Pricing for customers with probabilistic valuations as a continuous knapsack problem

Michael Benisch,Norman Sadeh,James Andrews

doi:10.1145/1151454.1151475

Abstract

In this paper, we examine the problem of choosing discriminatory prices for customers with probabilistic valuations and a seller with indistinguishable copies of a good. We show that under certain assumptions this problem can be reduced to the continuous knapsack problem (CKP). We present a new fast e-optimal algorithm for solving CKP instances with asymmetric concave reward functions. We also show that our algorithm can be extended beyond the CKP setting to handle pricing problems with overlapping goods (e.g.goods with common components or common resource requirements), rather than indistinguishable goods.We provide a framework for learning distributions over customer valuations from historical data that are accurate and compatible with our CKP algorithm, and we validate our techniques with experiments on pricing instances derived from the Trading Agent Competition in Supply Chain Management (TAC SCM). Our results confirm that our algorithm converges to an e-optimal solution more quickly in practice than an adaptation of a previously proposed greedy heuristic.

Highlights

In this paper we study a ubiquitous pricing problem: a seller with finite, indistinguishable copies of a good attempts to optimize profit in choosing discriminatory, take-it-or-leaveit offers for a set of customers
The results confirm the intuition of Pardoe and Stone in [8] that the greedy heuristic finds near optimal solutions on continuous knapsack problem (CKP) instances generated from Trading Agent Competition in Supply Chain Management (TAC SCM)
We have modeled this problem as a Probabilistic Pricing Problem with Indistinguishable Goods (P3ID) and formally shown its equivalence the Continuous Knapsack Problem (CKP)

Summary

Introduction

In this paper we study a ubiquitous pricing problem: a seller with finite, indistinguishable copies of a good attempts to optimize profit in choosing discriminatory, take-it-or-leaveit offers for a set of customers. Each customer draws a valuation from some probability distribution known to the seller, and decides whether or not they will accept the seller’s offers (we will refer to this as a probabilistic pricing problem for short). This setting characterizes existing electronic markets built around supply chains for goods or services. In such markets, sellers can build probabilistic valuation models for their customers, e.g.to capture uncertainty about prices offered by competitors, or to reflect the demand of their own customers.