A Neural Network Based Choice Model for Assortment Optimization
Discrete-choice models are used in economics, marketing and revenue
management to predict customer purchase probabilities, say as a function of
prices and other features of the offered assortment. While they have been shown
to be expressive, capturing customer heterogeneity and behaviour, they are also
hard to estimate, often based on many unobservables like utilities; and
moreover, they still fail to capture many salient features of customer
behaviour. A natural question then, given their success in other contexts, is
if neural networks can eliminate the necessity of carefully building a
context-dependent customer behaviour model and hand-coding and tuning the
estimation. It is unclear however how one would incorporate assortment effects
into such a neural network, and also how one would optimize the assortment with
such a black-box generative model of choice probabilities. In this paper we
investigate first whether a single neural network architecture can predict
purchase probabilities for datasets from various contexts and generated under
various models and assumptions. Next, we develop an assortment optimization
formulation that is solvable by off-the-shelf integer programming solvers. We
compare against a variety of benchmark discrete-choice models on simulated as
well as real-world datasets, developing training tricks along the way to make
the neural network prediction and subsequent optimization robust and comparable
in performance to the alternates.
Hanzhao Wang, Zhongze Cai, Xiaocheng Li, Kalyan Talluri
# A Neural Network Based Choice Model for Assortment Optimization

###### Abstract

Discrete-choice models are used in economics, marketing and revenue management to predict customer purchase probabilities, say as a function of prices and other features of the offered assortment. While they have been shown to be expressive, capturing customer heterogeneity and behaviour, they are also hard to estimate, often based on many unobservables like utilities; and moreover, they still fail to capture many salient features of customer behaviour. A natural question then, given their success in other contexts, is if neural networks can eliminate the necessity of carefully building a context-dependent customer behaviour model and hand-coding and tuning the estimation. It is unclear however how one would incorporate assortment effects into such a neural network, and also how one would optimize the assortment with such a black-box generative model of choice probabilities. In this paper we investigate first whether a single neural network architecture can predict purchase probabilities for datasets from various contexts and generated under various models and assumptions. Next, we develop an assortment optimization formulation that is solvable by off-the-shelf integer programming solvers. We compare against a variety of benchmark discrete-choice models on simulated as well as real-world datasets, developing training tricks along the way to make the neural network prediction and subsequent optimization robust and comparable in performance to the alternates.

Imperial College Business School, Imperial College London

(h.wang19, z.cai22, xiaocheng.li, kalyan.talluri)@imperial.ac.uk

## 1 Introduction

What goes on in the consumer's mind during the purchase process? How, and why, do they decide to purchase? What features of the product or environment encourage them to purchase? Given the heterogeneity and idiosyncratic thought processes of customers, these questions are unlikely to be answered definitely. Nevertheless these are fundamental questions of interest to marketers, behavioural psychologists, economists and operations researchers. Hence, for operational and algorithmic purposes, a number of stylized or reduced-form models have been proposed in the literature to explain individual-level decision outcomes as well as collective aggregate purchase behavior.

Discrete-choice models are one such class of models, widely used in marketing, economics, transportation and revenue management, especially suitable for situations where customers choose at most one of several alternatives. They are typically parsimonious models, based on micro-economic utility maximization principles in a stochastic framework, and expressive enough to incorporate product, customer and environment features, and tractable enough for estimation and incorporation into optimization models. The classical Multinomial Logit choice model (MNL) is an early and prominent example that, along with mixed-logit, nested-logit and probit, has found widespread application in many fields, both in theoretical models as well as in empirical studies. New alternatives to MNL, that are more flexible or more expressive, such as Markov chain choice (Blanchet et al., 2016) and Exponential (Alptekinoglu and Semple, 2016), have been proposed recently.
One could potentially design more elaborate choice models, folding in many known aspects of the purchase behavior. However, this line of development has some natural limitations as, one, the models have to be specialized and tuned to a particular industry or even to a firm's purchase funnel; two, would require developing and hand-coding its estimation which may be difficult both in theory and practice; and finally, using the model in an assortment optimization method may be computationally infeasible.

