Discrete Choice Models

Item demand forecasting is an integral component of store assortment optimization. Existing literature focuses on learning a suitable customer choice model and using this model to determine the value of an objective function (i.e. expected demand) with respect to an assortment proposal. However, for large item universe with many categories, this approach can prove inefficient, needing a separate demand forecast for every possible item assortment. An alternate approach exists whereby we combine the efficiency of forecasting item demand independently, while at the same time applying adjustments to the independent forecasts that account for the relations between item demand and the availability of other similar items on the shelf. Central to this approach is the estimation of Demand Transfer (DT) coefficients. These DT coefficients represent the percent of a particular target item's (item that the customer walked in the store to buy) demand that is redirected to each other item in the universe should the target item be removed from the shelf. We introduce an approach that allows us to compute these DT coefficients on large item universes (assortments having 1 million+ items). Experiments on data as well as historical transaction data for multiple locations within categories demonstrate that when certain reasonable assumptions about substitution behavior are satisfied, our procedure is able to accurately estimate underlying DT coefficients and lead to improvements in demand forecasting.

Discrete choice modeling is a common tool used for preference elicitation during policy-making, but this is typically done through parametric models. Machine learning can push the boundaries of discrete choice modeling for policy-based preference elicitation by adopting a data-driven approach or learning individual preferences. However, there is limited knowledge of how well machine learning methods can estimate individual discrete choice rules under individual heterogeneity, especially in the context of challenges often experienced during preference elicitation. This study evaluates four machine learning models (multinomial logistic regression, generalized additive model, twinned neural network, and Gaussian process) with respect to their capacity to learn and predict five choice rules that are important in the behavioral and social sciences (linear strong utility, monotonic strong utility, ideal point, lexicographic semiorder, and multiattribute linear ballistic accumulator). Monte Carlo experiments were performed to assess model performance when increasing a) the number of attributes in the choice alternatives, b) the number of training choice sets, and c) the choice rule's determinism. The simulation results demonstrated that semi-parametric and non-parametric models generally outperform parametric models across all choice rules and experimental contexts. Model performance also generally improves by 6% to 96% and 0% to 55%, respectively, with an increase in training choice sets and choice rule determinism. A case study using real energy policy preference data was also conducted, where TNN performed best with a BIC of 13.351. This work demonstrated the viability and limitations of semi-parametric and non-parametric models in the context of policy-centric discrete choice modeling and showed how the choice task context should drive model selection.

The softmax policy $π(a \mid s) \propto \exp(βQ(s,a))$ is the default model of stochastic choice in reinforcement learning (RL). Various justifications based on robustness, exploration, and optimization have been offered in the RL literature, but none uniquely derives the softmax form from first principles. This leaves a basic tension unresolved: the entropy bonus in the soft Bellman equation violates the Independence axiom that underwrites the Markov decision process (MDP) reward structure. We dissolve this tension by distinguishing two kinds of randomness: chance and choice. By restricting von Neumann-Morgenstern (VNM) Independence to environmental lotteries over base prospects, we show that imposing independence of irrelevant alternatives (IIA) and monotonicity on the policy and value functions at choice nodes uniquely determines the Boltzmann policy, the entropy-regularized representation, and the soft Bellman equation. The choice between the soft and hard Bellman equations thus reduces to a design decision: whether the agent values its own ability to choose. We develop RL-specific consequences, including return monotonicity and convergence under generalized discounting, and synthesize the independent lines from economics and information theory that arrive at the same structure, offering a normative assessment of when IIA is appropriate for agent design.

Tabular foundation models (TFMs) generate predictions on structured data via in-context learning, without task-specific estimation. We ask whether TFMs can be effectively applied to discrete choice, a central demand estimation framework in marketing and operations, and find that directly applying TFMs yields limited performance. The gap is structural: TFMs assume row-independent observations, whereas discrete choice is inherently set-valued and subject to persistent consumer preference heterogeneity. We propose a reformulation that encodes both choice-set dependence and individual heterogeneity within a row-based learning framework. Evaluated on a yogurt scanner panel, individual-level heterogeneity encoding is the dominant driver of predictive accuracy. The best reformulation outperforms hierarchical Bayesian estimation by 8% in holdout log-likelihood and 3.6% in hit rate, running 16 times faster, a practical advantage for large-scale demand estimation. The advantage is largest in the medium-data regime (10--40 purchase occasions per consumer), where parametric Bayesian shrinkage most distorts estimates for atypical consumers. Fine-tuning on population choice data provides additional gains for consumers with shallow purchase histories, where in-context learning has limited individual-specific signal to condition on. These results establish a principled approach for applying foundation models to consumer choice problems more broadly.

