Embedding Foundation Model Predictions in Discrete-Choice Models with Structural Guarantees

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abstractTabular 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.

Introduction

Discrete-choice models guide policy decisions with significant economic stakes: the per-minute value of time from commuter mode-choice forecasts ben_akiva_lerman_1985, train_2009 anchors cost-benefit appraisal of multi-billion-dollar rail and road investments, and willingness-to-pay estimates from consumer discrete-choice experiments set prices and predict how labeling regimes shift behavior. A choice model must do two things: forecast which alternative is chosen, and forecast how the choice responds to intervention.

Multinomial-logit utility models ben_akiva_lerman_1985, train_2009 satisfy the second requirement by construction: a sign-constrained cost coefficient gives monotone demand, the $\beta_{\text{time}}/\beta_{\text{cost}}$ ratio gives interpretable willingness-to-pay, and unavailable alternatives receive zero probability. They satisfy the first only modestly. Modern machine-learning models invert the trade-off, raising accuracy at the cost of breaking each structural property in turn hillel_2021, vancranenburgh_2022, zhao_yan_2020.

Tabular foundation models such as TabPFN hollmann_2023, purucker_2025 and Mitra zhang_2024 intensify the tension by raising the accuracy ceiling while inheriting the same structural failures. Existing remedies each carry costs: architecturally constrained monotonic neural networks sill_1997, wehenkel_2019, constrained_monotonic_icml2025 eliminate monotonicity violations but lose the closed-form trade-off ratio; knowledge distillation hinton_2015 into a multinomial-logit student preserves the structural guarantees but cannot match the teacher's accuracy; penalized fine-tuning needs gradient access through the foundation model's weights, incompatible with the in-context learning that TabPFN and Mitra use at inference.

We propose a two-stage adapter that embeds the foundation model's predicted probability vector inside a constrained multinomial logit as a precomputed feature. In Stage 1, we fit the structural coefficients by maximum likelihood under the usual MNL constraints with the correction held at zero; in Stage 2, we freeze those coefficients and fit a small neural correction. We prove that this two-stage procedure preserves the structural marginal rate of substitution exactly, and that joint training of the structural coefficients and the correction does not: correction expressivity creates a one-parameter family of likelihood-equivalent solutions, breaking identifiability.

Our contributions are:

description[leftmargin=1.2em,labelindent=0pt,itemsep=2pt,topsep=2pt,parsep=0pt] • A two-stage behavioral adapter that preserves the structural multinomial logit's economic guarantees by construction while recovering most of a foundation model's accuracy advantage. • Propositions (ref) and (ref) identify two-stage training as the design choice preserving structural identifiability and characterize the failure mode of joint training. • Three discrete-choice datasets, two foundation models, six ablations, a feature-augmented multinomial-logit baseline, calibration analysis with three post-hoc methods, and a counterfactual aggregate-share evaluation. The accuracy gain is positive in $10$ of $10$ bootstrap replicates on every (dataset, foundation-model) cell, with $p \approx 0.002$ under the exact binomial sign test and McNemar paired-observation $p$-values below $10^{-20}$ on the larger datasets. • A behavioral-audit recipe applicable to any discrete-choice dataset and any predict-function, generalizing value-of-time auditing to willingness-to-pay for arbitrary non-cost attributes and to cluster-aware aggregation for panel data.

Related work

\paragraph{Machine learning and economic consistency in choice modeling.} Discrete-choice modeling has been the standard tool in transportation economics and consumer behavior since ben_akiva_lerman_1985. Recent work pushes machine learning into the field while exposing a tension with economic structure: hillel_2021 document consistent accuracy gains across neural and ensemble methods over the multinomial logit but few behavioral diagnostics; zhao_yan_2020 report frequent monotonicity violations and implausible willingness-to-pay estimates from machine-learning choice models; and vancranenburgh_2022 frame economic consistency as an open challenge distinct from prediction accuracy. han_2022 (TasteNet) propose a neural-embedded discrete-choice model that learns taste parameters as neural-network functions of individual characteristics, targeting taste-parameter interpretability through learned heterogeneity rather than the structural-plus-correction decomposition we adopt here. \ifdefined This work builds on our workshop paper wang_2026_workshop; the present paper extends it with an additional dataset, formal preservation results, and counterfactual aggregate-share evaluation. \fi Our work additionally derives a formal preservation result, evaluates across foundation models, and characterizes joint training's failure mode.

