On the limitations of data-based price discrimination

TL;DR

This paper investigates data-based pricing for third-degree price discrimination (3PD) when the joint distribution of valuations and covariates, $F_{Y,X}$, is unknown and only i.i.d. samples are available. It introduces the $K$-markets ERM strategy, proving that its revenue deficiency relative to the true $F_{Y,X}$-optimal 3PD decays at rate $O(n^{-1/2})$ when optimally tuned with $K\asymp n^{1/4}$, while the uniform ERM strategy converges at rate $O(n^{-2/3})$ to the true uniform optimum. The authors also establish information-theoretic lower bounds showing minimax deficiencies of $\Omega(n^{-1/2})$ for 3PD and $\Omega(n^{-2/3})$ for uniform pricing, implying the rates are tight in the worst case. Numerical experiments on real (eBay) and simulated data illustrate the trade-offs: with small samples, uniform pricing can outperform discriminative pricing, but as $n$ grows, $K$-market ERM increasingly approaches the $F_{Y,X}$-optimal 3PD, highlighting the curse of dimensionality in learning covariate-driven prices. The work also discusses small-sample complications and poses open questions about finite-sample regimes where data-based 3PD may never outperform uniform pricing.

Abstract

The classic third degree price discrimination (3PD) model requires the knowledge of the distribution of buyer valuations and the covariate to set the price conditioned on the covariate. In terms of generating revenue, the classic result shows that 3PD is at least as good as uniform pricing. What if the seller has to set a price based only on a sample of observations from the underlying distribution? Is it still obvious that the seller should engage in 3PD? This paper sheds light on these fundamental questions. In particular, the comparison of the revenue performance between 3PD and uniform pricing is ambiguous overall when prices are set based on samples. This finding is in the nature of statistical learning under uncertainty: a curse of dimensionality, but also other small sample complications.

Figures (6)

• Figure 1: Revenue under uniform and $K$-markets ERM strategies
• Figure 2: Data-based revenue deficiency under uniform and $K$-markets ERM strategies (with $K\asymp n^{1/4}$).
• Figure 3: Uniform and $K$-markets revenue for the case of $X$ and $Y$ uniform on $[0,1]$ and independent of each other.
• Figure B.1: Perturbation functions $\phi_{Y}$ and $\phi_{X}$.
• Figure B.2: Density functions $f_1$ and $f_2$.

Theorems & Definitions (29)

• Remark
• Theorem 1
• Remark
• Corollary 1
• Theorem 2
• Theorem 3: Lower bounds for 3PD, deficiency in pointwise revenue
• Theorem 4: Lower bounds for 3PD, deficiency in expected revenue
• Remark
• Theorem 5
• proof : Proof of Corollary \ref{['thm:upper-bound-unif']}