We propose a framework for the Markov chain (MC) choice model with panel data, including parameter estimation, personalized choice prediction, and personalized assortment optimization. In contrast to the traditional setting, which assumes that each transaction is independently drawn from a random utility model, our framework accounts for dependencies among transactions for the same customer in historical data, captured by partial-ordering preference information. To the best of our knowledge, our framework initiates the study of choice modeling with panel data under MC. As our primary result, we propose novel expectation-maximization (EM) algorithms for MC parameter estimation by incorporating partial-ordering-based customer preference information. On synthetic datasets and the sushi dataset, our EM algorithms outperform the traditional EM algorithm of Simsek and Topaloglu (Operations Research, 66, 2018) and multinomial-logit-based partial-order benchmarks adapted from Jagabathula and Vulcano (Management Science, 64, 2018). As our secondary contribution, we present hardness and computational results for conditional choice prediction and assortment optimization problems. These results complement our estimation framework and clarify the computational landscape of conditional choice and assortment optimization, which may be of independent interest.

Tabular foundation models achieve strong accuracy on choice prediction tasks, but their predictions often violate the economic logic those tasks require: raising a price can increase predicted demand, implied willingness-to-pay estimates are frequently negative or implausible, and unavailable alternatives receive nonzero probability. We propose a two-stage adapter that takes a foundation model's predicted choice probabilities as a precomputed feature and embeds them inside a multinomial logit's utility. In Stage 1, we fit the multinomial logit's structural coefficients by maximum likelihood with sign constraints; in Stage 2, we freeze those coefficients and fit a small neural correction operating on the foundation model's predictions. We prove that this composition exactly preserves the multinomial logit's marginal rate of substitution, so analytically computable value-of-time becomes a mathematical guarantee rather than an empirical accident. Across three datasets and two foundation models, the adapter gains 6.4 percentage points (pp) of test accuracy on average over the multinomial logit and up to 12.8 pp, maintains 100% cost monotonicity, and produces values of time within the published transportation-economics range on the transportation datasets. Performance degrades gracefully under foundation-model context restriction, retaining at least 6 pp of accuracy gain even at 10% of the original foundation-model context.

Travelers increasingly ask large language model (LLM) assistants which hotel to book, making these systems gatekeepers of property visibility -- yet what moves their recommendations is undocumented. We conduct a pre-specified algorithm audit using a randomized choice-based conjoint: across personas, prompt templates, and twelve open-weight and proprietary models, assistants choose among five hotels whose guest rating, review volume and recency, management response, chain affiliation, price, eco-certification, and list position are independently randomized. We estimate the average marginal component effect of each signal on the probability of recommendation. Guest rating and price dominate (a top rating raises selection by 31.6 percentage points; a high price lowers it by 30.0), reproducing human valence-and-price primacy but over-weighting eco-certification and ignoring management response. List position -- a content-free artifact -- shifts recommendations causally, worth about $12 per night. Stated reasons track revealed weights imperfectly. The findings ground generative engine optimization and the accountability of AI infomediaries in causal evidence.

Cities deliver basic services through mixed public-private facility networks, including schools, clinics, transit providers, and subsidized service points. In these systems, planners often observe where households go, but not the latent cost function through which they trade off factors such as distance, price, and institutional access. We study this urban problem through school choice in the Philippines, where the country's largest national education subsidy is intended to redirect learners from congested public schools to participating private schools. Treating school-to-school enrollment flows as an entropic optimal transport plan, we recover latent choice costs using two complementary inverse optimal transport models: an interpretable distance-banded model with a subsidy term, and a neural cost model trained through a differentiable Sinkhorn forward pass. Applied to 283{,}016 learner trips across 23{,}820 observed flows in the most populated region, the framework estimates a subsidy-equivalent distance, $λ^{(k)}$, interpreted as the kilometers of perceived travel cost offset by the subsidy. The case demonstrates how administrative origin-destination data can be transformed into interpretable planning metrics for accessibility-aware subsidy design, facility siting, and urban service allocation.

