Splitting Rent with Triangles | Infinite Series

TL;DR

Sperner's Lemma can be used to fairly divide rent among roommates based on their preferences, using triangles subdivided into smaller triangles.

Transcript

Larry, Curly, and Moe are moving into a new apartment. Both Moe and Larry want the room with the balcony, but Curly isn't as picky. He mostly just wants a cheap room. Taking into account all their subjective preferences, how should they fairly decide who gets which room and how much each person should pay? To help these wise guys fairly divide ren... Read More

Key Insights

• 👪 Sperner's Lemma can be used to divide rent fairly among roommates based on their preferences.
• ➗ The fair division is represented by a fully labeled triangle, where each vertex corresponds to a rent division and room preferences.
• 🥹 Sperner's Lemma also holds for simplices in higher dimensions, enabling fair division for larger groups of roommates.
• 🧚 The precision of the fair division increases with the number of subdivisions in the triangle.
• 💦 The method works even if one roommate's preferences are secret, ensuring fairness for everyone involved.
• ➗ Sperner's Lemma is applicable to various scenarios beyond rent division, such as cake division and chore allocation.

Questions & Answers

Q: How does Sperner's Lemma help in fairly dividing rent?

Sperner's Lemma provides a method to subdivide a triangle into smaller triangles, each representing a rent division. One of these triangles will be fully labeled, allowing for a fair rent distribution.

Q: Can Sperner's Lemma be applied to larger groups of roommates?

Yes, Sperner's Lemma can be expanded to higher-dimensional triangles to accommodate more roommates. Each additional roommate requires an additional simplex.

Q: What happens if one roommate's preferences are secret?

In such cases, the modified method is used, where all roommates indicate their preferences, including the one with the secret preferences. Sperner's Lemma is then applied to find a fully labeled triangle, ensuring a fair division.

Q: How precise is the fair division of rent using Sperner's Lemma?

The precision depends on the number of layers or subdivisions in the triangle. Though the price distributions may differ slightly at the corners, being fair within a small margin is considered acceptable.

Summary & Key Takeaways

• The fair division of rent among roommates is solved using Sperner's Lemma, a geometric mathematical concept.

• Sperner's Lemma guarantees the existence of a fully labeled triangle in which each vertex represents a rent division and room preferences.

• The rent can be divided fairly by assigning rooms based on the preferences indicated in the fully labeled triangle.