Revenue-Optimal Pricing for Budget-Constrained Buyers in Data Markets

TL;DR

This 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.

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.

Figures (8)

• Figure 1: A pricing function $\textcolor{red!75!textHeavy}{\bm{p}}$ (solid red) and its convex hull $\textcolor{blue!75!textHeavy}{\widetilde{\bm{p}}}$ (dashed blue). Note that $\bm{p}(x) \neq \widetilde{\bm{p}}(x)$ for $x \in (0.4, 0.8)$, and $\widetilde{\bm{p}}$ is linear in that interval, with slope 5.
• Figure 2: Pricing functions for CE and SE of \ref{['ex:inapprox_n_ce_se']}, where the prices are in red
• Figure 3: Pricing function and net utility for \ref{['ex:no-util-max']}.
• Figure 4: Plots for \ref{['ex:non-convex-price']}.
• Figure 5: Let there be a single dataset and 3 buyers having values 2, 3, and 5 for the dataset. The first plot shows the left derivative of a pricing function $p$. The second plot shows the left derivative of the piecewise-linearization of $p$.

Theorems & Definitions (139)

• Theorem 1
• Example 1
• Example 2
• Theorem 2
• Example 3: Non-submodularity of $r(\bm{p})$
• Theorem 3
• Theorem 4
• Definition 1: Separable function
• Example 4: Separable and inseparable pricing functions
• Definition 2: Revenue of a pricing function