📈 Ranking Systems with Linear Equations – TP1

This repository contains the first practical assignment (Trabajo Práctico 1) for the course Numerical Methods at the University of Buenos Aires (UBA), Faculty of Exact and Natural Sciences, completed during the second semester of 2021.

The project explores and compares different algorithms for generating rankings of teams or individuals in sports competitions using linear systems, specifically the Colley Matrix Method, Win Percentage, and Traditional Football Scoring.

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📌 Problem Summary

Given the results of matches between teams or players, the goal is to construct a numerical ranking that orders participants based on performance. The assignment explores three ranking methods:

1. Colley Matrix Method (CMM) – Solves a system of linear equations derived from match history.
2. Win Percentage (WP) – Ranks based on the ratio of wins to total games played.
3. Traditional Football Scoring (Fulbo) – Uses standard 3/1/0 point system per win/draw/loss.

The algorithms are analyzed in terms of logic, complexity, fairness, and numerical stability.

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🧠 Topics Covered

• Gaussian Elimination (with and without pivoting)
• System of linear equations
• Numerical stability and error analysis
• Complexity analysis: O(T³ + P) for CMM
• Evaluation of ranking fairness using real and synthetic data (e.g., ATP 2015)

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📊 Experiments

The report includes:

• Absolute error analysis and floating-point stability
• Real data comparison: ATP 2015 rankings vs. algorithm outputs
• Fairness tests (how outcomes affect non-involved players)
• Strategy analysis: how to best rank #1 depending on the scoring method

See informe.pdf for detailed graphs, tables, and test case results.

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📄 Report

The full report (in Spanish) is included here:
📎 [informe.pdf](./Informe - TP1.pdf)

It contains:

• Formal algorithm definitions and pseudocode
• Implementation notes and complexity proofs
• Benchmark results and ranking comparisons
• References to academic materials and real-world datasets

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🧾 Authors

Developed by:

• Federico Alliani
• Manuel Nores
• Lucas Raposeiras
• Mariano Oca

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🎓 Academic Context

• Course: Numerical Methods (Métodos Numéricos)
• Institution: Universidad de Buenos Aires (UBA), Facultad de Ciencias Exactas y Naturales
• Semester: 2nd semester, 2021
• Language: Python / C (depending on implementation)

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🪪 License

This project is for academic and educational purposes only. Please cite the original authors if reused.