Resource-Constrained Adaptive Inference for Sequential Pricing

Short observational pricing panels often contain many data points but very few actual price changes. This paper shows that this sparsity creates a hidden source of error that standard statistical methods miss. When estimating price effects, most of the uncertainty does not come from sample size within a panel, but from the specific history of price movements observed. Standard confidence intervals fail because they only measure variation within the panel, ignoring this broader design-level error. Using simulations, we find that this cross-design variation accounts for most of the estimation error, causing standard methods to significantly understate uncertainty. First, we show that cross-design error decreases predictably as the total volume of price variation increases. Second, adding more data from regions that share the same price trends does not fix the issue; true precision improves only when combining data across units with independent price trajectories. Third, applying a simple variance-component adjustment across independently priced units restores accurate statistical coverage. We confirm these findings in real-world store scanner data, showing that products and pricing zones behave as if they have far fewer independent price movements than their raw counts suggest. Ultimately, reliable inference in passive pricing data requires genuine, independent variation, which can be achieved through controlled regional price testing.

We introduce a causal aware foundation-model framework for real time optimal decision making in discrete choice environments. We propose a constrained triple-head price optimization (C3PO) network to solve a bilevel decision problem in which a service provider selects an optimal assortment while heterogeneous users make personalized acceptance or rejection choices optimizing their own personalized preferences. C3PO integrates imitation learning of prices, multi-task learning of revenue responses, and in context learning of price elasticity to generate pricing recommendations while adhering to business constraints. During inference, frontier model prompting retrieves an enhanced elasticity prior for new products from behavioral economics literature, improving pricing effectiveness. We demonstrate strong in context learning performance using simulated, synthetic, and real-world datasets. C3PO is trained on simulated data generated from multiple classical discrete choice models in economics. The model is trained on data comprising simulated customer segments and counterfactual action and outcome pairs and evaluated on randomly generated choice environments with no access to the underlying preference structure. The trained model consistently improves the pricing KPIs, with gains increasing as customer price sensitivity increases. We also deploy the tuned foundation model for optimal pricing in real-world applications such as healthcare, tender pricing, airline ancillary pricing, and other domains, achieving substantial gains across multiple products, markets, and divisions.

This article considers an online version of conformal inference, called adaptive conformal inference [ACI] and introduced by Gibbs and Candès (2021): prediction sets are issued sequentially, after observing features and before the outcomes are revealed. These sets are evaluated both in terms of validity (the fraction of rounds where the outcome was lying in the prediction set) and efficiency (the average lengths of the prediction sets). The two criteria point to different directions (validity favors larger sets). We also target a wide range of scenarios, with exchangeable data and arbitrary data (lack of any stochastic guarantees) as two extremes. A series of existing strategies for ACI typically guarantee that empirical coverage converges to the desired level for arbitrary sequences, but they generally lack simultaneous efficiency guarantees. To provide a unified study, we first formulate ACI as a repeated two-player game with finite action sets and vector-valued payoffs encoding validity and efficiency. Building on this reformulation, we introduce a strategy based on Blackwell approachability and on its opportunistic extension by Bernstein et al. (2014) that ensures validity while adapting the efficiency of the prediction intervals to the underlying degree of stochasticity of the opponent player. The resulting guarantee is "best of many worlds": it recovers the relevant efficiency guarantees in exchangeable and adversarial settings, and provides guarantees in intermediate settings that arise in typical applications such as the forecasting of time series.