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Embedding Foundation Model Predictions in Discrete-Choice Models with Structural Guarantees
\pagestyle{plain}
\maketitle
\thispagestyle{plain}
\begin{abstract}
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.
\end{abstract}
\section{Introduction}
\label{sec:intro}
Discrete-choice models guide policy decisions with significant
economic stakes: the per-minute value of time from commuter
mode-choice forecasts \citep{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 \citep{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
\citep{hillel_2021, vancranenburgh_2022, zhao_yan_2020}.
Tabular foundation models such as TabPFN \citep{hollmann_2023,
purucker_2025} and Mitra \citep{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 \citep{sill_1997,
wehenkel_2019, constrained_monotonic_icml2025} eliminate
monotonicity violations but lose the closed-form trade-off ratio;
knowledge distillation \citep{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:
\begin{description}[leftmargin=1.2em,labelindent=0pt,itemsep=2pt,topsep=2pt,parsep=0pt]
\item[Adapter.] 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.
\item[Theory.] Propositions~\ref{prop:mrs-preservation}
and~\ref{prop:joint-collapse} identify two-stage training as the
design choice preserving structural identifiability and characterize
the failure mode of joint training.
\item[Evaluation.] 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.
\item[Audit pipeline.] 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.
\end{description}
\section{Related work}
\label{sec:related}
\paragraph{Machine learning and economic consistency in choice modeling.}
Discrete-choice modeling has been the standard tool in transportation
economics and consumer behavior since \citet{ben_akiva_lerman_1985}.
Recent work pushes machine learning into the field while exposing a
tension with economic structure: \citet{hillel_2021} document
consistent accuracy gains across neural and ensemble methods over
the multinomial logit but few behavioral diagnostics;
\citet{zhao_yan_2020} report frequent monotonicity violations and
implausible willingness-to-pay estimates from machine-learning choice
models; and \citet{vancranenburgh_2022} frame economic consistency
as an open challenge distinct from prediction accuracy.
\citet{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
\citep{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 \citep{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 \citep{hollmann_2023,
purucker_2025} and Mitra \citep{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
\citep{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.
\section{Method}
\label{sec:method}
\subsection{Setup and notation}
\label{sec:setup}
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
$P_k = \exp V_k / \sum_{j \in \mathcal{K}_i} \exp V_j$
\citep{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.
\subsection{Behavioral audit}
\label{sec: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.
\textbf{Intervention protocol.} For adapter and feature-augmented MNL
we use the \emph{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
\emph{recomputed-$\mathbf{q}$} protocol re-runs the foundation
model on perturbed inputs; we use it for the raw foundation-model
counterfactual evaluation (Section~\ref{sec:results-counterfactual})
only, and forfeit the structural guarantees in that case.
The three functionals are:
(1) \emph{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) \emph{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) \emph{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.
\subsection{Two-stage behavioral adapter}
\label{sec:adapter-arch}
\begin{description}[leftmargin=1.2em,labelindent=0pt,itemsep=2pt,topsep=2pt,parsep=0pt]
\item[Architecture.] 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^{\text{struct}}\text{ (economic structure)}}
+ \underbrace{g_k(\mathbf{q}(\mathbf{x}_i))}_{\text{foundation-model correction}},
\label{eq:adapter}
\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
$\beta = -\exp(\theta)$.
\item[Two-stage training.] We fit the model in two sequential stages.
\emph{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
$\boldsymbol{\beta}^\ast$. \emph{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 \citep{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{prop:mrs-preservation}--\ref{prop:joint-collapse}
make this precise and identify joint training as the failure mode.
\end{description}
\subsection{Propositions}
\label{sec:propositions}
We use $\boldsymbol{\beta}^\ast$ for the Stage~1 maximum-likelihood
estimate and reuse the notation of \eqref{eq:adapter}.
\begin{proposition}[Marginal-rate-of-substitution preservation under
two-stage training, fixed-$\mathbf{q}$ protocol]
\label{prop:mrs-preservation}
Let $\boldsymbol{\beta}^\ast$ be the Stage~1 maximum-likelihood
estimate, let $g$ be any Stage~2 parameters in \eqref{eq:adapter},
and operate under the fixed-$\mathbf{q}$ protocol of
Section~\ref{sec:audit}. 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.
