Dynamic Pricing in the Linear Valuation Model using Shape Constraints (2502.05776v3)

Abstract: We propose a shape-constrained approach to dynamic pricing for censored data in the linear valuation model eliminating the need for tuning parameters commonly required by existing methods. Previous works have addressed the challenge of unknown market noise distribution $F_0$ using strategies ranging from kernel methods to reinforcement learning algorithms, such as bandit techniques and upper confidence bounds (UCB), under the assumption that $F_0$ satisfies Lipschitz (or stronger) conditions. In contrast, our method relies on isotonic regression under the weaker assumption that $F_0$ is $\alpha$-H\"older continuous for some $\alpha \in (0,1]$, for which we derive a regret upper bound. Simulations and experiments with real-world data obtained by Welltower Inc (a major healthcare Real Estate Investment Trust) consistently demonstrate that our method attains lower empirical regret in comparison to several existing methods in the literature while offering the advantage of being tuning-parameter free.

• The paper introduces a dynamic pricing approach that uses isotonic regression to estimate the market noise CDF without relying on tuning parameters.
• The method operates under a weaker α-Hölder continuity assumption, broadening its applicability compared to traditional Lipschitz-based methods.
• The proposed strategy achieves asymptotic regret bounds that match state-of-the-art performance, offering a simpler and competitive implementation.

This document details a dynamic pricing approach within the linear valuation model, utilizing shape constraints to circumvent the need for tuning parameters often present in alternative methods. The focus is on scenarios with censored demand data, where only purchase decisions (sale or no-sale) are observed, not the customer's full valuation.

Problem Formulation and Background

In the standard linear valuation model for dynamic pricing with contextual information, a customer's valuation $V_t$ for a product at time $t$ is modeled as:

$V_t = \beta_0^T x_t + \epsilon_t$

where $x_t \in \mathbb{R}^d$ is a vector of observable customer/product features, $\beta_0 \in \mathbb{R}^d$ is an unknown vector of parameters representing the linear relationship between features and valuation, and $\epsilon_t$ is unobservable market noise, assumed to be drawn independently from a distribution with CDF $F_0$.

The seller sets a price $p_t$ at time $t$. A sale occurs if $V_t \ge p_t$. The seller only observes the binary outcome $y_t = \mathbb{I}(V_t \ge p_t)$, not $V_t$ itself. The probability of a sale, given $x_t$ and $p_t$, is:

$$\mathbb{P}(y_t = 1 | x_t, p_t) = \mathbb{P}(\beta_0^T x_t + \epsilon_t \ge p_t) = \mathbb{P}(\epsilon_t \ge p_t - \beta_0^T x_t) = 1 - F_0(p_t - \beta_0^T x_t)$$

The objective in dynamic pricing is to sequentially choose prices $p_1, \dots, p_T$ to maximize cumulative revenue $\sum_{t=1}^T p_t y_t$, which involves learning the unknown parameters $\beta_0$ and the noise distribution $F_0$ from the observed data $(x_t, p_t, y_t)$. A key challenge lies in estimating the unknown, potentially non-parametric, noise distribution $F_0$.

Previous approaches often rely on kernel density estimation or kernel regression to estimate $F_0$ or its derivatives, which necessitates selecting tuning parameters like bandwidths. Other methods employ reinforcement learning or bandit algorithms, frequently assuming Lipschitz continuity (or stronger conditions) on $F_0$ to construct confidence bounds (e.g., UCB algorithms) or ensure convergence (2109.07340, 1604.07463). These tuning parameters or stringent assumptions can limit practical applicability.

Shape-Constrained Estimation using Isotonic Regression

The proposed method leverages the inherent shape constraint of the noise CDF $F_0$ – namely, that it is a non-decreasing function. This allows for the use of isotonic regression, a non-parametric technique specifically designed for estimating monotone functions.

The core idea is to estimate $F_0$ without resorting to methods requiring explicit tuning parameters. Given an estimate $\hat{\beta}$ of $\beta_0$ at time $t$, one can define residuals or transformed variables $z_{i} = p_i - \hat{\beta}^T x_i$ for past observations $i=1, \dots, t-1$. The observed outcomes $y_i$ provide censored information about $F_0$ at these points: $y_i=1$ suggests $F_0(z_i) \le 1 - \mathbb{P}(y_i=1 | x_i, p_i)$ and $y_i=0$ suggests $F_0(z_i) > 1 - \mathbb{P}(y_i=1 | x_i, p_i)$.

