worst case scenario for stable matching algorithm visualized

Guri Bee

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21-10-2024

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The Worst Case Scenario for Stable Matching Algorithms: A Visualized Guide

The Stable Matching Problem, often exemplified by the "Gale-Shapley Algorithm," is a fascinating area in computer science and game theory. It addresses the question of how to pair two sets of elements (like students and colleges, or men and women) in a way that avoids any "instability," meaning nobody would prefer another partner over their current match. While the algorithm guarantees a stable solution, there's a fascinating worst-case scenario that's worth understanding.

The Problem:

Imagine a world of five students (A, B, C, D, E) and five colleges (1, 2, 3, 4, 5). Each student has a preference list for the colleges, and each college has a preference list for students. Here's a possible scenario where the Stable Matching Algorithm might yield a seemingly "unfair" outcome:

Preference Lists:

• Student A: 1 > 2 > 3 > 4 > 5

• Student B: 1 > 2 > 3 > 4 > 5

• Student C: 1 > 2 > 3 > 4 > 5

• Student D: 1 > 2 > 3 > 4 > 5

• Student E: 1 > 2 > 3 > 4 > 5

• College 1: E > D > C > B > A

• College 2: E > D > C > B > A

• College 3: E > D > C > B > A

• College 4: E > D > C > B > A

• College 5: E > D > C > B > A

The Unfair Outcome:

Notice how all students have the same preference list, and all colleges have the same preference list. This extreme case leads to a situation where Student E (the most preferred student for all colleges) gets matched with College 1 (their most preferred college), while the remaining students get matched with their least preferred colleges (Student A with College 5, Student B with College 4, etc.).

Visualizing the Unfairness:

We can visualize this scenario using a simple diagram:

     Student A - College 5 (Least Preferred)
     Student B - College 4 (Least Preferred)
     Student C - College 3 (Least Preferred)
     Student D - College 2 (Least Preferred)
     Student E - College 1 (Most Preferred)

This outcome, although technically stable (no one can swap partners to improve their situation), appears highly unfair. Students A through D are essentially stuck with their least preferred colleges, while Student E gets the best possible match.

Why Does This Happen?

The Stable Matching Algorithm operates on the principle of "proposing and rejecting." Colleges propose to their top choices, and students accept the best offer they've received. Since all colleges prefer Student E, they keep proposing to her, pushing the other students down the preference list. This results in the last student remaining with the least preferred college.

Analyzing the Worst Case:

While the worst case scenario might seem like a theoretical edge case, it highlights some key aspects of the algorithm:

• Priority: The algorithm prioritizes the "better" (more preferred) parties, often at the expense of the "lesser" parties.
• Stability Over Fairness: The algorithm ensures a stable solution, but it doesn't guarantee a fair distribution of matches.
• Context is Key: The algorithm's effectiveness depends heavily on the structure of preference lists. In real-world applications, preference lists are rarely uniform.

Practical Implications:

Understanding the worst-case scenario helps us appreciate the limitations of Stable Matching algorithms and their sensitivity to the structure of preferences. In practical settings, it's important to:

• Design fair preference lists: Incorporate factors beyond just "preference" to ensure a balanced and equitable outcome.
• Use different algorithms: There are variations of the Stable Matching Algorithm that aim to address fairness concerns.
• Consider the context: The choice of algorithm and its implementation should be tailored to the specific application and its unique requirements.

By understanding both the strengths and weaknesses of Stable Matching algorithms, we can better utilize them in diverse real-world applications, ensuring both stability and fairness in matching processes.