Stable matching algorithms are used to clear many markets — most notably, for residencies and public schools. However, the relationship between the input to these algorithms (preferences, priorities, and capacities) and aggregate outcome measures (such as the number of agents matched to their first, second, and third choices) is poorly understood. 

Analyses of large finite markets make restrictive assumptions on priorities and preferences, for instance by assuming that they are drawn uniformly at random. Prior continuum models of stable matching allow for general priorities and preferences, but apply only in settings with large capacities. This paper introduces a model that allows for arbitrary preferences, priorities and capacities. Markets in this model always have a unique stable outcome, making it well-suited for generation and evaluation of predictions. The model accurately predicts simulation outcomes from the literature, even for markets with as few as ten or or twenty participants on each side. 

We illustrate the usefulness of the model by proving two new results. First, we compare matching size under different lottery procedures. Second, we study how the number of students receiving their first choice under a common lottery depends on the popularity of different schools.

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