\end{proposition}
\begin{itemize}[leftmargin=2em,itemsep=2pt,topsep=2pt,parsep=0pt]
\item \textbf{Corollary (probability-derivative MRS).}
\label{cor:prob-deriv-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{sec:proof-prop1}).
\item \textbf{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{sec:proof-prop1}.
\end{itemize}
\begin{proposition}[Joint training breaks structural identifiability]
\label{prop:joint-collapse}
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 (\emph{cost-recoverability assumption}); the correction
class $\mathcal{G}$ is dense in
$C^0(\Delta^{K-1}, \mathbb{R}^K)$~\citep{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
$\boldsymbol{\beta}^\ast$.
\end{proposition}
\begin{itemize}[leftmargin=2em,itemsep=2pt,topsep=2pt,parsep=0pt]
\item \textbf{Note (cost-recoverability).}
The assumption is dataset-dependent; partial-recoverability
cases in Appendix~\ref{sec:prop2-partial}.
\item \textbf{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{sec:ablations}) is the empirical illustration; full proof
in Appendix~\ref{sec:proof-prop2}.
\end{itemize}
\section{Experimental setup}
\label{sec:setup-experiments}
\paragraph{Datasets.}
We evaluate on three discrete-choice datasets. Swissmetro
\citep{bierlaire_2001}: $10{,}719$ stated-preference commuter choices
among rail, the proposed Swissmetro, and car. LPMC
\citep{hillel_2018}: $81{,}086$ revealed-preference London trips among
walk, cycle, public transport, and drive. IoT-Wearables
\citep{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
(\citealp{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:
\begin{itemize}[leftmargin=2em,itemsep=2pt,topsep=2pt,parsep=0pt]
\item \textbf{Swissmetro:} per-alternative travel time, cost,
headway, availability indicators; respondent age, income,
season-ticket holding, luggage, trip purpose.
\item \textbf{LPMC:} per-alternative duration, transit and driving
costs, trip distance; respondent age, sex, license, car
ownership.
\item \textbf{IoT-Wearables:} per-alternative price, functional
features, security label; respondent age, education, sex,
security-behavior score, condition fixed effects.
\end{itemize}
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{sec:audit} for the adapter.
\paragraph{Models.}
Each cell evaluates five primary models, reported in
Table~\ref{tab:headline}:
\begin{itemize}[leftmargin=2em,itemsep=2pt,topsep=2pt,parsep=0pt]
\item Multinomial logit (Stage~1 of the adapter).
\item Raw foundation model: Mitra \citep{zhang_2024} or TabPFN
\citep{hollmann_2023, purucker_2025}.
\item Architecturally constrained monotonic neural network
\citep{constrained_monotonic_icml2025}.
\item Feature-augmented multinomial logit, with $\mathbf{q}$
appended to the structural feature set.
\item Simplified two-stage adapter
$V_k = V^{\text{struct}}_k + g_k(\mathbf{q})$.
\end{itemize}
Three additional variants appear in prose only: a masked foundation
model (Swissmetro, \S\ref{sec:results-fm-fails}); a convex ensemble
$\alpha P_{\text{MNL}} + (1{-}\alpha) P_{\text{FM}}$
(\S\ref{sec:results-adapter}); 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{sec:adapter-arch}).
\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{tab:headline}).
Cross-fitting shifts adapter test accuracy by at most $-0.6$ pp
(Appendix~\ref{sec:cross-fit-details}).
\paragraph{Metrics and significance.}
We report:
\begin{itemize}[leftmargin=2em,itemsep=2pt,topsep=2pt,parsep=0pt]
\item Test-set accuracy.
\item Per-row monotonicity rate (cluster-aware aggregation on
IoT-Wearables).
\item Trade-off ratio: value-of-time on transportation datasets;
willingness-to-pay for indicated non-cost attributes on
IoT-Wearables.
\item Availability leak (Swissmetro only).
\item Expected calibration error (ECE), post-calibration with
$K=15$ equal-weight bins.
\end{itemize}
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
\citep{guo_2017}, vector temperature scaling, and isotonic
regression) are fit on validation and evaluated on test.