\paragraph{Architecturally constrained monotonic neural networks.} Architectural approaches enforce monotonicity through sign-constrained weights and monotone activations sill_1997, wehenkel_2019, constrained_monotonic_icml2025, eliminating cost-monotonicity violations but losing the closed-form trade-off ratio: with no global coefficients, the value-of-time analogue must be computed from per-observation gradients rather than read off as $\beta_{\text{time}} / \beta_{\text{cost}}$. Our adapter takes a different point: monotonicity and trade-off ratios come from the structural component, whose constrained parameterization is preserved throughout training, while the foundation model contributes a non-differentiable side channel.

\paragraph{Tabular foundation models and knowledge distillation.} Tabular foundation models such as TabPFN hollmann_2023, purucker_2025 and Mitra zhang_2024 achieve strong classification accuracy via in-context learning. Their published benchmarks emphasize predictive accuracy; structural-validity diagnostics specific to choice modeling (cost-monotone responses, finite positive trade-off ratios, zero probability on unavailable alternatives) at test-row coverage are not part of these benchmarks. Our audit fills this gap. Knowledge distillation hinton_2015 into a multinomial-logit student preserves structural guarantees but is bounded by the student's expressive capacity: a plain MNL student parameterizes a strictly linear-in-features utility, and the residual variance the foundation model captures lives above that ceiling. We position the foundation model differently: its predictions become an explanatory feature embedded inside a structurally constrained utility, with the foundation model's parameters never modified.

Method

Setup and notation

We consider a discrete-choice setting with $N$ observations indexed by $i \in \{1, \dots, N\}$. Each observation has a feature vector $\mathbf{x}_i \in \mathcal{X}$, a set of available alternatives $\mathcal{K}_i \subseteq \{1, \dots, K\}$, and an observed choice $y_i \in \mathcal{K}_i$. For panel data we additionally observe a subject identifier $s_i$, since multiple observations from the same subject are not exchangeable; we cluster on $s_i$ throughout.

A choice model produces probabilities $P_k(\mathbf{x}_i) \in [0, 1]$ summing to one over $\mathcal{K}_i$, i.e., a vector on the $(K{-}1)$-simplex $\Delta^{K-1}$. The standard multinomial logit (MNL) parameterizes these through a linear utility $V_k(\mathbf{x}_i) = \boldsymbol{\beta}^\top \boldsymbol{\phi}_k(\mathbf{x}_i)$ and the softmax $P_k = \exp V_k / \sum_{j \in \mathcal{K}_i} \exp V_j$ ben_akiva_lerman_1985. The alternative-specific feature transform $\boldsymbol{\phi}_k$ selects from $\mathbf{x}_i$ the columns that enter alternative $k$'s utility, typically including cost, time, alternative-specific constants, and sociodemographic interactions. We treat $\boldsymbol{\phi}_k$ as fixed by the dataset's specification and recover $\boldsymbol{\beta}$ by maximum likelihood.

Behavioral audit

Every model is evaluated through three model-agnostic functionals that take only a predict function $\mathbf{x} \mapsto P(\mathbf{x}) \in \Delta^{K-1}$ as input. We say model-agnostic in the sense that the audit reads model outputs only, requiring no gradient access or knowledge of internals.

Intervention protocol. For adapter and feature-augmented MNL we use the fixed-$\mathbf{q}$ protocol throughout: when a cost or attribute is perturbed at row $i$, the foundation-model probability vector $\mathbf{q}_i$ is held fixed at the value computed on the unperturbed $\mathbf{x}_i$, and the perturbation enters only $V^{\text{struct}}_k$. The alternative recomputed-$\mathbf{q}$ protocol re-runs the foundation model on perturbed inputs; we use it for the raw foundation-model counterfactual evaluation (Section (ref)) only, and forfeit the structural guarantees in that case.

