Two-sided Assortment Optimization: Adaptivity Gaps and Approximation Algorithms

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.