\section{Main results}
\label{sec:results}
\begin{table*}[!htbp]
\centering
\footnotesize
\setlength{\tabcolsep}{4pt}
\caption{\textbf{Headline results} across three datasets and two
foundation models (Mitra, TabPFN). Adapter row reports the
simplified two-stage adapter (ablation A2, paper's primary).
Trade-off units: Swissmetro CHF/hr value-of-time (VOT); LPMC GBP/hr
VOT (public-transport / drive); IoT-Wearables USD willingness-to-pay
(function / label feature). Means across $10$ bootstrap replicates
(std omitted). $^{\dagger}$~TabPFN-LPMC adapter uses in-sample
train $\mathbf{q}$; other adapter cells use cross-fitted ($k{=}5$)
train $\mathbf{q}$ (\S\ref{sec:setup-experiments}). Monotonic NN
does not use a foundation model (so its Mitra and TabPFN columns
are identical) and has no closed-form trade-off ratio. \textbf{Bold} marks behavioral-validity failures:
monotonicity below $100\%$, trade-off ratio with wrong sign or
implausibly large magnitude, or accuracy substantially below the
multinomial-logit baseline. Swissmetro availability leak:
${<}\,10^{-9}$ for MNL/adapter; $5{\times}10^{-4}$ for raw TabPFN;
$2{\times}10^{-3}$ for raw Mitra.}
\label{tab:headline}
\resizebox{\textwidth}{!}{
\begin{tabular}{llcccccccc}
\toprule
& & \multicolumn{2}{c}{Acc.\ (\%)} & \multicolumn{2}{c}{Mono.\ (\%)} & \multicolumn{2}{c}{Trade-off} & \multicolumn{2}{c}{ECE (\%)} \\
\cmidrule(lr){3-4} \cmidrule(lr){5-6} \cmidrule(lr){7-8} \cmidrule(lr){9-10}
Dataset & Model & Mitra & TabPFN & Mitra & TabPFN & Mitra & TabPFN & Mitra & TabPFN \\
\midrule
Swissmetro & MNL & 63.6 & 63.6 & 100 & 100 & 84.4 & 84.4 & 4.0 & 4.0 \\
& raw FM & 77.7 & 78.0 & --- & \textbf{96.4} & --- & 72.5 & --- & 15.7 \\
& monotonic NN & 63.1 & 63.1 & 100 & 100 & --- & --- & --- & --- \\
& feat-aug & 77.4 & 77.4 & 100 & 100 & \textbf{1509} & \textbf{678} & 9.6 & 16.8 \\
& adapter & 76.3 & 76.4 & 100 & 100 & 84.4 & 84.4 & 8.4 & 17.0 \\
\midrule
LPMC & MNL & 69.9 & 69.9 & 100 & 100 & 1.8/16.6 & 1.8/16.6 & 2.2 & 2.2 \\
& raw FM & 74.2 & 74.4 & --- & \textbf{28.9} & --- & \textbf{4.1/-14.5} & --- & 0.9 \\
& monotonic NN & \textbf{52.6} & \textbf{52.6} & 100 & 100 & --- & --- & --- & --- \\
& feat-aug & 74.0 & 74.3 & 100 & 100 & 10.3/6.7 & 7.8/4.9 & 0.7 & 0.7 \\
& adapter & 72.8 & 72.9$^{\dagger}$ & 100 & 100 & 1.8/16.6 & 1.8/16.6 & 1.3 & 1.3 \\
\midrule
IoT-Wearables & MNL & 62.5 & 62.5 & 100 & 100 & 26.1/10.0 & 26.1/10.0 & 3.4 & 3.4 \\
& raw FM & 67.1 & 67.0 & \textbf{45.6} & \textbf{40.8} & \textbf{0.0/-0.1} & \textbf{-10.5/-8.8} & 4.2 & 4.3 \\
& monotonic NN & 64.1 & 64.1 & 100 & 100 & --- & --- & --- & --- \\
& feat-aug & 67.3 & 67.4 & 100 & 100 & 14.9/5.4 & 14.2/5.3 & 3.2 & 4.1 \\
& adapter & 65.7 & 66.5 & 100 & 100 & 26.1/10.0 & 26.1/10.0 & 2.7 & 3.6 \\
\bottomrule
\end{tabular}
}
\end{table*}
Table~\ref{tab:headline} 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{sec:setup-experiments}. The results break into three
threads, mirroring the subsection structure below:
\begin{itemize}[leftmargin=2em,itemsep=2pt,topsep=2pt,parsep=0pt]
\item \S\ref{sec:results-fm-fails}: the raw foundation models fail
behavioral validity on three diagnostics (monotonicity, trade-off
ratios, availability).