So, given the fundamental indeterminacy in how and why a customer chooses to purchase, our inability to observe all the data required by some of these models, and as the process itself is opaque and liable to change by industry and context, a reasonable research proposition is to search for a robust universal model that requires minimal tuning or expert modeling knowledge and works as well as behavioural models across all contexts and industries. Naturally, machine learning methods, specifically neural networks, come to mind. They have been applied with great success for many prediction tasks in industry precisely because they do not require fine-grained modeling specific to a situation and industry, are robust, and come with standardized training algorithms that work off-the-shelf. With enough data they also have proved to be very good performers in many areas, sometimes proving to be superior even to hand-crafted models.

In this paper we develop a neural network based choice model suitable for assortment optimization. The firm has to offer an assortment of products, potentially customized to each individual customer, and the customer either chooses one of the offered products or decides not to purchase. This problem arises in a wide variety of industries where customers choose at most one item, such as hotel rooms, airline seats, or expensive one-off purchases in a single category such as automobiles, laptops or mobile phones. (We leave extensions to multi-item purchases for future research.)

The challenges are the following: First, the data is not at a scale typically required for neural networks training; a hotel for instance has only a few months of seasonal data it can use for predictions. Second, the neural network predictions have to be used subsequently in an assortment optimization problem--essentially revenue-optimal subset-selection--and a neural network output promises no structure to make this optimization efficient. Consequently we want a network as compact as possible so the optimization--in an integer programming formulation that we develop--is tractable for a reasonably large number of products.

Our contributions in this paper are the following:

1. We tackle the above-mentioned challenges and develop a neural network based choice model under two settings: (i) a feature-free setting where no additional features on products or customers are used in prediction and (ii) a feature-based setting that can utilize both product and customer features.
2. We formulate an integer-program over the trained neural network to find an assortment that maximizes expected revenue.
3. We perform extensive numerical simulations that compare the predictive and optimization performance under the following scenarios: 1. Synthetic simulations where the ground-truth is generated by a panoply of models in the choice literature--MNL, Markov chain, non-parametric, mixed-logit--choice models and cross-validate them, both on raw prediction of choice probabilities, as well as the quality of the optimized assortment. 2. Situations where the customer behavior does not follow the models' script or rational economic behaviour, but is documented experimentally and empirically in the literature.
3. Real-world assortment and sales transactions data from the following important industries: physical fashion-retail, airline, hotel, and transportation, and compare the predictive performances of the popular discrete-choice models vs. our neural network.
4. We device a meta-learning trick that allows us to obtain good robust prediction performance from a compact neural network even when trained on limited transactional purchase data. Moreover, we show a well-trained network can be warm-started when adding new products to the pool.

Our numerical results suggest that the neural network gives comparable performance to the best choice model in most cases, both in simulations as well as on real data. Some of the parametric models do perform better in simulations, unsurprisingly, when the ground-truth data-generation coincides with the model. We find a similar pattern for real-world data, where on a hold-out sample our neural network gives robust predictions across the different industries. Moreover, we find that a one-layer neural choice model can capture what may be considered "irrational" purchase behavior--documented to occur in experiments and empirical studies--that cannot be explained by random utility models. In summary, shallow neural networks enjoy both prediction power and computational tractability for assortment optimization.

## 2 Literature Review

Probabilistic choice models originated in economics based on a stochastic utility model of the consumer. The oldest and a still widely used discrete-choice model is the MNL, with alternates and generalizations proposed to overcome its known limitations, such as the nested-logit and mixed-logit. We refer the reader to the seminal paper McFadden and Train (2000) and the book Ben-Akiva and Lerman (1985) for back-ground on discrete-choice models and their economic motivation and estimation from data.

We review the literature in Computer Science and Operations Research next on neural network modeling of discrete-choices and assortment optimization.

### Neural Network Models in Utility Modeling

In the Computer Science community, Bentz and Merunka (2000) use a neural network to capture the non-linear effects of features on the latent utility. A few recent works Wang et al. (2020); Han et al. (2020); Sifringer et al. (2020); Wong and Farooq (2021); Aouad and Desir (2022); Arkoudi et al. (2023) study different application contexts and explore different neural network architectures for the feature-utility mapping. In addition to the product and customer features, Gabel and Timoshenko (2022) also encode history purchase information. Chen et al. (2021); Chen and Misic (2022) take a different route and consider the random forest model as the mapping function. Van Cranenburgh et al. (2022) discuss the potential avenues for further integrating machine learning into choice modeling with a comprehensive review of the existing literature.