In the forward reinforcement-learning problem, the reward is fixed and known; the learner is asked to find a good policy or value function. Here we turn the question around. Given offline data generated by an expert, can we recover the reward the expert was optimizing? This is the inverse reinforcement learning problem, and remarkably, two communities, structural econometricians studying dynamic discrete choice (DDC) and machine learners studying entropy-regularized IRL, have been working on exactly the same probabilistic model under different names. We begin by proving their equivalence. We then develop the classical identification result of Magnac and Thesmar and the classical computational paradigms that grew out of it: Rust's nested fixed-point algorithm, the conditional-choice-probability approach of Hotz and Miller, and the two temporal-difference approaches of Adusumilli and Eckardt: linear semi-gradient TD and approximate value iteration. Each route has its limits: dimensionality, transition-kernel estimation, the deadly triad, or projected fixed-point bias. We then walk through the modern ML/IRL strand: adversarial IRL, occupancy matching, IQ-Learn, and offline ML-IRL, deriving each method's actual objective and stating precisely what it does and does not identify. We close with the empirical-risk-minimization framework of Kang et al., which yields a gradient-based estimator for offline IRL/DDC.

Tabular foundation models achieve strong accuracy on choice prediction tasks, but their predictions often violate the economic logic those tasks require: raising a price sometimes increases predicted demand, and implied willingness-to-pay estimates are frequently negative or implausible. We propose a two-stage adapter that embeds foundation model predictions within a utility-maximization framework. In the first stage, we estimate a standard choice model whose parameters are constrained to obey economic theory. In the second stage, we freeze those parameters and train a correction term that incorporates the foundation model's predictions as additional information. The result is a model that inherits the foundation model's accuracy gains while guaranteeing monotonic price-demand relationships under policy perturbation and producing analytically computable trade-off measures. On two transportation datasets, the adapter recovers up to 13 percentage points of accuracy over a standard logit model while maintaining perfect economic consistency, something neither the raw foundation models nor conventional distillation achieve.

LLMs are increasingly used to make or support high-stakes decisions under uncertainty, where alignment depends not only on factual accuracy but on how models weigh tradeoffs between different outcomes. We present an empirical pipeline for estimating the implied preferences that an LLM's observed choices optimize: we elicit the model's probability distribution over unknowns along with the choice it would make for the decision task and then fit a discrete choice model to recover the cost function that best rationalizes the model's decisions. We show how this revealed-preference description allows rigorous evaluation of whether models behave in a consistently goal-directed way, whether they can verbalize a description of their objectives which matches their revealed decision policy, and whether prompting can reliably steer those policies to implement a user-specified cost function. We apply this evaluation across four medical diagnosis domains and multiple frontier and open-source models. We find that while many models have a nontrivial degree of internal coherence, they also have significant weaknesses in faithfully reporting or adopting preferences in response to user direction.

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.

Misaligned artificial agents might resist shutdown. One proposed solution is to train agents to lack preferences between different-length trajectories. The Discounted Reward for Same-Length Trajectories (DReST) reward function does this by penalizing agents for repeatedly choosing same-length trajectories, and thus incentivizes agents to (1) choose stochastically between different trajectory-lengths (be NEUTRAL about trajectory-lengths), and (2) pursue goals effectively conditional on each trajectory-length (be USEFUL). In this paper, we use DReST to train deep RL agents and fine-tune Qwen3-8B and Llama-3.1-8B-Instruct to be NEUTRAL and USEFUL. We find that these DReST models generalize to being NEUTRAL and USEFUL in unseen contexts at test time. Indeed, DReST RL agents achieve 11% (PPO) and 18% (A2C) higher USEFULNESS on our test set than default agents, and DReST LLMs achieve near-maximum USEFULNESS and NEUTRALITY. We also test our LLMs in an out-of-distribution setting where they can pay costs to influence when shutdown occurs. We find that DReST training roughly halves the mean probability of influencing shutdown (from 0.62 to 0.30 for Qwen and from 0.42 to 0.23 for Llama). DReST training also almost entirely eliminates the share of prompts on which influencing shutdown is the most likely option (from 0.59 to 0.01 for Qwen and from 0.53 to 0.00 for Llama). Our results thus provide some early evidence that DReST could be used to train more advanced agents to be useful and shutdownable.

Discrete choice models are fundamental tools in management science, economics, and marketing for understanding and predicting decision-making. Logit-based models are dominant in applied work, largely due to their convenient closed-form expressions for choice probabilities. However, they impose restrictive assumptions on the stochastic utility component, constraining our ability to capture realistic substitution patterns. We propose an amortized inference approach that relies on a neural network emulator to approximate choice probabilities for general error distributions, including those with correlated errors. We develop a specialized neural network architecture designed to respect the invariance properties of discrete choice models. We provide group-theoretic foundations for the architecture, including a proof of universal approximation given a minimal set of invariant features. Once trained, the emulator enables rapid likelihood evaluation and gradient computation. We use Sobolev training, augmenting the likelihood loss with a gradient-matching penalty, so that the emulator learns both choice probabilities and their derivatives. We show that emulator-based maximum likelihood estimators are consistent and asymptotically normal under mild approximation conditions, and we provide sandwich standard errors that remain valid for a psuedo-true parameter even with imperfect likelihood approximation. Simulations show significant gains over the GHK simulator in accuracy and speed.

