Two-sided Assortment Optimization: Adaptivity Gaps and Approximation Algorithms

TL;DR

This work studies two-sided assortment optimization on bipartite matching platforms under general discrete-choice preferences, introducing policy classes that range from Fully Static to Fully Adaptive. It rigorously analyzes adaptivity gaps between policy classes, establishing exact gaps: One-sided Static vs One-sided Adaptive equals $\frac{e}{e-1}$ and One-sided Adaptive vs Fully Adaptive equals $2$, with worst-performing policies showing simultaneous exposure to all agents. It also delivers approximation algorithms: a $1/4$-approximation for Fully Adaptive and a $0.067$-approximation for Fully Static under MNL preferences, plus extensions to cardinality constraints and a comprehensive computational study. The results offer principled guidance on when adaptive vs non-adaptive platform designs are beneficial and how to design efficient algorithms within a fixed policy class for practical two-sided markets.

Abstract

To address efficiency and design challenges in choice-based matching platforms, 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 $e/(e-1)$. 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 $2$. We also note that the worst policies are those who simultaneously show assortments to all the agents. For (2), we first design a polynomial time algorithm that achieves a $1/4$ approximation factor within the class of policies that adaptively show assortments to agents one by one. Furthermore, when agents' preferences are governed by multinomial-logit models, we show that a 0.067 approximation factor can be obtained within the class of policies that show assortments to all agents at once. We further generalize our results to constrained assortment settings, where we impose an upper bound on the size of the displayed assortments. Finally, we present a computational study to evaluate the empirical performance of our theoretical guarantees.

Figures (2)

• Figure 1: Main results for the unconstrained setting (in blue). Each adaptivity gap (presented below a bracket) corresponds to the ratio between the value on the right over the value on the left, for example, the adaptivity gap between (One-sided Static) and (One-sided Adaptive) is $e/(e-1)$. On the top, we present our guarantees for each class of problems (all guarantees are obtained in polynomial-time except the pseudo-polynomial-time sampling method in Corollary \ref{['coro:approx_onesided_adaptive']}).
• Figure 2: Main results for the two-way constrained setting (in blue). The adaptivity gaps are presented below the brackets. On the top, we present our guarantees for each class of problems.

Theorems & Definitions (61)

• Definition 1: Adaptivity Gap
• Remark 1
• Proposition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Remark 2
• Lemma 1
• Lemma 2
• Lemma 3: agrawal2010correlation