We term this line of works as deep learning based utility modeling instead of choice modeling, because these works mainly focus on predicting the unobserved product utilities with the observed features and applying an MNL-style operator for predicting purchase probabilities (except Aouad and Desir (2022), who use a mixture of a set of estimated utilities to mimic the mixed-logit model). What is missing is the _assortment effect_ on utilities--the interactions between the products within (and possibly outside) the set of offered products. This has two negative implications: First, some of the neural network models require all the training samples to have a fixed-size assortment. This requirement is simply not met in many application contexts such as hotels, airlines or fashion retail where the assortments change frequently. Second, it fails to capture the effect that the assortment has on the product utilities.
Intuitively, a product's utility is not only determined by product and customer features, but also affected by the assortment offered to the customer which is a distinguishing feature of our neural network.

Another common drawback of the existing deep learning approaches is that they all require the availability of product or customer features. When there are no features available, most of the models degenerate to a uniform choice probability. In contrast, our neural network based choice model is designed to handle the feature-free setting. Importantly, we make a distinction between the feature-free model and the feature-based model not by the availability of the features, but by whether the downstream application is interested in a population-level or personalized choice model for its assortment optimization.

### Assortment Optimization

Assortment optimization is an important problem in operations management, used in the airline, hotel, and fashion retail e-commerce. For more on the history and applications of assortment optimization, we refer readers to the books Talluri et al. (2004); Gallego and Topaloglu (2019). Here we briefly review the complexity of the problem under some of the aforementioned choice models:

**Multinomial Logit Model (MNL):** Under the MNL model Talluri and Van Ryzin (2004) show that the optimal assortment is a revenue-ordered nested assortment and thus the optimal policy is to order the items by their revenues to form a nested set of assortments and choose the assortment with the largest expected revenue in that nested set. Rusmevichientong and Topaloglu (2012) show such revenue-ordered policy (RO) is robust against the uncertainty in MNL parameters and solvable under a cardinality constraint.

**Markov Chain Choice Model (MCCM):** Blanchet et al. (2016) give a polynomial-time algorithm for finding the optimal assortment under this model by repeatedly using a Bellman operator until convergence. Feldman and Topaloglu (2017) further prove that the optimal assortment can be found by solving a linear programming. With constraints, Desir et al. (2020) prove that this assortment optimization problem is NP-hard and introduce algorithms with a provable worst-case approximation ratio.

**Mixed-Multinomial Logit model (MMNL):** Bront et al. (2009) show that even without constraints, the assortment optimization problem under MMNL is NP-hard and then provide a column-generation method to solve the problem. Rusmevichientong et al. (2014) show that a revenue-ordered policy can obtain the optimal assortment in MMNL when the choice model has some special structures. Mendez-Diaz et al. (2014) propose a branch-and-cut algorithm for solving the optimization problem.

**A General Choice Model:** When the revenue function underlying a choice model is a black-box function, algorithms for finding the optimal assortment cannot be tailored by utilizing structure. Jagabathula (2014) purposes the ADXOpt algorithm which repeatedly finds the best action in adding (A), deleting (D), and exchanging (X) based on the current assortment with provable approximation ratio under some conditions. Udwani (2023) defines submodular order functions and purposes several heuristics for assortment optimization under various types of constraints. With mild conditions, the heuristics have bounded approximation ratios.

In this paper we show that in the context of assortment optimization, (1) a shallow network is enough to recover existing choice models and (2) good performance and robustness is achievable with a standard neural network and integer programming when dealing with different choice models compared to customized heuristics.