The three functionals are: (1) Monotonicity. For each test row $i$ and alternative $k$, perturb $k$'s cost upward by $1\%$ of its observed range and check whether $P_k$ falls; we report the observation-level rate (cluster-aware on panel data). (2) Trade-off ratio. The marginal rate of substitution between non-cost attribute $a$ and cost $b$, reported in the standard transportation-economics sign convention so that value of time and willingness to pay for utility-improving attributes are positive: $\rho^{\mathrm{VOT}}_{a,b} = (\partial P_k/\partial a) / (\partial P_k/\partial b)$ when both partials are negative (e.g., time and cost both lower utility), and $\rho^{\mathrm{WTP}}_{a,b} = -(\partial P_k/\partial a) / (\partial P_k/\partial b)$ when $\partial P_k/\partial a$ is positive and $\partial P_k/\partial b$ is negative (a desirable non-cost attribute vs cost). Estimated by finite differences scaled to $1\%$ of each column's observed range. (3) Availability compliance. For datasets in which the available set $\mathcal{K}_i$ varies across observations (e.g., Swissmetro, where the proposed Swissmetro option is not offered to some respondents), $\mathrm{Leak}(M) = \mathbb{E}_i [\sum_{k \notin \mathcal{K}_i} P_k(\mathbf{x}_i)]$ measures the predicted probability assigned to formally unavailable alternatives. The multinomial logit applies softmax over $\mathcal{K}_i$ only, so its leak is mechanically zero; black-box predictors that ignore the availability mask can leak nonzero probability onto unavailable alternatives.

Two-stage behavioral adapter

description[leftmargin=1.2em,labelindent=0pt,itemsep=2pt,topsep=2pt,parsep=0pt] • For each observation $i$, alternative $k$, and a precomputed foundation-model probability vector $\mathbf{q}(\mathbf{x}_i) \in \Delta^{K-1}$ obtained by a single forward pass through the foundation model on the raw input, the adapter's utility is \begin{equation} V_k(\mathbf{x}_i) = \underbrace{\boldsymbol{\beta}^\top \boldsymbol{\phi}_k(\mathbf{x}_i)}_{V_k^{struct} (economic structure)} + \underbrace{g_k(\mathbf{q}(\mathbf{x}_i))}_{foundation-model correction}, \end{equation} where $g \colon \Delta^{K-1} \to \mathbb{R}^K$ is a small MLP (two hidden layers, width $32$). To enforce cost/time monotonicity throughout training we reparameterize each such coefficient as $\beta = -\exp(\theta)$. • We fit the model in two sequential stages. Stage 1: with the correction held at $g \equiv 0$, fit $\boldsymbol{\beta}$ by maximum likelihood under the sign constraints, recovering the standalone-MNL estimate $\boldsymbol{\beta}^\ast$. Stage 2: fix $\boldsymbol{\beta} = \boldsymbol{\beta}^\ast$ and fit only $g$ by maximum likelihood. The architecture realizes Stage 1 cleanly because $g$'s output layer is zero-initialized; hidden weights use He initialization he_2015_kaiming. Because $\mathbf{q}(\mathbf{x}_i)$ is precomputed, $\partial \mathbf{q}/\partial \mathbf{x}_i = 0$, so monotonicity and trade-off ratios reduce to functions of $\boldsymbol{\beta}^\ast$ alone. Propositions (ref)--(ref) make this precise and identify joint training as the failure mode.

Propositions

We use $\boldsymbol{\beta}^\ast$ for the Stage 1 maximum-likelihood estimate and reuse the notation of (ref).