Isotonic regression is applied to estimate the non-decreasing function $F_0$ based on the pairs $(z_i, y_i)$. Specifically, it finds the non-decreasing function $\hat{F}_t$ that minimizes a weighted least squares criterion subject to the monotonicity constraint. The Pool Adjacent Violators Algorithm (PAVA) provides an efficient way to compute the isotonic regression estimate.

This approach offers a significant advantage: the estimation of $F_0$ is entirely data-driven and avoids the need to specify tuning parameters like kernel bandwidths. The only underlying assumption required on $F_0$ for the theoretical analysis is $\alpha$-Hölder continuity.

Theoretical Guarantees under Hölder Continuity

A central theoretical contribution is the analysis under the assumption that $F_0$ is $\alpha$-Hölder continuous for some $\alpha \in (0, 1]$. Recall that a function $f$ is $\alpha$-Hölder continuous if there exists a constant $C$ such that $|f(x) - f(y)| \le C |x-y|^\alpha$ for all $x, y$ in its domain. This is a weaker condition than Lipschitz continuity, which corresponds to the case $\alpha=1$. Many existing theoretical analyses for dynamic pricing with unknown non-parametric demand rely on the stronger Lipschitz assumption.

The paper (2502.05776) derives an upper bound on the asymptotic expected regret using this shape-constrained approach. The regret measures the expected difference in revenue compared to an oracle policy that knows $\beta_0$ and $F_0$ perfectly. The derived regret bound is shown to match the existing state-of-the-art bounds in the literature for the special case of $\alpha=1$ (Lipschitz continuity). This demonstrates that the proposed method achieves comparable theoretical performance to existing methods under standard assumptions, while crucially relying on a weaker, more general condition ($\alpha$-Hölder continuity) and eliminating tuning parameters.

Comparison to Existing Methods and Advantages

The primary advantages of the shape-constrained isotonic regression approach compared to alternatives are:

1. Tuning-Parameter Free: Unlike kernel-based methods requiring bandwidth selection or certain RL algorithms requiring tuning of exploration parameters, this method avoids such hyperparameters for the estimation of $F_0$. This simplifies implementation and removes the sensitivity to potentially suboptimal parameter choices. (arxiv.org)
2. Weaker Assumptions: The theoretical guarantees hold under the relatively weak assumption of $\alpha$-Hölder continuity for $F_0$, broadening the applicability compared to methods requiring Lipschitz continuity or specific parametric forms.
3. Competitive Theoretical Bounds: The asymptotic regret bound matches existing results for the commonly studied Lipschitz case ($\alpha=1$), indicating no loss in theoretical performance despite the weaker assumptions and lack of tuning parameters.
4. Strong Empirical Performance: Simulations and experiments on real-world data (Welltower Inc. healthcare REIT data) reportedly show that the method achieves lower empirical regret compared to several benchmark algorithms from the literature. This suggests practical benefits beyond the theoretical advantages. (arxiv.org)

While the paper focuses on estimating $F_0$, a complete dynamic pricing algorithm would typically involve interleaving the estimation of $\beta_0$ (e.g., using maximum likelihood estimation based on the current estimate of $F_0$, potentially resembling a GLM estimation) and the isotonic estimation of $F_0$. The pricing policy itself would likely involve balancing exploration and exploitation, possibly using optimism based on confidence bounds derived for both $\hat{\beta}$ and $\hat{F}_0$, though the non-parametric nature of $\hat{F}_0$ requires careful construction of these bounds under the Hölder assumption.

Conclusion

The dynamic pricing method using shape constraints, specifically isotonic regression for estimating the market noise CDF $F_0$, offers a compelling alternative to existing approaches in the linear valuation model. By leveraging the natural monotonicity of the CDF, it eliminates the need for tuning parameters associated with non-parametric estimation, simplifies implementation, and relies on weaker theoretical assumptions ($\alpha$-Hölder continuity). Theoretical analysis confirms its asymptotic regret performance matches prior results under stronger assumptions, while empirical evaluations demonstrate superior performance in practice. This makes it a promising approach for dynamic pricing applications where the noise distribution is unknown and potentially non-smooth.