2024/11/13 by Kenny Peng, Peng, Kenny
Biochemistry, Genetics and Molecular Biology · Economics, Econometrics and Finance · Mathematics · #Combinatorics (math.CO) #Diffusion and Search Dynamics #FOS: Mathematics #Game Theory and Voting Systems #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2411.09716
openalex publication_date 2024/11/13 · openalex created_date 2024/11/21 · openalex updated_date 2026/07/28
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 (1)/(9) + ((1)/(2))n((2n)/(3) - (8)/(9) + (2)/(n)) for odd n and (1)/(9) - ((1)/(2))n((2n)/(3) - (8)/(9)) for even n. This probability approaches 1/9 as n→ ∞. We also show that the probability every person is matched in equilibrium is 0 for odd n and \frac3n/2-12n-1 for even n.