[Paper Review] Revenue-Optimal Pricing for Budget-Constrained Buyers in Data Markets

Bhaskar Ray Chaudhury, Jugal Garg+2 more|arXiv (Cornell University)|Feb 14, 2026

Advanced Bandit Algorithms Research0 citations

TL;DR

The paper studies revenue-maximizing pricing in data markets with budget-constrained buyers, showing nonlinear pricing is solvable in polynomial time while linear pricing is APX-hard, with practical approximation algorithms.

ABSTRACT

We study revenue-optimal pricing in data markets with rational, budget-constrained buyers. Such a market offers multiple datasets for sale, and buyers aim to improve the accuracy of their prediction tasks by acquiring data bundles. For each dataset, the market sets a pricing function, which maps the number of records purchased from the dataset to a non-negative price. The market's objective is to set these pricing functions to maximize total revenue, considering that buyers with quasi-linear utilities choose their bundles optimally under budget constraints. We analyze optimal pricing when each dataset's pricing function is only required to be monotone and lower-continuous. Surprisingly, even with this generality, optimal pricing has a highly structured form: it is piecewise linear and convex (PLC) and can be computed efficiently via an LP. Moreover, the total number of kinks across all pricing functions is bounded by the number of buyers. Thus, when datasets far outnumber buyers, most pricing functions are effectively linear. This motivates studying linear pricing, where each record in a dataset is priced uniformly. Although competitive equilibrium gives revenue-optimal linear prices in rivalrous markets with quasi-linear buyers, we show that revenue maximization under linear pricing in data markets is APX-hard. Hence, a striking computational dichotomy emerges: fully general (nonlinear) pricing admits a polynomial-time algorithm, while the simpler linear scheme is APX-hard. Despite the hardness, we design a 2-approximation algorithm when datasets arrive online, and a $(1-1/e)^{-1}$-approximation algorithm for the offline setting. Our framework lays the groundwork for exploring more general pricing schemes, richer utility models, and a deeper understanding of how market structure -- rivalrous versus non-rivalrous -- shapes revenue-optimal pricing.

Motivation & Objective

• Model a centralized data marketplace with m datasets and n buyers with budgets.
• Characterize optimal pricing functions under monotone lower-continuous pricing.
• Show that the optimal nonlinear pricing is piecewise linear and convex (PLC) and computable via LP.
• Establish a complexity dichotomy: nonlinear pricing solvable in polynomial time vs. linear pricing APX-hard.

Proposed method

• Formulate buyer utility with data-accuracy value and dataset-priced costs.
• Prove optimal ML C pricing functions are obtained via convex envelopes and PLC approximation.
• Derive an LP to compute optimal PLC pricing and show structural properties (kinks bounded by n).
• Show that in the linear pricing regime, revenue maximization is APX-hard.
• Provide approximation algorithms: 2-approx online and (1-1/e)^{-1} offline.

Experimental results

Research questions

• RQ1 How should a data marketplace price datasets to maximize total revenue under budget-constrained, quasi-linear buyers?
• RQ2 What is the structural form of revenue-optimal pricing under monotone lower-continuous pricing?
• RQ3 Can nonlinear pricing be efficiently computed, and how does linear pricing compare in complexity and performance?
• RQ4 What approximations are possible for linear pricing in online versus offline settings?

Key findings

• Optimal solutions to the pricing problem can be computed in polynomial time.
• The optimal nonlinear pricing is piecewise linear and convex (PLC) with at most n breakpoints per dataset on average, and the total number of kinks across all datasets is bounded by the number of buyers.
• The problem exhibits a dichotomy: nonlinear pricing is polynomial-time solvable, while linear pricing is APX-hard.
• When datasets are priced linearly, the revenue maximization problem admits a 2-approximation in online arrival and a (1-1/e)^{-1} approximation in the offline setting.
• A closed-form revenue expression exists: r(p) = sum_i min(b_i, sum_{j: tau_{i,j} >= p_j} p_j), reflecting per-buyer contributions without inter-buyer competition due to non-rivalry.