Stable Tables
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...
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Zusammenfassung: | 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 $n$. |
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DOI: | 10.48550/arxiv.2411.09716 |