  This paper introduces a unified framework for stable matching, which nests
the traditional definition of stable matching in finite markets and the
continuum definition of stable matching from Azevedo and Leshno (2016) as
special cases. Within this framework, I identify a novel continuum model, which
makes individual-level probabilistic predictions.
  This new model always has a unique stable outcome, which can be found using
an analog of the Deferred Acceptance algorithm. The crucial difference between
this model and that of Azevedo and Leshno (2016) is that they assume that the
amount of student interest at each school is deterministic, whereas my proposed
alternative assumes that it follows a Poisson distribution. As a result, this
new model accurately predicts the simulated distribution of cutoffs, even for
markets with only ten schools and twenty students.
  This model generates new insights about the number and quality of matches.
When schools are homogeneous, it provides upper and lower bounds on students'
average rank, which match results from Ashlagi, Kanoria and Leshno (2017) but
apply to more general settings. This model also provides clean analytical
expressions for the number of matches in a platform pricing setting considered
by Marx and Schummer (2021).
