  Assignment markets involve matching with transfers, as in labor markets and
housing markets. We consider a two-sided assignment market with agent types and
stochastic structure similar to models used in empirical studies, and
characterize the size of the core in such markets. Each agent has a randomly
drawn productivity with respect to each type of agent on the other side. The
value generated from a match between a pair of agents is the sum of the two
productivity terms, each of which depends only on the type but not the identity
of one of the agents, and a third deterministic term driven by the pair of
types. We allow the number of agents to grow, keeping the number of agent types
fixed. Let $n$ be the number of agents and $K$ be the number of types on the
side of the market with more types. We find, under reasonable assumptions, that
the relative variation in utility per agent over core outcomes is bounded as
$O^*(1/n^{1/K})$, where polylogarithmic factors have been suppressed. Further,
we show that this bound is tight in worst case. We also provide a tighter bound
under more restrictive assumptions. Our results provide partial justification
for the typical assumption of a unique core outcome in empirical studies.
