# Constrained Triple-Head Price Optimization (C3PO) Network for Discrete-Choice Models

arXiv: https://arxiv.org/abs/2605.06941 · HTML: https://mlsanity.com/paper/2605.06941/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 45 references that are in the map.
- First posted on arXiv 7 May 2026.

## Builds on (most cited first, 14 of 14)

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- [TabPFN: A Transformer That Solves Small Tabular Classification Problems in a Second](https://mlsanity.com/paper/2207.01848/) (2022)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Fantastic Generalization Measures and Where to Find Them](https://mlsanity.com/paper/1912.02178/) (2019)
- [Transformers Can Do Bayesian Inference](https://mlsanity.com/paper/2112.10510/) (2021)
- [An Introduction to Bi-level Optimization: Foundations and Applications in Signal Processing and Machine Learning](https://mlsanity.com/paper/2308.00788/) (2023)
- [On the Power of Foundation Models](https://mlsanity.com/paper/2211.16327/) (2022)
- [TabPFGen -- Tabular Data Generation with TabPFN](https://mlsanity.com/paper/2406.05216/) (2024)
- [Representing Random Utility Choice Models with Neural Networks](https://mlsanity.com/paper/2207.12877/) (2022)
- [Avoiding spurious sharpness minimization broadens applicability of SAM](https://mlsanity.com/paper/2502.02407/) (2025)
- [Constrained Prescriptive Trees via Column Generation](https://mlsanity.com/paper/2207.10163/) (2022)
- [Predicting Drivers' Route Trajectories in Last-Mile Delivery Using A Pair-wise Attention-based Pointer Neural Network](https://mlsanity.com/paper/2301.03802/) (2023)

## Abstract (arXiv)

We propose a Constrained Triple-Head Price Optimization (C3PO) network for real-time bi-level decision making in discrete-choice environments. In these settings, a seller selects product or service offerings to achieve business objectives, while heterogeneous users make personalized acceptance or rejection decisions based on their preferences, ultimately determining seller outcomes. Existing approaches that fine-tune pre-trained tabular foundation models on synthetic data are limited by their imitation-learning paradigm and inability to enforce common pricing rules, including cross-product and global constraints. C3PO addresses these limitations through a novel transformer-based architecture that jointly integrates price imitation learning, multi-task revenue prediction, and in-context elasticity learning to generate robust pricing recommendations across domains without retraining. A lightweight differentiable constraint layer enables generated prices to satisfy business constraints while optimizing the revenue objective. We demonstrate strong out-of-distribution performance on simulated, synthetic, and real-world datasets. Across simulated benchmarks where the calibrated MNL model is misspecified relative to the ground-truth choice model, C3PO improves expected revenue by 1.4 to 7.9 percentage points over a calibrated MNL baseline. The constraint layer reduces average absolute constraint violations by more than 50%. We further deploy the model in real-world applications, including healthcare tender pricing and airline ancillary pricing, where C3PO improves directional pricing effectiveness, measured by price-increase and price-decrease recall, by 18--34 percentage points depending on the baselines considered, including calibrated MNL, classical optimization, and off-the-shelf tabular foundation-models.