High School Course Choice Equilibrium Models

ISEF Category: Mathematics

Subcategory: Other  ·  Difficulty: Advanced  ·  Setup: Home Setup  ·  Time: Full Year

The Hook

Your schedule choices can act like a crowd. If enough students chase the same class, the whole pattern shifts. That makes course selection a real math problem, not just a school admin problem. You can model it, test it, and compare your predictions with public enrollment data.

What Is It?

This project studies how students choose classes when their choices affect each other. A mean-field game is a math model for many people making similar decisions at the same time. Instead of tracking each student one by one, you track the overall pattern. Think of it like traffic flow. You do not follow every car. You study the average behavior of the crowd.

In this topic, you use public enrollment data to build a model of course-selection equilibria. An equilibrium is a stable outcome where no one has a strong reason to change choices by themselves. Monotonicity means that if one factor increases, the response moves in one direction too. In plain terms, if more students want a class, the model should react in a predictable way. Your goal is to see whether the model gives one clear equilibrium and whether that equilibrium matches historical enrollment patterns.

Why This Is a Good Topic

This is a strong science fair topic because the core question is testable with public data and clear math tools. You can compare model predictions to real enrollment patterns, check whether the equilibrium is unique, and measure how well simple assumptions fit school behavior. The project connects to real problems like course scheduling, equity in access, and resource planning. You can learn applied modeling, optimization, and data analysis without needing a wet lab.

Research Questions

• How does the strength of monotonicity affect whether the model gives one equilibrium or multiple equilibria?
• What is the effect of course capacity limits on the predicted enrollment equilibrium?
• Does adding peer preference terms improve the match between predicted and historical course enrollments?
• To what extent do public school enrollment patterns fit a mean-field game model better than a simpler choice model?
• Which classes or grade levels show the largest gap between equilibrium predictions and observed enrollments?
• How does changing the payoff structure alter the stability of the equilibrium solution?

Basic Materials

• Laptop or desktop computer with internet access.
• Public course enrollment data from a school district, state education site, or school report.
• Spreadsheet software such as Google Sheets or Excel.
• Python installed with NumPy, pandas, SciPy, and Matplotlib.
• Graph paper or a notebook for model setup and proof sketches.
• Calculator for checking small numeric examples.

Advanced Materials

• Laptop or desktop computer with internet access.
• Public or anonymized district-level enrollment data.
• Python with NumPy, pandas, SciPy, Matplotlib, and statsmodels.
• Jupyter Notebook for model development and reproducible analysis.
• Symbolic math software such as SymPy for equilibrium derivations.
• Version control software such as Git for tracking model revisions.

Software & Tools

• Python: Builds the model, runs simulations, and compares predicted equilibria to data.
• Jupyter Notebook: Keeps equations, code, and plots in one place for easy revision.
• Google Sheets: Organizes enrollment data and helps you do quick sanity checks.
• GeoGebra: Helps you graph payoff functions and visualize where equilibrium points appear.
• ImageJ: Not needed for this topic, but useful if you later turn charts or figures into annotated visuals.

Experiment Steps

1. Define the decision setting by choosing one grade level, one course cluster, and one measurable outcome such as enrollment share.
2. Translate the student choice problem into a utility model with a clear payoff for each option.
3. State the monotonicity assumption and write down the condition you expect to guarantee uniqueness.
4. Collect public enrollment data and decide how you will turn it into comparable categories for the model.
5. Build a baseline equilibrium solver and check whether it matches the historical pattern.
6. Test alternative assumptions, then compare how much each change improves or worsens prediction error.

Common Pitfalls

• Using enrollment totals with mismatched grade bands, which makes the model compare different populations.
• Treating public data as if it were already an equilibrium, which hides the gap you want to study.
• Making the payoff function too vague, which prevents you from proving or testing uniqueness.
• Fitting the model to one year only, which makes the result look stronger than it really is.
• Ignoring capacity or schedule constraints, which can create fake equilibrium solutions that schools could never offer.

What Makes This Competitive

A stronger project would not stop at a basic simulation. You would prove a clean mathematical result, then test it against several years of real data. You could also compare your model with a simpler baseline and use a real error metric, not just visual fit. If you show where the model works, where it fails, and why, your project starts to look like original applied math research.

Project Variations

• Use AP course enrollment data instead of general course choices to test whether elite course demand follows the same equilibrium pattern.
• Compare one school district against several districts to see whether the same monotonicity assumptions hold across settings.
• Replace the full equilibrium solver with a simpler best-response model and measure which one predicts enrollment more accurately.

Learn More

• MIT OpenCourseWare, Game Theory and Applied Mathematics materials: Search MIT OpenCourseWare for game theory, optimization, and mathematical modeling lectures.
• NIH PubMed, review articles on choice models and equilibrium: Search PubMed for review articles on discrete choice, equilibrium models, and social interaction effects.
• JSTOR, mathematical modeling papers: Search for accessible articles on mean-field games, equilibrium, and applied optimization.
• Springer Open, free math articles and book chapters: Search for open-access papers on mean-field games and equilibrium analysis.
• US Department of Education, public enrollment and school data: Use federal education data portals to find district, school, and course-access reports.