A Two-Level Plackett-Luce Model for preference modeling in smart mobility platforms

Abstract: The Plackett-Luce model is widely used to deal with probabilities in discrete choice settings. This paper introduces a novel two-level Plackett-Luce model combined with a multinomial logistic scheme that provides the basis for the route choice module in a smart mobility platform. For this, we develop Bayesian inference and prediction mechanisms to capture consumers' preferences for personalized route recommendations. The model is empirically tested, allowing for refinements and discussion of its applicability. We also illustrate its practical relevance through several use cases, including relevant route selection, coordinated car pooling, incentive design and synthetic data generation.

• The paper develops a two-level Plackett–Luce–multinomial logit model that maps user and trip features to interpretable route-attribute preferences for time, cost, and walking distance.
• The model reaches 16.56% top-1 accuracy on 200,000 validation observations, improving on the 12.5% uniform baseline while revealing substantial unexplained behavioral variation.
• The sequential Bayesian pipeline combines Gaussian-mixture priors, particle filtering, and decay-weighted likelihoods to support behavior adaptation in route offers, car-pooling, incentives, and synthetic data generation.

Overview and motivation

This paper develops a preference model for personalized route choice within an AI-driven smart mobility platform, combining a Plackett–Luce (PL) discrete choice model with a multinomial logit (MNL) regression layer that maps user and contextual features to route-attribute weights (2605.06236). The motivation is operational: the platform generates $K=8$ Pareto-optimal routes per trip request via OpenTripPlanner (OTP2), each characterized by travel time, monetary cost, and walking time, and must predict which route a given user will select. The authors position the model as an interpretable alternative to black-box predictors, retaining a parametric structure while supporting sequential Bayesian updating as interaction data arrive daily.

The data pipeline exploits two user interactions: registration (age group, postcode-derived socio-economic level, disability level) and trip request (time window as urgency proxy, weather forecast indicator). Exploratory analysis shows only weak marginal effects—for example, severe-disability users exhibit slightly lower walking times in the distribution tails—motivating a joint model over all covariates rather than univariate analysis.

Model specification

The model is two-level. At the first level, the probability that route $l_0$ is chosen from $K$ candidates follows a PL/softmax form with worth $\lambda_l = \exp(v(\mathbf{r}_l))$, where $v$ is a linear additive value function $v(\mathbf{r}_l) = -\sum_i w_i r_l^i$ over standardized attributes. The weights are interpretable through odds ratios: $w_i$ is the log-odds change per unit difference in attribute $i$. At the second level, the weights are themselves functions of the feature vector $\mathbf{z}$ via a softmax MNL link, $$w_i = \exp((\mathbf{a}^i)^\top \mathbf{z}) / \sum_j \exp((\mathbf{a}^j)^\top \mathbf{z})$$, with $l_0$0 imposed to resolve the standard shift identifiability of the softmax parameterization. With $l_0$1 attributes and seven features (three one-hot age indicators, socio-economic level, weather, urgency, disability), the model has 14 free parameters—a deliberately low-dimensional design.

The authors note the architecture can be read as a small neural network ($l_0$2 inputs, softmax hidden layer, linear layer, softmax output), but its value lies in interpretability: posterior means directly quantify how features shift relative importance across time, cost, and walking time.

Inference: static and dynamic schemes

Posterior inference uses NUTS implemented in NumPyro/JAX. Two regimes are distinguished:

Static model: full-batch inference on all accumulated data $l_0$3, used for benchmarking and interpretation. With 200,000 observations, posteriors are approximately Gaussian, so MAP and posterior mean coincide, and point estimates are read from posterior means. The learned hierarchy is substantively sensible: e.g., adult age group shifts weight away from walking time toward time and cost; urgency ($l_0$4) strongly reduces the cost weight ($l_0$5, 90% CI $l_0$6); adverse weather increases the walking-time weight ($l_0$7, CI $l_0$8).