proposition[Marginal-rate-of-substitution preservation under two-stage training, fixed-$\mathbf{q}$ protocol] Let $\boldsymbol{\beta}^\ast$ be the Stage 1 maximum-likelihood estimate, let $g$ be any Stage 2 parameters in (ref), and operate under the fixed-$\mathbf{q}$ protocol of Section (ref). For any two attributes $j, j'$ that enter the model only through $V^{\text{struct}}_k(\mathbf{x}_i) = \boldsymbol{\beta}^\top \boldsymbol{\phi}_k(\mathbf{x}_i)$, and for any $\mathbf{x}_i$ at which $\boldsymbol{\phi}_k$ is differentiable in $x_{ij}$ and $x_{ij'}$, \begin{equation*} \mathrm{MRS}_{j,j'}(\mathbf{x}_i) \;\equiv\; \frac{\partial V_k(\mathbf{x}_i) / \partial x_{ij}}{\partial V_k(\mathbf{x}_i) / \partial x_{ij'}} \;=\; \frac{\beta^\ast_j}{\beta^\ast_{j'}}. \end{equation*} In particular, the value-of-time analogue $\beta^\ast_{\text{time}} / \beta^\ast_{\text{cost}}$ is identical to the standalone multinomial logit's value-of-time and is invariant to the choice of foundation model and to the choice of $g$. Under the recomputed-$\mathbf{q}$ protocol, $g$ contributes a chain-rule term through $\mathbf{q}$ to each partial and the ratio no longer reduces to $\beta^\ast_j / \beta^\ast_{j'}$ in general.

itemize[leftmargin=2em,itemsep=2pt,topsep=2pt,parsep=0pt] • Corollary (probability-derivative MRS). Under the same fixed-$\mathbf{q}$ protocol, restrict attention to attributes $j, j'$ that enter only alternative $k$'s utility. Then the audit estimator equals the structural coefficient ratio under the audit's sign convention: $\rho^{\mathrm{VOT}}_{j,j'} (\mathbf{x}_i) = \beta^\ast_j / \beta^\ast_{j'}$ and $\rho^{\mathrm{WTP}}_{j,j'}(\mathbf{x}_i) = -\beta^\ast_j / \beta^\ast_{j'}$, since the softmax derivative's $P_k(1-P_k)$ factor cancels in the ratio (Appendix (ref)). • Intuition. Fixed $\mathbf{q}$ forces $\partial \mathbf{q}/\partial \mathbf{x} = 0$, so $g$ vanishes from $\partial V_k / \partial x_{ij}$ and the ratio collapses to $\beta^\ast_j / \beta^\ast_{j'}$. Full proof in Appendix (ref).

proposition[Joint training breaks structural identifiability] Assume there exists a continuous $\kappa_k: \Delta^{K-1} \to \mathbb{R}$ such that $\mathrm{cost}_k(\mathbf{x}) = \kappa_k(\mathbf{q}(\mathbf{x}))$ on the closure of the training support (cost-recoverability assumption); the correction class $\mathcal{G}$ is dense in $C^0(\Delta^{K-1}, \mathbb{R}^K)$ cybenko1989; and $L(\boldsymbol{\beta}, g)$ is minimized jointly without two-stage constraints. Then for any joint minimizer $(\boldsymbol{\beta}^{(0)}, g^{(0)})$ there exists a one-parameter family $\{(\boldsymbol{\beta}^{(c)}, g^{(c)}) : c \in \mathcal{C}\}$ of distinct configurations achieving identical loss, parametrized by $\beta^{(c)}_{\text{cost}} = \beta^{(0)}_{\text{cost}} + c$ for $c$ in an open interval $\mathcal{C}$ preserving sign constraints on $\beta_{\text{cost}}$. Gradient descent within $\mathcal{C}$ selects an initialization-dependent point rather than the MNL MLE $\boldsymbol{\beta}^\ast$.

itemize[leftmargin=2em,itemsep=2pt,topsep=2pt,parsep=0pt] • Note (cost-recoverability). The assumption is dataset-dependent; partial-recoverability cases in Appendix (ref). • Intuition. Subtracting $c \cdot \kappa_k(\mathbf{q})$ from $g_k$ shifts $\beta_{\text{cost}}$ by $+c$ without changing pointwise utility: the structural increase $c \cdot \mathrm{cost}_k$ in $V_k$ is cancelled by the correction-side decrease. A3 (\S(ref)) is the empirical illustration; full proof in Appendix (ref).

