Subscription services typically offer duration discounts, rewarding longer commitments with lower per-period costs. The “menu” of contract plan prices must be balanced to encourage potential customers to select into subscription overall and to nudge those that do to more profitable contracts. Because subscription menu prices change infrequently, they are difficult to optimize using historical pricing data alone.

We propose that firms solve this problem via an experiment and a model that jointly accounts for whether to opt-in and, conditionally, which plan to choose. To that end, we conduct a randomized online pricing experiment that orthogonalizes the “elevation” and “steepness” of price menus for a major dating pay site. Users’ opt-in and plan choice decisions are analyzed using a novel model for correlated binary selection and multinomial conditional choice, estimated via Hamiltonian Monte Carlo. Benchmark comparisons suggest that common models of consumer choice may systematically misestimate price substitution patterns, and that a key consideration is the distinctiveness of the opt-out (e.g., non-subscription) option relative to others available.

Our model confirms several anticipated pricing effects (e.g., on the margin, raising prices discourages both opt-in overall and choice of any higher-priced plans), but also some that alternative models fail to capture, most notably that across-the-board pricing increases have a far lower negative impact than standard random-utility models would imply. Joint optimization of the price menu’s component prices suggests the firm has set them too low overall, particularly for its longest-duration plan, and adjusting the price menu accordingly should lead to roughly 10% greater revenue.

Managers of online services are interested in both short- and long- term profitability, and how they are affected by customer-facing service attributes. This is especially true in subscription settings where different contracts, characterized by length, price, and other attributes, can affect long-run profits. A commonplace customer acquisition strategy utilized by these services involves offering price promotions targeted at new customers to induce initial conversion, while reverting to standard unpromoted prices for acquired customers. As the initial contract choice of customers can have long-run impact on their subsequent renewal choices, setting new member pricing represents a tradeoff between running promotions steep enough to convert new customers while attracting customers into contract options that maximize customer lifetime value (CLV). However, problematic to conventional methods for measuring long-run effects of initial price promotions, e.g., longitudinal A/B tests, is their time-consuming nature, as these require extended trial periods in order for long-run resubscription behaviors to play out. 

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To address this shortcoming, we introduce a Bayesian nonparametric data fusion framework that enables inference on the long-run effects of initial price promotions using only a parsimonious ‘one-shot’ experiment on initial conversions, augmented with resubscription patterns found in longitudinal customer (CRM) databases. We apply our proposed framework to impute the long-run trajectory of resubscription choices for subjects from a A/B price test whose observations are limited solely to that of the initial conversions. We develop a class of Gaussian process (GP) prior data fusion models that utilize Bayesian regularization as the mechanism for the sharing of information across the datasets at the customer-choice -occasion level. The degree to which the longitudinal data regularize, or inform, the experimental subjects’ likely renewal trajectories is given by the GP’s Automatic Relevance Detection (ARD) kernel, which allows for differential degrees of regularization based on the distances between observations in the joint space of the customers’ characteristics, price offers, and time trends. Beyond the specific application to estimating the long-run CLV of the ‘one-shot’ experimental subjects, the data fusion framework can be generalized to any customer-base analytics employing a discrete-choice hazard function.

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As GP computationally scales cubically with observations, conventional gradient- and MCMC- based estimation strategies are intractable given the large-scale nature of the fused data sources, both in terms of runtime and memory. To overcome these hurdles, we leverage the sparse ‘inducing point’ GP approach (Titsias 2009) for stochastic variational inference (SVI), in a first-of-its-kind application of this highly scalable Bayesian estimation strategy in marketing, with applicability to a broad class of choice, response, and latent class models.
 

Credit scores play a vital role in reducing the risk of lending, insuring, and renting to consumers. Accurate scoring aids financial and other institutions, who rely on a portfolio of interrelated credit scores to vet prospective customers and set attractive risk premiums. Like all forecasts, credit scores are vulnerable to data missingness, such as unbalanced, incomplete, or thin credit files. These can in turn lead to non-random gaps in the score history that reduce the reliability of the scores as metrics to target profitable borrowers and identify cross-selling opportunities for financial products and services. To address this problem, we develop a Bayesian nonparametric data fusion model to impute credible intervals for gaps in individual score histories within a portfolio of dynamically and contemporaneously interrelated scores. We apply this model to novel data from a leading credit bureau on scores for a segment of the U.S. population from 2011-2015, along with credit file decision attributes used to generate them. To address the high-dimensionality of both the model and feature spaces, we apply the “Gaussian Integral Trick” to decorrelate prior distributions, enabling scalable and efficient model estimation using stochastic mean-field variational inference. We find that credit score portfolios with imputed missingness are more accurate in predicting consumer delinquencies and bankruptcies than existing scores.