  To address the challenge of choice congestion in matching markets, in this
work, we introduce a two-sided assortment optimization framework under general
choice preferences. The goal in this problem is to maximize the expected number
of matches by deciding which assortments are displayed to the agents and the
order in which they are shown. In this context, we identify several classes of
policies that platforms can use in their design. Our goals are: (1) to measure
the value that one class of policies has over another one, and (2) to
approximately solve the optimization problem itself for a given class. For (1),
we define the adaptivity gap as the worst-case ratio between the optimal values
of two different policy classes. First, we show that the gap between the class
of policies that statically show assortments to one-side first and the class of
policies that adaptively show assortments to one-side first is exactly $1-1/e$.
Second, we show that the gap between the latter class of policies and the fully
adaptive class of policies that show assortments to agents one by one is
exactly $1/2$. We also note that the worst policies are those who
simultaneously show assortments to all the agents, in fact, we show that their
adaptivity gap even with respect to one-sided static policies can be
arbitrarily small. For (2), we first show that there exists a polynomial time
policy that achieves a $1/4$ approximation factor within the class of policies
that adaptively show assortments to agents one by one. Finally, when agents'
preferences are governed by multinomial-logit models, we show that a 0.082
approximation factor can be obtained within the class of policies that show
assortments to all agents at once.
