Tiered Random Matching Markets: Rank is Proportional to Popularity

TL;DR

This paper analyzes the stable marriage problem in markets with agents divided into tiers based on public scores, showing how these scores influence match quality and rank distributions.

Contribution

It introduces a model with tiered preferences in stable matching, providing explicit calculations of average ranks and concentration results in large markets.

Findings

Average rank depends on agent tiers and public scores.

Public scores influence match quality but not the probability of matching to a tier.

Results generalize uniform preference models to heterogeneous attractiveness.

Abstract

We study the stable marriage problem in two-sided markets with randomly generated preferences. We consider agents on each side divided into a constant number of "soft tiers", which intuitively indicate the quality of the agent. Specifically, every agent within a tier has the same public score, and agents on each side have preferences independently generated proportionally to the public scores of the other side. We compute the expected average rank which agents in each tier have for their partners in the men-optimal stable matching, and prove concentration results for the average rank in asymptotically large markets. Furthermore, we show that despite having a significant effect on ranks, public scores do not strongly influence the probability of an agent matching to a given tier of the other side. This generalizes results of [Pittel 1989] which correspond to uniform preferences. The results quantitatively demonstrate the effect of competition due to the heterogeneous attractiveness of agents in the market, and we give the first explicit calculations of rank beyond uniform markets.