Stable Tables

TL;DR

This paper analyzes the probability of individuals being unmatched in stable matchings in a circular table setting, deriving exact formulas for odd and even numbers of participants and showing asymptotic behavior.

Contribution

It provides explicit formulas for the likelihood of unmatched and fully matched individuals in equilibrium for circular neighbor matchings, a novel analytical result.

Findings

Probability of a person being unmatched approaches 1/9 as n increases.

Probability all persons are matched is zero for odd n, positive for even n.

Derived exact formulas for both odd and even n cases.

Abstract

We consider equilibrium one-on-one conversations between neighbors on a circular table, with the goal of assessing the likelihood of a (perhaps) familiar situation: sitting at a table where both of your neighbors are talking to someone else. When $n$ people in a circle randomly prefer their left or right neighbor, we show that the probability a given person is unmatched in equilibrium (i.e., in a stable matching) is $$\frac{1}{9} + \left(\frac{1}{2}\right)^n\left(\frac{2n}{3} - \frac{8}{9} + \frac{2}{n}\right)$$ for odd $n$ and $$\frac{1}{9} - \left(\frac{1}{2}\right)^n\left(\frac{2n}{3} - \frac{8}{9}\right)$$ for even $n$. This probability approaches $1/9$ as $n\rightarrow \infty$. We also show that the probability \textit{every} person is matched in equilibrium is $0$ for odd $n$ and $\frac{3^{n/2}-1}{2^{n-1}}$ for even $n$.