\item \S\ref{sec:results-adapter}: the adapter recovers behavioral
validity while keeping most of the accuracy gain. Calibration is
competitive with one documented exception.
\item \S\ref{sec:results-counterfactual}: under counterfactual cost
perturbations, the raw foundation models violate aggregate
monotonicity in $6$ of $16$ scenarios while the adapter never
does.
\end{itemize}
\subsection{The foundation models fail behavioral validity}
\label{sec:results-fm-fails}
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. \textbf{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 $100\%$. \textbf{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{sec:discussion}.
\textbf{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.
\subsection{The adapter recovers behavioral validity while keeping most of the accuracy gain}
\label{sec:results-adapter}
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{tab:headline}): 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 \citep{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{prop:mrs-preservation}) 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{prop:mrs-preservation}), 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{tab:headline});
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{sec:discussion}. Full NLL, Brier, and
uncalibrated ECE in Appendix~\ref{sec:cal-metrics} mirror this
pattern.
\subsection{Counterfactual aggregate-share evaluation}
\label{sec:results-counterfactual}
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{tab:counterfactual} reports the full result matrix.
\begin{table}[h]
\centering
\footnotesize
\setlength{\tabcolsep}{6pt}
\caption{Counterfactual aggregate-share change (pp) under $+10\%$
cost perturbation. Adapter: mean across $10$ replicates
(std $<0.15$ pp). \textbf{Bold} marks monotonicity violations
($\Delta$share $>0$ under a cost \emph{increase}).}
\label{tab:counterfactual}
\begin{tabular}{llcccccc}
\toprule
& & Stage~1 & \multicolumn{2}{c}{Adapter} & \multicolumn{2}{c}{raw FM} \\
\cmidrule(lr){3-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7}
Dataset & Scenario & MNL & Mitra & TabPFN & Mitra & TabPFN \\
\midrule
Swissmetro & train & $-0.43$ & $-0.11$ & $-0.07$ & $-1.73$ & $-1.80$ \\
& SM & $-1.59$ & $-0.67$ & $-0.24$ & $-6.06$ & $-3.87$ \\
& car & $-1.10$ & $-0.47$ & $-0.16$ & $-1.91$ & $-1.51$ \\
LPMC & PT & $-0.43$ & $-0.31$ & $-0.31$ & $\mathbf{+0.09}$ & $-0.05$ \\
& drive & $-0.38$ & $-0.25$ & $-0.24$ & $\mathbf{+0.51}$ & $\mathbf{+0.41}$ \\
IoT-Wearables & alt1 & $-8.30$ & $-7.47$ & $-7.46$ & $\mathbf{+0.37}$ & $\mathbf{+3.24}$ \\
& alt2 & $-9.06$ & $-8.82$ & $-8.78$ & $-1.49$ & $-0.98$ \\
& alt3 & $-5.92$ & $-5.39$ & $-5.42$ & $\mathbf{+0.72}$ & $-0.71$ \\
\bottomrule
\end{tabular}
\end{table}
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{tab:counterfactual}); the gap
reflects softmax saturation under the foundation-model correction,
not a violation of Proposition~\ref{prop:mrs-preservation}.
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{sec:results-fm-fails} propagating to policy-relevant
aggregates: in $6$ of $16$ scenarios the raw foundation model
predicts a positive aggregate-share change under a cost
\emph{increase} (bolded in Table~\ref{tab:counterfactual}). 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.