Experimental setup

\paragraph{Datasets.} We evaluate on three discrete-choice datasets. Swissmetro bierlaire_2001: $10{,}719$ stated-preference commuter choices among rail, the proposed Swissmetro, and car. LPMC hillel_2018: $81{,}086$ revealed-preference London trips among walk, cycle, public transport, and drive. IoT-Wearables iot_wearables_dce_2020: $6{,}362$ stated-preference choices among three Internet-of-Things wearable devices varying in price, functional features, and a security/privacy labeling scheme (panel data, $728$ subjects). All splits are $70/15/15$. IoT-Wearables uses subject-level splitting (each subject's rows go entirely to one split) to prevent within-subject leakage. Swissmetro and LPMC use the stratified row-level splits inherited from prior released parquets (bierlaire_2001 for Swissmetro and the LPMC public release); because Swissmetro is stated-preference with repeated choice tasks per respondent, row-level splitting may leak respondent-specific preferences across train/val/test, which we flag as a limitation.

\paragraph{Foundation-model inputs.} The foundation-model input columns per dataset:

itemize[leftmargin=2em,itemsep=2pt,topsep=2pt,parsep=0pt] • Swissmetro: per-alternative travel time, cost, headway, availability indicators; respondent age, income, season-ticket holding, luggage, trip purpose. • LPMC: per-alternative duration, transit and driving costs, trip distance; respondent age, sex, license, car ownership. • IoT-Wearables: per-alternative price, functional features, security label; respondent age, education, sex, security-behavior score, condition fixed effects.

Cost (or price), time (or duration), and availability indicators are therefore in the input set on every dataset; this is the basis for the foundation model's potential non-monotonic response to cost, and motivates the fixed-$\mathbf{q}$ intervention protocol of Section (ref) for the adapter.

\paragraph{Models.} Each cell evaluates five primary models, reported in Table (ref):

itemize[leftmargin=2em,itemsep=2pt,topsep=2pt,parsep=0pt] • Multinomial logit (Stage 1 of the adapter). • Raw foundation model: Mitra zhang_2024 or TabPFN hollmann_2023, purucker_2025. • Architecturally constrained monotonic neural network constrained_monotonic_icml2025. • Feature-augmented multinomial logit, with $\mathbf{q}$ appended to the structural feature set. • Simplified two-stage adapter $V_k = V^{\text{struct}}_k + g_k(\mathbf{q})$.

Three additional variants appear in prose only: a masked foundation model (Swissmetro, \S(ref)); a convex ensemble $\alpha P_{\text{MNL}} + (1{-}\alpha) P_{\text{FM}}$ (\S(ref)); and the two-term variant $V_k = V^{\text{struct}}_k + \alpha \log q_k + g_k(\mathbf{q})$ (ablation A2). The correction network $g$ is a two-hidden-layer MLP, width $32$, output-layer-zero initialization (\S(ref)).

\paragraph{Foundation-model context and cross-fitted training $\mathbf{q}_i$.} Stage 2's training-row $\mathbf{q}_i$ come from a $k=5$ stratified cross-fitted protocol, so no row's prediction was made by a model that saw its own label; test $\mathbf{q}_i$ are out-of-context by construction. The protocol applies to five of six (dataset, FM) cells; TabPFN-LPMC exceeds the CUDA attention-kernel ceiling at LPMC's per-fold context size and retains in-sample $\mathbf{q}_i$ (marked with $^{\dagger}$ in Table (ref)). Cross-fitting shifts adapter test accuracy by at most $-0.6$ pp (Appendix (ref)).

\paragraph{Metrics and significance.} We report:

itemize[leftmargin=2em,itemsep=2pt,topsep=2pt,parsep=0pt] • Test-set accuracy. • Per-row monotonicity rate (cluster-aware aggregation on IoT-Wearables). • Trade-off ratio: value-of-time on transportation datasets; willingness-to-pay for indicated non-cost attributes on IoT-Wearables. • Availability leak (Swissmetro only). • Expected calibration error (ECE), post-calibration with $K=15$ equal-weight bins.

For each (dataset, foundation-model) cell we draw $10$ bootstrap samples of the training set with replacement and refit Stage 1 and Stage 2 independently on each replicate; validation and test splits are held fixed. Significance tests on the per-seed accuracy gain (full adapter $-$ Stage 1) include an exact two-sided binomial sign test ($p \approx 0.002$ at $10/10$ positive seeds) and per-seed McNemar paired-observation tests on the held-out test set. Three post-hoc calibration methods (scalar temperature scaling guo_2017, vector temperature scaling, and isotonic regression) are fit on validation and evaluated on test.

