Linear Programming Based Near-Optimal Pricing for Laminar Bayesian Online Selection

31 Pages Posted: 5 Aug 2019 Last revised: 11 Sep 2020

See all articles by Nima Anari

Nima Anari

Stanford University - Computer Science Department

Rad Niazadeh

University of Chicago - Booth School of Business

Amin Saberi

Stanford University - Department of Management Science & Engineering

Ali Shameli

Stanford University, Management Science & Engineering

Date Written: July 31, 2019

Abstract

In the Bayesian online selection problem, the goal is to design a pricing scheme for a sequence of arriving buyers that maximizes the expected social-welfare (or revenue) subject to different types of structural constraints. Inspired by applications in operations management, the focus of this paper is on the cases where the set of served customers is characterized by a laminar matroid.

We give the first Polynomial-Time Approximation Scheme (PTAS) for the problem when the laminar matroid has constant depth. Our approach is based on rounding the solution of a hierarchy of linear programming relaxations that approximate the optimum online solution with any degree of accuracy plus a concentration argument that shows the rounding incurs a small loss. We also study another variation, which we call the production constrained problem, for which the allowable set of served customers is characterized by a collection of production and shipping constraints forming a certain form of laminar matroid. Using a similar LP-based approach, we design a PTAS for this problem even when the depth of the laminar matroid is not constant. The analysis exploits the negative dependency of the optimum selection rule in the lower-levels of the laminar family.

Finally, we conclude with a discussion of the linear programming based approach employed in the paper and re-derive some of the classic prophet inequalities known in the literature.

Keywords: Bayesian online selection; Prophet inequalities; PTAS; Dynamic programming

Suggested Citation

Anari, Nima and Niazadeh, Rad and Saberi, Amin and Shameli, Ali, Linear Programming Based Near-Optimal Pricing for Laminar Bayesian Online Selection (July 31, 2019). Chicago Booth Research Paper No. 20-24, Available at SSRN: https://ssrn.com/abstract=3430156 or http://dx.doi.org/10.2139/ssrn.3430156

Nima Anari

Stanford University - Computer Science Department ( email )

353 Serra Mall
Stanford, CA 94305
United States

Rad Niazadeh (Contact Author)

University of Chicago - Booth School of Business ( email )

5807 S Woodlawn Ave
Chicago, IL 60637

HOME PAGE: http://radniazadeh.github.io/

Amin Saberi

Stanford University - Department of Management Science & Engineering ( email )

473 Via Ortega
Stanford, CA 94305-9025
United States

Ali Shameli

Stanford University, Management Science & Engineering ( email )

473 Via Ortega
Stanford, CA 94305-9025
United States

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