## 3 Problem Setup

The firm has a set of \(n\) products \(\mathcal{N}=\{1,2,...,n\}\) that can be offered to customers. An assortment \(\mathcal{S}\subseteq\mathcal{N}\) is a subset of \(\mathcal{N}\) that the firm decides to present to the customer. In practice, the assortment can
be determined by the product availability or the seller may decide the assortment to maximize profits by limiting the choice of the customer. The choice model gives a prediction of the probability of the customer choosing product \(i\) conditional on the assortment \(\mathcal{S}\):

\[\mathbb{P}(i|\mathcal{S})\text{ for all }i\in\mathcal{N}\text{ and }\mathcal{S}\subseteq \mathcal{N}.\]

In particular, we assume that \(\mathbb{P}(i|\mathcal{S})=0\) for \(i\notin\mathcal{S}\), i.e., the customer cannot choose a product not offered in the assortment. In this way, a choice model

\[\mathcal{M}=\{\mathbb{P}(i|\mathcal{S}):\mathcal{S}\subseteq\mathcal{N}\} \tag{1}\]

dictates \(2^{n}-1\) probability distributions, each of which corresponds to one possible nonempty assortment. In all applicable business contexts, there is usually a "no-purchase" option where the customer chooses not to purchase any of the items from the offered assortment. The no-purchase option can be captured by one product (say, indexed as \(n\)) in \(\mathcal{N}\) and we can always have \(n\in\mathcal{S}\).

A tabular parameterization of the choice model \(\mathcal{M}\) generally requires \(\Omega(n\cdot 2^{n})\) parameters. Additional structures are therefore imposed to facilitate efficient estimation and revenue optimization.

### Random Utility Models (RUM) and Beyond

The _random utility model_ (RUM) is the dominant framework for choice models in practice. It assigns a random utility for each product and models the consumer's choice behavior according to the principle of utility maximization. Specifically, the utility of the \(i\)-th product

\[U_{i}=u_{i}+\epsilon_{i}\text{ for }i\in\mathcal{N}.\]

Here \(u_{i}\) is deterministic and represents the mean utility (to the customer) of purchasing the \(i\)-th product among the population. The random quantities, \(\epsilon_{i}\)'s are assumed to have a zero mean and can be correlated among products, to model unobservable factors as well as heterogeneity across products. We note that customer heterogeneity is also modeled by assuming the random part is specific to product as well as the customer. Under the RUM,

\[\mathbb{P}(i|\mathcal{S})\coloneqq\mathbb{P}\left(U_{i}=\max_{j\in\mathcal{S}} U_{j}\right)\quad\text{for }i\in\mathcal{S}. \tag{2}\]

For simplicity, we assume ties are broken randomly. Different distributions of \(\epsilon_{i}\)'s specialize in this general RUM framework into different choice models as in the following.

**Multinomial Logit Model (MNL):** The MNL choice model assumes \(\epsilon_{i}\)'s are i.i.d. from a mean-zero Gumbel distribution with scale parameter 1. Then the choice probability for \(i\in\mathcal{S}\) has a nice closed-form solution:

\[\mathbb{P}(i|\mathcal{S})\coloneqq\frac{\exp(u_{i})}{\sum_{j\in\mathcal{S}} \exp(u_{j})}. \tag{3}\]

When there is a feature vector \(\mathbf{f}_{i}\) associated with each product, one can also represent the deterministic utility \(u_{i}=\mathbf{\theta}^{\top}\mathbf{f}_{i}\) and obtain the linear-in-attributes MNL model. Several recent works model \(u_{i}\) with a neural network function of the feature \(\mathbf{f}_{i}\)(Bentz and Merunka, 2000; Sifringer et al., 2020; Wang et al., 2020; Arkoudi et al., 2023).

Although MNL model is simple to estimate and optimize for an assortment, the model's assumptions
make it restricted to the _independence of irrelevant alternatives_ (IIA) property: for any \(i,j\in\mathcal{S}\subset\mathcal{S}^{\prime}\),

\[\frac{\mathbb{P}(i|\mathcal{S})}{\mathbb{P}(j|\mathcal{S}^{\prime})}=\frac{ \mathbb{P}(i|\mathcal{S})}{\mathbb{P}(j|\mathcal{S}^{\prime})}.\]

This means the ratio of two products' choice probabilities is independent of the other products in the assortment. We illustrate the effect of this for prediction and contrast it with the other methods.