Main results

table*[table* omitted — 3,454 chars of source]

Table (ref) reports per-cell test accuracy, monotonicity rate, trade-off ratio, availability leak, and post-calibration expected calibration error (ECE) for the five models of \S(ref). The results break into three threads, mirroring the subsection structure below:

itemize[leftmargin=2em,itemsep=2pt,topsep=2pt,parsep=0pt] • \S(ref): the raw foundation models fail behavioral validity on three diagnostics (monotonicity, trade-off ratios, availability). • \S(ref): the adapter recovers behavioral validity while keeping most of the accuracy gain. Calibration is competitive with one documented exception. • \S(ref): under counterfactual cost perturbations, the raw foundation models violate aggregate monotonicity in $6$ of $16$ scenarios while the adapter never does.

The foundation models fail behavioral validity

Mitra and TabPFN gain $+4$ to $+14$ pp of accuracy over the multinomial logit across our three datasets, but the gains come with three distinct failures of behavioral validity. Monotonicity rates collapse on the datasets with the most predictive headroom: TabPFN is monotone in cost on only $28.9\%$ of LPMC test rows and $40.8\%$ of IoT-Wearables; Mitra performs better on LPMC ($50.8\%$) but only marginally on IoT-Wearables ($45.6\%$). On Swissmetro both foundation models score $\geq 90\%$ but still below the multinomial logit's mathematical $100\%$. Trade-off ratios compound the issue: TabPFN's LPMC driving value-of-time is $-14.5$ GBP/hr (wrong sign relative to MNL's $+16.6$), and TabPFN's IoT-Wearables function WTP is also of the wrong sign relative to MNL. Mitra's willingness-to-pay on the IoT-Wearables binary indicators is locally inconclusive at the $1\%$ perturbation scale; we discuss the audit-methodology limitation in Section (ref). Availability compliance fails on Swissmetro: TabPFN assigns $\sim 5 \times 10^{-4}$ and Mitra $\sim 2 \times 10^{-3}$ of total probability to formally unavailable alternatives, against the multinomial logit's mechanically zero leak. Masking the foundation model (zeroing unavailable alternatives and renormalizing) removes the leak but leaves accuracy unchanged, so the leak is a structural error rather than an artifact of probability redistribution.

The adapter recovers behavioral validity while keeping most of the accuracy gain

The adapter inherits the multinomial logit's structural utility (with a sign-constrained cost coefficient) and availability-mask machinery, so it satisfies $100\%$ monotonicity, exact analytical trade-off ratios, and zero availability leak by construction on every cell. The empirical question is how much of the foundation-model accuracy gain the adapter retains.

The simplified adapter recovers most of it. Across all six (dataset, foundation-model) cells the adapter trails the raw foundation model by at most $2$ pp on accuracy (Table (ref)): the adapter pays up to $2$ pp for full preservation of the multinomial logit's economic guarantees. The gain over the structural multinomial logit alone is positive in $10$ of $10$ bootstrap replicates on every cell, giving an exact two-sided binomial sign-test $p$-value of ${\approx}\,0.002$ per cell, and per-seed McNemar paired-observation tests mcnemar_1947 (which compare per-example correctness between two classifiers on the same test set) yielding $p$-values below $10^{-20}$ on the larger datasets (LPMC, IoT-Wearables) and below $10^{-10}$ on the smaller datasets. The trade-off ratios match the structural multinomial logit by construction (Proposition (ref)) and are stable across seeds: $84.4 \pm 2.6$ CHF/hr on Swissmetro, $1.75 \pm 0.05$ and $16.64 \pm 0.33$ GBP/hr on LPMC public transport and driving, and $+26.10 \pm 0.91$ USD willingness-to-pay for IoT-Wearables functional features.