Generalized extreme value models capture dependence among choice alternatives in discrete choice modeling, but require this dependence to be predefined, symmetric, and shared uniformly across individuals. Recent efforts to synergize discrete choice models with deep neural networks have improved predictive performance but still cannot explicitly represent alternative dependence within neural architectures. To address these gaps, we introduce the alternative graph -- a graph in which nodes represent choice alternatives and edges encode their dependence -- and propose Alternative Graph Neural Networks (Alt-GNNs), a family of GNN-based discrete choice models that embed alternative dependence within a unified framework. Theoretically, Alt-GNNs incorporate multinomial logit, nested logit, and ASU-DNN as special cases and enable innovative model designs, including Nested Alt-GNN, Complete Alt-GNN, and Attention Alt-GNN. Alt-GNNs are consistent with random utility maximization theory, enforce behavioral constraints through alternative graphs, and offer a novel graph-based interpretation of utility functions. Empirically, on two travel mode choice datasets from London and Chicago, Alt-GNNs significantly improve predictive performance over all benchmark models in mode choice modeling because of their flexible alternative graph design and vast hyperparameter space. Even the simplest Alt-GNN variant -- Nested Alt-GNN -- generalizes the nested logit model while preserving its unique two-layer substitution properties, enabling graph-based behavioral constraints over otherwise unconstrained behavioral patterns from deep neural networks.

Discrete choice models (DCMs) are used to analyze individual decision-making in contexts such as transportation choices, political elections, and consumer preferences. DCMs play a central role in applied econometrics by enabling inference on key economic variables, such as marginal rates of substitution, rather than focusing solely on predicting choices on new unlabeled data. However, while traditional DCMs offer high interpretability and support for point and interval estimation of economic quantities, these models often underperform in predictive tasks compared to deep learning (DL) models. Despite their predictive advantages, DL models remain largely underutilized in discrete choice due to concerns about their lack of interpretability, unstable parameter estimates, and the absence of established methods for uncertainty quantification. Here, we introduce a deep learning model architecture specifically designed to integrate with approximate Bayesian inference methods, such as Stochastic Gradient Langevin Dynamics (SGLD). Our proposed model collapses to behaviorally informed hypotheses when data is limited, mitigating overfitting and instability in underspecified settings while retaining the flexibility to capture complex nonlinear relationships when sufficient data is available. We demonstrate our approach using SGLD through a Monte Carlo simulation study, evaluating both predictive metrics--such as out-of-sample balanced accuracy--and inferential metrics--such as empirical coverage for marginal rates of substitution interval estimates. Additionally, we present results from two empirical case studies: one using revealed mode choice data in NYC, and the other based on the widely used Swiss train choice stated preference data.

Problem definition: This paper studies the problem of estimating consumer preferences from bundle sales data. Product bundling is a widely used pricing strategy in retail markets. To set profitable bundle selection and prices, the seller needs to learn the distribution of consumers' valuations for individual products from the transaction data. When customers purchase bundles or multiple products, classical methods such as discrete choice models cannot be used to estimate consumers' valuations. In this paper, we propose an approach to learn the distribution of consumers' valuations toward the products using bundle sales data. Methodology/results: Our approach is to define a utility model for customer choices and estimate the parameters of a valuation distribution that maximizes the likelihood of observing the transaction data. Our approach reduces this problem to an estimation problem where the samples are censored by polyhedral regions on the valuation space of customers. Using the EM algorithm and Monte Carlo simulation, our approach can recover the distribution of consumers' valuations. We extend the framework to allow for unobserved no-purchases, clustered market segments and to incorporate non-additive bundle utilities with synergy effects. In addition, we provide theoretical results on the identifiability of the probability model and sufficient conditions for local convergence of the EM algorithm. Moreover, the performance of the approach is also demonstrated numerically with synthetic and real datasets. Managerial implications: This study demonstrates the challenge to leverage the transaction data of bundle sales to learn customers' preferences. The proposed algorithm provides a practical guidance for retailers.