Table 1 is an example to illustrate the limitation of the IIA property: In case I there is one product \(A\) with a no-purchase option. In Case II, when we add another product \(A^{\prime}\) identical to the product \(A\), we assume the customer who would purchase product \(A\) in Case I will have equal probability to purchase \(A\) or \(A^{\prime}\) in Case II. From the predicted probabilities, we can see the estimated MNL model cannot capture this purchase behavior due to the IIA property.

The following three RUM models (along with many others) were developed to mitigate this.

**Mixed-Multinomial Logit Model (MMNL):** The MMNL choice model (McFadden and Train, 2000) is a generalization of MNL to multiple segments. It assumes there are several customer types and each type has its own deterministic utilities; the random terms are still assumed to be i.i.d. and Gumbel distruted. We denote \(\mathcal{C}\) as the set of customer types and \(\alpha_{c}\) as the probability (for a customer) of being \(c\)'s type. Then the choice probability for \(i\in\mathcal{S}\) is:

\[\mathbb{P}(i|\mathcal{S})\coloneqq\sum_{c\in\mathcal{C}}\alpha_{c}\frac{\exp(u _{c,i})}{\sum_{j\in\mathcal{S}}\exp(u_{c,j})}.\]

For a continuous distribution of customer types, the summation is replaced by an integral. McFadden and Train (2000) show any RUM can be represented by an MMNL choice model under mild conditions and thus MMNL also has a large model capacity. However, the estimation of an MMNL is challenging and many of the methods proposed in the literature fail to recover the latent types' parameters (Jagabathula et al., 2020; Hu et al., 2022).

**Markov Chain Choice Model (MCCM):** The MCCM (Blanchet et al., 2016) defines a discrete-time Markov chain on the space of products \(\mathcal{N}\) and models each customer's choice through the realization of a sample path. Specifically, it assumes a customer arrives to product \(i\) with probability \(\lambda_{i}\) (interpreted as the initial state of the Markov chain). Then if the product is in the assortment \(\mathcal{S}\), the customer purchases it and leaves. Otherwise, the customer transitions from product \(i\) to product \(j\) with probability \(\rho_{ij}\). Naturally, \(\sum_{i\in\mathcal{N}}\lambda_{i}=1\) and \(\sum_{j\in\mathcal{N}}\rho_{ij}=1\) for all \(i\in\mathcal{N}.\) Let \(X_{t}\in\mathcal{N}\) denote the product (state) that the customer visits at time \(t\), and define the first hitting time \(\tau=\{t\geq 0:X_{t}\in\mathcal{S}\}.\) Then, under the

\begin{table}
\begin{tabular}{c c|c|c c c} \hline \hline  & Product & True Prob. & \multicolumn{4}{c}{Pred. Prob.} \\  & & NN & MNL-MLE & MCCM-EM \\ \hline \multirow{2}{*}{Case I} & No-purchase &.40 &.42 &.46 &.40 \\  & A &.60 &.58 &.54 &.60 \\ \hline \multirow{2}{*}{Case II} & No-purchase &.40 &.42 &.32 &.38 \\  & A &.30 &.28 &.38 &.32 \\ \multicolumn{2}{c}{} & A’ &.30 &.30 &.30 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Example illustrating the effects of the IIA property of MNL. Column True Prob. is the true choice probabilities in each case and Pred. Prob. is the predicted probabilities from each method: NN is a one-layer Gated-Assort-Net neural choice model (that we introduce later), MNL-MLE is the MNL model calibrated via Maximum-Likelihood, while MCCM-EM implements the expectation-maximization algorithm (Simsek and Topaloglu, 2018) to fit a MCCM model. All choice models are estimated based on 8000 training samples where each sample’s assortment is randomly picked with equal probability.
MCCM,

\[\mathbb{P}(i|\mathcal{S})\coloneqq\mathbb{P}(X_{\tau}=i).\]

The MCCM is a generalization of MNL, in the sense MNL arises for a specific form of the \(\lambda\)'s and \(\rho\)'s. It has more parameters, hence has a larger model capacity to capture more complex choice behavior. Berbeglia (2016) establishes MCCM as a RUM through a random walk argument.