Dynamic model: yesterday's posterior serves as today's prior, with three refinements. First, since HMC requires continuous priors, the discrete posterior sample is approximated by a mixture of Gaussians centered at the particles, with a diagonal bandwidth matrix from first-order derivative density estimation theory; this avoids the degeneracy that would arise if the prior had support only at sample points. Second, a particle-filtering step reweights particles by their likelihood under the new day's data (with median-shifted, numerically stable softmax weights) followed by systematic residual resampling. Third, a weighted likelihood with exponential temporal decay $l_0$9 accommodates behavioral drift, capped by a maximum observation age $K$0 and a cap $K$1 on retained observations to bound runtime.

Empirical performance

The headline accuracy result should be stated plainly: on a validation set of 200,000 observations, top-1 accuracy against the uniform baseline of $K$2 reaches only 16.56%, consistent with predicted top-route probabilities of roughly 15–25%. The authors attribute the gap to substantial residual variability from unobserved contextual or personality factors—an honest concession that limits how much personalization the current feature set can deliver. Accuracy improves monotonically with likelihood size, at logarithmically increasing computational cost, which informs the choice of $K$3 for overnight inference.

For the dynamic scheme, experiments use 500–2,500 observations per iteration. More observations yield higher median accuracy and lower variance across iterations. Adding the filtering step improves long-run mean accuracy and reduces variance relative to no filtering. The decay factor $K$4 exhibits a clear trade-off: small $K$5 adapts quickly after a forced distributional shift but sacrifices asymptotic accuracy; large $K$6 behaves oppositely, leading the authors to recommend $K$7 and to propose—but not implement—adaptive selection of $K$8 based on detected change points.

Use cases

Four applications illustrate deployment, all driven by the acceptance probability or equivalently the value function $K$9 evaluated at posterior means (full posterior integration being infeasible in real time):

• Route offer selection: candidate routes are clustered into $\lambda_l = \exp(v(\mathbf{r}_l))$0 groups (k-means on $\lambda_l = \exp(v(\mathbf{r}_l))$1), and the highest-value route per cluster is displayed, ensuring diversity among offered alternatives.
• Coordinated car-pooling: seat allocation follows a three-tier priority—reduced-mobility riders ranked by increasing walking time, then candidates ranked by emission savings, then remaining candidates ranked by preference-model value—with veto constraints and cancellation penalties.
• Incentive design: the closed-form minimum incentive $\lambda_l = \exp(v(\mathbf{r}_l))$2 makes a detour route indifferent to the driver's baseline, scaling with the driver's personalized time-to-cost weight ratio. This extends to maximizing expected social utility over multiple users via a product-of-acceptance-probabilities objective.
• Synthetic data generation: correlated route attributes are generated through a shared latent factor $\lambda_l = \exp(v(\mathbf{r}_l))$3 plus Gaussian noise, dominated routes are removed, and choices are sampled from the model probabilities, producing realistic noisy training data without privacy exposure (the released code uses synthetic data for this reason).

Limitations and open questions

Several limitations are acknowledged explicitly. The modest absolute accuracy indicates that the seven-feature set leaves most choice variability unexplained; identifying additional explanatory variables is left open. The weighting scheme handles recency but not seasonal effects, which the authors flag as future work. The Gaussian-mixture prior approximation introduces sensitivity to the bandwidth choice, whose calibration depends on sample size and dimension in ways the paper discusses qualitatively rather than resolving theoretically. The recommendation of adaptive $\lambda_l = \exp(v(\mathbf{r}_l))$4 remains unimplemented. Finally, extrapolation of the observed error behavior to production-scale datasets rests on the assumption that patterns from ~2,500-observation runs carry over when compute allows larger likelihoods—a claim asserted but not demonstrated empirically.

Conclusion

The paper contributes a compact, interpretable two-level PL–MNL model with a practical sequential Bayesian inference pipeline (Gaussian-mixture priors, particle filtering, decay-weighted likelihoods) tailored to nightly retraining in a mobility platform. Its empirical results are honest about the ceiling imposed by available features—16.56% top-1 accuracy against a 12.5% baseline—and its main strength lies in the closed-form, personalized decision mechanisms (route ranking, car-pooling prioritization, incentive computation) that the parametric structure enables. The central open question is whether enriching the feature space can close the gap between predicted and observed choice behavior without sacrificing the interpretability that motivates the approach.