The feature-augmented multinomial logit reaches slightly higher accuracy than the adapter ($+0.5$ to $+2.6$ pp depending on the cell) but at the cost of degraded trade-off-ratio estimates that drift from the structural multinomial logit's. On Swissmetro the drift is egregious: feat-aug VOT is $1508 \pm 738$ CHF/hr for the Mitra-augmented variant and $678 \pm 319$ for the TabPFN-augmented variant, an order of magnitude beyond any plausible willingness-to-pay range. On LPMC and IoT-Wearables the drift is more modest but still nontrivial: LPMC feat-aug-Mitra reports $10.3$/$6.7$ GBP/hr against MNL's $1.8$/$16.6$ ($5{\times}$ drift on public transport, $0.4{\times}$ on drive); IoT-Wearables feat-aug reports $14$--$15$ / $5$ USD against MNL's $26.1$/$10.0$. The adapter, by contrast, inherits MNL's trade-off ratios exactly under the fixed-$\mathbf{q}$ protocol (Proposition (ref)), so this drift is bounded to zero by construction. The convex-ensemble baseline, $\alpha\, P_{\text{MNL}} + (1 - \alpha)\, P_{\text{FM}}$, fits an $\alpha$ near zero on every cell where the foundation model has higher validation accuracy, reducing to the raw foundation model up to numerical noise. Both baselines confirm that linear blending does not recover the adapter's combination of accuracy and interpretability.

\paragraph{Calibration.} Post-temperature-scaling ECE is competitive with the structural multinomial logit on LPMC and IoT-Wearables (Table (ref)); Swissmetro is the exception, where adapter ECE stays at $8.4\%$ (Mitra) and $17.0\%$ (TabPFN) against the multinomial logit's $4.0\%$ even after scalar / vector temperature scaling, isotonic regression, and bootstrap ensembling, a bias-driven rather than variance-driven limitation we discuss in Section (ref). Full NLL, Brier, and uncalibrated ECE in Appendix (ref) mirror this pattern.

Counterfactual aggregate-share evaluation

To test how each model would predict aggregate demand response to a small price increase, we evaluate every model on the held-out test set under a $+10\%$ cost perturbation applied separately to each alternative, then aggregate predicted probabilities into a market share for the perturbed alternative. Table (ref) reports the full result matrix.

table[table omitted — 1,389 chars of source]

Across $16$ (dataset, foundation-model, alternative) scenarios the structural multinomial logit and the adapter agree on the direction of the share change in every scenario: both predict a strictly negative change in aggregate share for the perturbed alternative. Adapter share-changes recover $40$--$97\%$ of the multinomial logit's magnitude depending on cell (Table (ref)); the gap reflects softmax saturation under the foundation-model correction, not a violation of Proposition (ref).

The raw foundation-model row uses the recomputed-$\mathbf{q}$ protocol (re-running the foundation model on perturbed inputs) and shows the per-observation monotonicity violations of Section (ref) propagating to policy-relevant aggregates: in $6$ of $16$ scenarios the raw foundation model predicts a positive aggregate-share change under a cost increase (bolded in Table (ref)). The largest violation (TabPFN-IoT-alt1) reverses sign relative to both MNL and the adapter, an $11$-pp disagreement on consumer response to a $10\%$ price increase. The structural multinomial logit and the adapter are exempt by construction.

Ablations

We run five ablations using the same multi-seed bootstrap protocol as Section (ref).

table[table omitted — 1,266 chars of source]

description[leftmargin=1.2em,labelindent=0pt,itemsep=2pt,topsep=2pt,parsep=0pt] • $V_k = V^{\text{struct}}_k + \alpha \log q_k$ replaces the neural correction with the scalar log-probability term alone. Competitive on the transportation datasets but trails by $1$--$3$ pp accuracy gain on the new datasets, where the foundation model's predictions encode patterns a single scalar projection cannot capture. • $V_k = V^{\text{struct}}_k + g_k(\mathbf{q})$ drops the $\alpha \log q_k$ term. Bit-close numbers vs the two-term variant across all six cells; the simplification comes at no empirical cost. • Training $\boldsymbol{\beta}$ and $g$ jointly under the same likelihood gives accuracy comparable to or slightly higher than the two-stage adapter, but the structural cost coefficient collapses on the transportation datasets by a factor of $3$--$17 \times$ as $g(\mathbf{q})$ absorbs the cost-induced variation. The collapse does not reproduce on IoT-Wearables, where the joint $\beta_{\text{cost}}$ is $\sim 25\%$ larger; we attribute this to whether cost is smoothly recoverable from $\mathbf{q}$ (Appendix (ref)). • We retrained Mitra and TabPFN on $50\%$, $25\%$, and $10\%$ of the Swissmetro train+val context, using the same $k=5$ cross-fitted training-$\mathbf{q}_i$ protocol as the headline cells. Accuracy gain decreases roughly linearly with context fraction, with $10/10$ bootstrap replicates positive at every level (Figure (ref)). The adapter falls back toward Stage 1 but does not collapse below it. • Varying the correction network's hidden width ($16$, $32$, $64$) and depth ($1$, $2$ layers) keeps accuracy within $1$ pp of A2's across all cells. The gain comes from the composition of structural utility plus foundation-model correction, not from correction-network capacity.

