the same election

Many students want an example of a project where they can systematically compare voting rules. A strong approach is to take one set of ballots and apply several systems:

• Plurality (most first‑place votes wins)
• Runoff or instant runoff (ranked-choice voting)
• Borda count (points for each ranking position)
• Approval voting (voters mark all candidates they find acceptable)

For instance, you might collect ranked preferences for four hypothetical candidates in your class. Then, using the same raw data, compute the winner under each system. These examples of parallel calculations let you:

• Count how often the plurality winner differs from the Borda winner.
• Identify whether there is a Condorcet winner (one that beats everyone head‑to‑head) and see which systems pick that candidate.
• Analyze how sensitive each system is to small changes in the ballots.

You can turn this into one of the best examples of a math project by tying it to real debates in election reform. For instance, the Center for Election Science discusses approval voting and compares it to plurality and ranked-choice voting in real contexts (electionscience.org). Their discussions give you language and criteria—like voter satisfaction and spoiler resistance—that you can turn into measurable variables in your project.

Real examples of studying the mathematics of voting systems with public data

If you want real examples with actual election numbers, public data is your friend. Many governments and academic groups publish detailed election results that you can analyze.

Some starting points:

• The U.S. Census Bureau provides statistics on voter turnout, age groups, and demographics in federal elections (census.gov).
• The MIT Election Data and Science Lab hosts precinct‑level results, turnout data, and research tools (electionlab.mit.edu).
• Some city or state election offices publish ballot‑level data for ranked-choice elections.

A concrete example: you could take turnout data for U.S. presidential elections from 2000 to 2024 and examine whether states with closer margins tend to have higher turnout. That project blends statistics with voting theory, and you can discuss how small vote swings in a few states can change the Electoral College outcome, even when the national popular vote looks different. This gives you real examples of how a voting system (in this case, the Electoral College) can amplify some votes more than others.

Another example of a project using public data is to analyze how third‑party candidates affect outcomes. You can compare the 1992 and 1996 U.S. presidential elections, where Ross Perot ran as a strong third‑party candidate, or the 2000 election with Ralph Nader. Using state‑level vote totals from authoritative sources, you can estimate how results might have changed under ranked-choice voting, approval voting, or a Condorcet method. These examples of counterfactual analyses are mathematically rich and tie directly to real political history.

Examples of fairness criteria and Arrow’s theorem in student projects

If you want something more theoretical but still grounded in examples, you can build a project around fairness criteria. In the mathematics of voting, researchers often talk about properties like:

• Monotonicity: Ranking a candidate higher should not make them lose.
• Independence of irrelevant alternatives (IIA): Adding or removing a losing candidate should not change the winner.
• Non-dictatorship: No single voter always decides the outcome.

Arrow’s Impossibility Theorem, proved by economist Kenneth Arrow, states that there is no ranked voting system for three or more candidates that satisfies all of a reasonable set of fairness criteria at once. For a student project, you don’t need to reproduce the full proof. Instead, you can create small, concrete examples of elections where a system violates one criterion.

For instance, you can construct an example of a monotonicity failure in instant-runoff voting (IRV): a situation where giving a candidate more first‑place votes actually causes them to lose after the elimination rounds change. Real examples of this phenomenon have been documented in municipal elections and are discussed in academic papers and election‑reform reports.

Using a few carefully chosen examples of these violations, you can show that there is no perfect system and that every method involves trade‑offs. That message is very powerful on a science fair poster.

Modern examples of voting math beyond political elections

Not all examples of studying the mathematics of voting systems have to involve presidents or mayors. In 2024–2025, a lot of interesting voting happens online and in organizations:

• Online polls and rating sites: Platforms that let users upvote, like, or rate items are using simplified voting systems. You can model how different aggregation rules (average rating, median rating, Wilson score intervals) change which items rise to the top.
• Award shows and competitions: Some music and film awards use ranked ballots or multi‑round voting. You can research their rules and simulate alternative systems.
• Student councils and clubs: Many schools are experimenting with ranked-choice or approval voting for student government. If your school is one of them, you can get permission to analyze anonymized results.

For example, you might study how a streaming platform’s recommendation algorithm effectively “votes” on which shows to feature. While you may not have access to the internal algorithm, you can simulate several rules—like picking the top‑rated shows, the most‑watched shows, or a combination weighted by recent activity—and compare how each method favors different genres or audiences. These are modern, relatable examples of voting systems in action.

Another example of a 2024–2025 trend is participatory budgeting, where cities let residents vote on how to spend part of the local budget. Different cities use different voting rules—some use approval voting, others use ranked-choice, and some use point‑based systems where residents allocate a fixed number of points across projects. Analyzing one city’s rules and showing how they might change outcomes under an alternative method gives you a contemporary, real‑world project with clear mathematical structure.

Turning these examples into a science fair project

At this point, you’ve seen multiple examples of examples of studying the mathematics of voting systems: paradoxes, real election data, alternative rules, and modern online voting. To turn one of these into a polished project, you need three ingredients:

• A clearly stated question (for example: “How often does plurality fail to elect the Condorcet winner in simulated school elections?”).
• A defined set of voting rules you will compare.
• A source of data (simulated ballots, class surveys, or public election results).

You can then:

• Organize your data in a spreadsheet.
• Write formulas to compute winners under each system.
• Track metrics such as how often each system picks the Condorcet winner, how often a spoiler appears, or how satisfied voters are (you can measure satisfaction by assigning utility scores to rankings).

These steps transform scattered examples into a structured investigation. Along the way, you will naturally produce more examples of interesting behavior—for instance, elections where a candidate wins under Borda but loses badly under plurality.

The best examples of student projects in this area are the ones that do not just state that systems differ, but show exactly how and why, with real numbers and clear tables. That style will impress judges who care about mathematical reasoning.

FAQ: common questions and examples students ask about voting math

Q: What are some simple examples of voting system projects for middle school?
You can run a class election for favorite snack, movie, or field trip destination. Have students rank their choices, then compute the winner under plurality, Borda count, and instant-runoff voting. This gives you a basic example of how different rules can pick different winners from the same ballots.

Q: Can I use online data as an example of a voting system?
Yes. You can scrape or record data from public polls, rating sites, or crowdsourced rankings, as long as you follow your school’s rules and the site’s terms of use. Treat likes, upvotes, or stars as votes and compare how different mathematical rules for aggregating them change the ranking.

Q: Is there a famous example of mathematics changing how we think about voting?
Arrow’s Impossibility Theorem is the classic example of a mathematical result that reshaped how economists and political scientists think about voting. It shows that no ranked voting system can satisfy all of a set of fairness criteria at once. For a project, you can build small elections that illustrate where each criterion fails.

Q: Do I need advanced math to study voting systems?
Not really. Many strong projects rely on arithmetic, percentages, basic probability, and logical reasoning. If you want to go further, you can add statistics, simulations, or computer programming, but the core ideas are accessible with high school math.

Q: Where can I find more real examples and background on voting systems?
University sites and research labs are good starting points. The MIT Election Data and Science Lab, for example, provides data and explanations aimed at both researchers and the public. Political science departments at universities often post lecture notes and articles on voting rules and social choice theory.

By grounding your work in concrete, well‑explained examples of elections and clear mathematical rules, you can turn the abstract idea of “voting math” into a sharp, memorable science fair project.