Optimal Pricing for Submodular Valuations with Bounded Curvature

TL;DR

This paper investigates the computational complexity of optimal pricing with submodular valuations and proposes approximation algorithms for both unlimited and limited budget scenarios, extending solutions to multiple collaborating buyers.

Contribution

It demonstrates the hardness of the problem even with a single buyer and introduces new approximation algorithms for various budget settings.

Findings

The problem is computationally hard even with one buyer.

Proposed approximation algorithms for unlimited budget case.

Extended algorithms to multiple collaborating buyers with limited budgets.

Abstract

The optimal pricing problem is a fundamental problem that arises in combinatorial auctions. Suppose that there is one seller who has indivisible items and multiple buyers who want to purchase a combination of the items. The seller wants to sell his items for the highest possible prices, and each buyer wants to maximize his utility (i.e., valuation minus payment) as long as his payment does not exceed his budget. The optimal pricing problem seeks a price of each item and an assignment of items to buyers such that every buyer achieves the maximum utility under the prices. The goal of the problem is to maximize the total payment from buyers. In this paper, we consider the case that the valuations are submodular. We show that the problem is computationally hard even if there exists only one buyer. Then we propose approximation algorithms for the unlimited budget case. We also extend the algorithm for the limited budget case when there exists one buyer and multiple buyers collaborate with each other.