This work was presented at NeurIPS 2025 MLxOR workshop. You can also check out the full paper here.

Introduction

• Just as patients exhibit heterogeneous responses to medical treatments, consumers show highly heterogeneous responses to prices.
• Understanding and leveraging this individualized response is the fundamental goal of dynamic pricing.
• We consider the problem of learning personalized pricing policies by analyzing historical observational data to map customer features to optimal price points.

Practical impliation

• Explicit dynamic pricing faces public backlash.
• Still, dynamic pricing is widely used in practice by firm in implicit form
• A good example ad-supported subscription services such as Netflix and YouTube. Although these platforms typically offer only two plan types (free and paid), free users may receive a personalized level of advertising exposure, which can vary continuously. LLM tokens can be another example.
• Consumers seem to be accepting this type of implicit dynamic pricing.
• Therefore learning a good dynamic pricing policy from customer features is still an important problem.

Dangers of Policy Learning from Historical Data

Market Shift on customer profiles

• A pricing algorithm naturally achieves higher precision for customer profiles that are abundant in the historical dataset, as it has more observations to accurately estimate their conditional outcomes.

• However, when the market environment changes, such as when a firm enters a new geographic region or experiences demographic shifts, the feature distribution of the target market diverges from the historical data. A customer profile that was common in the historical data may become rare in the target market.

• Consequently, a policy trained on the historical data will not perform as effectively in the new market.

Under-exploration on price

• Even if the market does not shift, since we are not actively collecting data but relying on historical observational data, there is always risk of underexploration.
• To identify a profit-maximizing price for each customer ($\mathbf{x}$), the historical data must contain enough variability in price ($p$) to allow the algorithm to compare counterfactual outcomes.
• In practice, however, explicit pricing experimentation is prohibitively expensive and risky, and transactions are based on reasonable prices that followed some resonable policy, resulting in historical datasets with narrow price exploration.

Remedy: More Data, with Privacy

• One way to handle the two problems above is to collect more data: more data with wider price exploration and diverse customer profiles.
• Since we are not actively collecting data but relying on historical observational data, one way to collect more data is collaborations with other firms, sharing each other’s data.
• Data sharing raises privacy concerns. In adtech industry, firms created a secure environment called data clean room, where firms can share their privacy-protected data and run the analysis.
• Privacy protection is basically adding noise to the data or the analysis result. The amount of noise and the level of privacy is formalized by the notion called differential privacy.
• However, adding noise will hurt the accuracy of the data. This is the trade-off of privacy protection.

Goal of this paper: quantifying the trade-offs

• So, privacy protection can allevaiate market shift and under-exploration. But its noise injection hurts the accuracy of the data.
• So, we want to quantify the effect of these three factors to the performance of the pricing policy learning.
• To this end we first need to formalize and quantify the three factors

Market Shift: transfer exponent

• Let $P$ and $Q$ denote the distributions of customer features in the historical and target markets, respectively.
• We quantify this market shift using the transfer exponent, a standard concept in the transfer learning literature.
• Intuitively, for any customer segment of size $h \in (0,1)$, the transfer exponent $\alpha$ bounds the ratio between its proportion in historical and target markets by $h^\alpha$.
• A smaller $\alpha$ indicates stronger structural alignment between the markets.
• The bound naturally relaxes as segments become increasingly niche ($h \to 0$).

Price exploration: exploretion coefficient

• To quantify the quality of the logged data, we introduce a parameter $\zeta \in [0, 1]$ that captures the propensity of the past pricing algorithm to select near-optimal prices.
• Intuitively, this ensures that $P(T \approx t^\dagger \mid \mathbf{X}^P \approx \mathbf{x}) \ge \zeta$ for every context $\mathbf{x}$, where $t^\dagger$ denotes an optimal price (allowing for the possibility of multiple optima).

Privacy Protection: noise variance

• Local differntial privacy formalizes the notion of datapoint level privacy. Most of the time, local differential privacy is satisfied by adding iid noise to each datapoint.
• Intuitively, the privacy level corresponds to the inverse of the noise variance added to each datapoint.

Analysis without privacy

Unavoidable limit of any policy

We first demonstrate that for any policy $\pi$, its optimality gap relative to the optimal policy $\pi^\ast$—measured by expected profit in the target market—is lower bounded as:

\begin{equation}\label{regret_lower_bound} V({\pi^\ast}) - V(\pi) \gtrsim (\zeta n)^{-\frac{\beta }{2\beta+\alpha+d+1}}, \end{equation}

where $\beta$ represents the smoothness of the conditional outcome.

Reaching the limit by a policy learning algorithm

• We demonstrate that a policy learning algorithm based on the principle of pessimism achieves this theoretical lower bound.

• The algorithm utilizes a local average estimator to predict the conditional expected outcome.

• This estimator allows us to compute rigorous confidence intervals for the outcome associated with any specific customer profile and price.

• Rather than optimizing the point estimate of the outcome, the algorithm selects the price that maximizes the lower confidence bound (LCB).

• Through this pessimistic optimization, uncertainty stemming from lack of datapoints with respect to customer profile and price is naturally incorporated into the learning process.

• The policy learned via this algorithm achieves a regret bound that matches the lower bound up to a constant factor, proving that the lower bound is tight and rigorously quantifies the fundamental limits of the problem.

Analysis under privacy protection

• Working on this. The core is finding the confidence interval under noise injection. The pessimism is applied twice: once for the statistical uncertainty and once for the privacy-induced uncertainty.