Polynomial-Time Approximation Schemes via Utility Alignment: Unit-Demand Pricing and More

Abstract

This paper derives polynomial-time approximation schemes for several NP-hard stochastic optimization problems from the algorithmic mechanism design and operations research literatures. The problems we consider involve a principal or seller optimizing with respect to a subsequent choice by an agent or buyer. These include posted pricing for a unit-demand buyer with independent values (Chawla et al., 2007, Cai and Daskalakis, 2011), assortment optimization with independent utilities (Talluri and van Ryzin, 2004), and delegated choice (Khodabakhsh et al., 2024). Our results advance the state of the art for each of these problems. For unit-demand pricing with discrete distributions, our multiplicative PTAS improves on the additive PTAS of Cai and Daskalakis, and we additionally give a PTAS for the unbounded regular case, improving on the latter paper's QPTAS. For assortment optimization, no constant approximation was previously known. For delegated choice, we improve on both the $3$-approximation for the case with no outside option and the super-constant-approximation with an outside option. A key technical insight driving our results is an economically meaningful property we term utility alignment. Informally, a problem is utility aligned if, at optimality, the principal derives most of their utility from realizations where the agent's utility is also high. Utility alignment allows the algorithm designer to focus on maximizing performance on realizations with high agent utility, which is often an algorithmically simpler task. We prove utility alignment results for all the problems mentioned above, including strong results for unit-demand pricing and delegation, as well as a weaker but very broad guarantee that holds for many other problems under very mild conditions.