**Non-parametric (NP) Choice Model:** The NP choice model or random orderings model has origins in classical microeconomic demand theory based on preference rankings (Block and Marschak, 1959) and recently reintroduced with a new estimation proposal in (Farias et al., 2009, 2013). The model assumes that there exists a distribution \(\lambda:\mathrm{Perm}_{\mathcal{N}}\to[0,1]\) over the set \(\mathrm{Perm}_{\mathcal{N}}\) of all possible permutations of the products. There are \(n!\) customer types and each customer type has a preference list of the products corresponding to one permutation in \(\mathrm{Perm}_{\mathcal{N}}\). Customers always purchase the most preferred product in the preference list. The choice probability is given by

\[\mathbb{P}(i|\mathcal{S})\coloneqq\sum_{\sigma\in\mathrm{Perm}_{i}(\mathcal{S })}\lambda(\sigma)\]

where the set

\[\mathrm{Perm}_{i}(\mathcal{S})\coloneqq\{\sigma\in\mathrm{Perm}_{\mathcal{N} }:\sigma(i)<\sigma(j)\text{ for all }j\neq i\in\mathcal{S}\}\]

contains all customer types/permutations under which product \(i\) is the most preferable product in the assortment \(\mathcal{S}\).

The NP choice model has exponentially many parameters and thus a larger model capacity than both MNL and MCCM. Indeed, Block and Marschak (1959) show that any RUM choice model can be represented by a NP model, and vice versa. Moreover, the estimation method proposed by (Farias et al., 2009, 2013) needs to solve a linear programming with size \(n!\). Although the linear programming enjoys good theoretical properties, it is computationally infeasible when \(n\) is large and has to be further simplified or approximated by the sampling or representation method.

**Non RUM models:** RUM models have a _regularity_ property: if we add a new product to the assortment \(\mathcal{S}\), then the purchase probability of all currently offered products will never increase. However, as remarked by Chen and Misic (2022), there is an increasing body of experimental evidence which suggests that the aggregate choice behavior of customers is not always consistent with this regularity property, and thus often violates the premise of RUM. For instance, Seshadri et al. (2019) use the seminal experiment of (Tversky and Kahneman, 1985) and Chen and Misic (2022) quote the behavioral experiment of (Ariely and Jones, 2008) to motivate the development of choice models beyond the scope of RUMs. Here we use the examples from the above two behavioral experiments to show the limited capacity of RUM, while the neural choice model's potential to pick up such behaviour in the data.

* Table 2 is an implementation of the experiment in (Ariely and Jones, 2008): The price column shows the price of each option. In Case I, an assortment has two available products, while in Case II, a clearly inferior option of Print-Only is added. The addition of Print-Only twists customer utilities for the other two options, which is reflected by the change in the true choice probability.
* Table 3 is an implementation of the gambles example in (Tversky and Kahneman, 1985): The Win Prob. column shows the winning probability of each gamble and Payoff column indicates the winning payoff. There exists a preference cycle over three gambles A, B, C with different winning probabilities and payoffs, where A is preferred to B, B is preferred to C based on the expected payoff. However, C is preferred to A since the winning probability of C is significantly larger than
In sum, a parametric model that fits observed purchases in all situations is unlikely. Certainly more elaborate models of consumer search and decisions exist, but their estimation, based on unobservable factors and limited firm-level data, and subsequent optimization is daunting, and certainly will require a lot of development resources. Hence, it is worth striving for a flexible large-capacity learning engine to predict the purchase probabilities directly without worrying about getting the precise consumer behaviour modeling right.

With this in mind, we develop neural network models that allow the utilities to be dependent on the assortment with or without including customer and product features. Hence our neural choice model allows violation of the regularity property, letting the utilities change depending on the assortment that is offered.