\section{Ablations}
\label{sec:ablations}
We run five ablations using the same multi-seed bootstrap protocol
as Section~\ref{sec:results}.
\begin{table}[h]
\centering
\scriptsize
\setlength{\tabcolsep}{4pt}
\caption{\textbf{Ablation summary}: test accuracy (\%), mean
$\pm$ std across $10$ bootstrap replicates. A4 (graceful degradation
under foundation-model context restriction) has a different shape
(one row per context fraction) and is plotted separately in
Figure~\ref{fig:graceful_degradation}.}
\label{tab:ablations}
\resizebox{\textwidth}{!}{
\begin{tabular}{lcccccc}
\toprule
& \multicolumn{2}{c}{Swissmetro} & \multicolumn{2}{c}{LPMC} & \multicolumn{2}{c}{IoT-Wearables} \\
\cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7}
Variant & Mitra & TabPFN & Mitra & TabPFN & Mitra & TabPFN \\
\midrule
A1: $V_{\text{struct}} + \alpha \log q$ only & 76.42 $\pm$ 0.10 & 77.10 $\pm$ 0.05 & 71.89 $\pm$ 0.06 & 72.06 $\pm$ 0.04 & 63.25 $\pm$ 1.01 & 63.42 $\pm$ 0.93 \\
A2: adapter (paper's primary) & 76.44 $\pm$ 0.25 & 76.67 $\pm$ 0.27 & 72.94 $\pm$ 0.10 & 72.86 $\pm$ 0.11 & 66.43 $\pm$ 0.61 & 66.34 $\pm$ 0.54 \\
A3: joint training & 76.97 $\pm$ 0.23 & 77.00 $\pm$ 0.27 & 73.99 $\pm$ 0.07 & 74.19 $\pm$ 0.09 & 66.66 $\pm$ 0.54 & 66.61 $\pm$ 0.49 \\
A5: capacity sweep (range) & 76.45--76.49 & 76.67--76.73 & 72.89--72.90 & 72.78--72.88 & 65.63--66.65 & 66.16--66.47 \\
\bottomrule
\end{tabular}
}
\end{table}
\begin{description}[leftmargin=1.2em,labelindent=0pt,itemsep=2pt,topsep=2pt,parsep=0pt]
\item[A1: log-only.] $V_k = V^{\text{struct}}_k + \alpha \log q_k$
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.
\item[A2: simplified architecture (paper's primary).]
$V_k = V^{\text{struct}}_k + g_k(\mathbf{q})$ 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.
\item[A3: joint training.] 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{sec:prop2-partial}).
\item[A4: degraded foundation model.] 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{fig:graceful_degradation}). The adapter
falls back toward Stage~1 but does not collapse below it.
\item[A5: capacity sweep.] 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.
\end{description}
Table~\ref{tab:ablations} summarizes test accuracy across all five
variants. Four of the five preserve $100\%$ monotonicity by
construction; A3 is the empirical illustration of
Proposition~\ref{prop:joint-collapse}.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{figures/graceful_degradation.pdf}
\caption{\textbf{Graceful degradation (A4)} on Swissmetro:
adapter accuracy gain over Stage~1 (pp) as the foundation-model
context fraction is reduced. One panel per foundation model;
the dotted line marks the abstract's $\geq 6$ pp claim. Markers
are means across $10$ bootstrap replicates with $95\%$ CI
(cross-fitted protocol).}
\label{fig:graceful_degradation}
\end{figure}
\section{Discussion}
\label{sec:discussion}
\begin{description}[leftmargin=1.2em,labelindent=0pt,itemsep=2pt,topsep=2pt,parsep=0pt]
\item[Aggregate monotonicity violations on LPMC and IoT-Wearables.]
The per-row failures of Section~\ref{sec:results-fm-fails}
propagate to aggregate share predictions: raw foundation models
predict a $+10\%$ cost increase \emph{raises} aggregate share in
$6$ of $16$ scenarios. The structural multinomial logit and
adapter are exempt by Proposition~\ref{prop:mrs-preservation}.
\item[Calibration on Swissmetro.] 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.
\item[Audit methodology on discrete attributes.] 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.
\item[Proposition~\ref{prop:joint-collapse} is dataset-dependent.]
The joint-training cost-coefficient collapse (ablation A3,
\S\ref{sec:ablations}) 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.
\item[Adapter inherits Stage~1's MNL specification.] The adapter's
trade-off ratios are inherited from $\boldsymbol{\beta}^\ast$ by
construction (Proposition~\ref{prop:mrs-preservation}). 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.
\end{description}
\section{Conclusion}
\label{sec: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{prop:mrs-preservation} shows that the
two-stage procedure preserves the structural model's marginal rate of
substitution exactly; Proposition~\ref{prop:joint-collapse} 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.
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