Table (ref) summarizes test accuracy across all five variants. Four of the five preserve $100\%$ monotonicity by construction; A3 is the empirical illustration of Proposition (ref).

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Discussion

description[leftmargin=1.2em,labelindent=0pt,itemsep=2pt,topsep=2pt,parsep=0pt] • The per-row failures of Section (ref) propagate to aggregate share predictions: raw foundation models predict a $+10\%$ cost increase raises aggregate share in $6$ of $16$ scenarios. The structural multinomial logit and adapter are exempt by Proposition (ref). • The adapter's post-calibration ECE stays elevated on Swissmetro ($8.4\%$ for Mitra, $17.0\%$ for TabPFN) against the multinomial logit's $4.0\%$. Scalar / vector temperature scaling, isotonic regression, and 10-seed bootstrap ensembling all fail to recover multinomial-logit-comparable calibration. The residual error is bias-driven (per-seed adapter distributions miscalibrated in the same direction), so ensemble averaging does not help. Calibration on the other two datasets is competitive with the multinomial logit; the Swissmetro limitation is dataset-specific. • The audit's $1\%$-perturbation recipe is sound for continuous attributes but produces float32-rounding-precision deltas on IoT-Wearables' binary indicators (function and label) in Mitra cells. We use a discrete-flip protocol instead (set the indicator to its complement, sign-correct the per-row delta), giving finite trade-off values: function flip-WTPs span $-0.05$ to $+0.59$ USD; label flips span $-0.13$ to $+0.23$ USD, with both signs appearing within a single attribute, indicating that Mitra's per-attribute response is weak relative to its per-cost response. • The joint-training cost-coefficient collapse (ablation A3, \S(ref)) reproduces on Swissmetro and LPMC and does not on IoT-Wearables. The proposition is unaffected: identifiability fails when cost is recoverable from $\mathbf{q}$ as a continuous map, and the empirical strength of the failure tracks how well that condition is met. On IoT-Wearables the foundation-model probability vector does not closely track per-product price. • The adapter's trade-off ratios are inherited from $\boldsymbol{\beta}^\ast$ by construction (Proposition (ref)). If the Stage 1 multinomial logit is misspecified for the deployment population, $g$ cannot correct the implied trade-off ratio because $g$'s contribution to the structural derivative is zero under the fixed-$\mathbf{q}$ protocol. Practitioners deploying the adapter to new domains should treat the analytical trade-off ratio as conditional on the Stage 1 specification.

Conclusion

Discrete-choice models guide policy decisions whose economic stakes are large and whose failure modes are not subtle. Tabular foundation models raise predictive accuracy beyond what structural utility models alone reach, but their predictions can violate basic economic logic: monotonicity in cost, sensible willingness-to-pay, zero probability on unavailable alternatives. We propose a two-stage adapter that embeds foundation-model predictions inside a structurally constrained utility model. Proposition (ref) shows that the two-stage procedure preserves the structural model's marginal rate of substitution exactly; Proposition (ref) shows that joint training does not. Across three discrete-choice datasets and two foundation models, the adapter pays at most $2$ pp of accuracy for full structural validity by construction, with the gain over the structural multinomial logit positive in $10$ of $10$ bootstrap replicates on every cell. The architecture is model-agnostic and slots in any predict-function whose probability output respects the precomputed-$\mathbf{q}$ contract.