## 4 Neural Choice Model

### A Feature-free Neural Choice Model

In this section, we introduce our feature-free neural choice model defined simply as a neural network function that maps the assortment input \(\mathbf{S}\in\{0,1\}^{n}\) to the output \(\mathbf{Y}\in[0,1]^{n}\):

\[\mathbf{Y}=g(\mathbf{S};\mathbf{\theta})\]

where \(\mathbf{\theta}\) encapsulates all the model parameters. The input \(\mathbf{S}=(s_{1},...,s_{n})\) is the binary encoding of an assortment \(\mathcal{S}\) where \(s_{i}=1\) if \(i\in\mathcal{S}\) and \(s_{i}=0\) if \(i\notin\mathcal{S}\). The output \(\mathbf{Y}=(Y_{1},...,Y_{n})\) is a vector supported on the probability simplex with \(Y_{i}=\mathbb{P}(i|\mathcal{S})\).

We consider two neural network architectures illustrated in Figure 1. For both architectures, we let

\begin{table}
\begin{tabular}{c c c|c|c c c} \hline \hline  & \multicolumn{1}{c|}{Option} & \multicolumn{1}{c|}{Price} & \multicolumn{1}{c|}{True Prob.} & \multicolumn{4}{c}{Pred. Prob.} \\  & & & & NN & MNL-MLE & MCCM-EM \\ \hline \multirow{3}{*}{Case I} & No-purchase & – &.14 &.14 &.14 &.14 \\  & Internet-Only & \$59 &.57 &.58 &.43 &.43 \\  & Print-\&-Internet & \$125 &.29 &.28 &.43 &.43 \\ \hline \multirow{3}{*}{Case II} & No-purchase & – &.14 &.13 &.14 &.15 \\  & Internet-Only & \$59 &.29 &.31 &.43 &.43 \\  & Print-\&-Internet & \$125 &.57 &.55 &.43 &.42 \\  & Print-Only & \$125 &.00 &.01 &.00 &.00 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Example based on the behavioral experiment in (Ariely and Jones, 2008). The experiment setup is same as Table 1.

\begin{table}
\begin{tabular}{c c c c|c c c} \hline \hline  & \multicolumn{1}{c|}{\multirow{2}{*}{Gamble}} & \multicolumn{1}{c|}{\multirow{2}{*}{Win Prob.}} & \multirow{2}{*}{Payoff} & \multirow{2}{*}{True Prob.} & \multicolumn{4}{c}{Pred. Prob.} \\  & & & & NN & MNL-MLE & MCCM-EM \\ \hline \multirow{2}{*}{Case I} & A & 1/4 & \$ 6 &.75 &.76 &.42 &.62 \\  & B & 1/3 & \$ 4 &.25 &.24 &.58 &.38 \\ \hline \multirow{2}{*}{Case II} & B & 1/3 & \$ 4 &.75 &.74 &.43 &.62 \\  & C & 1/2 & \$ 2 &.25 &.26 &.57 &.38 \\ \hline \multirow{2}{*}{Case III} & A & 1/4 & \$ 6 &.20 &.20 &.36 &.24 \\  & C & 1/2 & \$ 2 &.80 &.80 &.64 &.76 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Example based on the behavioral experiment in (Tversky and Kahneman, 1985). The experiment setup is same as Table 1.
\(\mathbf{W}_{l}\in\mathbb{R}^{n_{l}\times n_{l-1}}\) and \(\mathbf{b}_{l}\in\mathbb{R}^{n_{l}}\) are the weight matrix and bias vector respectively for layer \(l\) with \(n_{l}\) nodes in the \(l\)-th layer, and \(n_{1}=n_{L}=n\). We let \(\mathbf{z}_{l}=(z_{l,1},...,z_{l,n_{l}})\in\mathbb{R}^{n_{l}}\) be the output of the \(l\)-th layer and the input for the next layer. The parameters \(L\) and \(\{n_{l},\ l=1,...,L\}\) need to be specified. (But we will show that \(L=1\) and \(n_{l}=n\) can perform well in both the real data and simulated data across different generative models.)

The final layer's output \(\mathbf{z}_{L}=(z_{L,1},...,z_{L,n})\) are to set the choice probabilities given by a gated operator

\[Y_{i}=\begin{cases}\frac{\exp(z_{L,i})}{\sum_{i^{\prime}\in\mathcal{S}}\exp(z _{L,i^{\prime}})},&i\in\mathcal{S},\\ 0,&i\notin\mathcal{S}.\end{cases}\]

While this may appear to be a standard MNL-like function, and hence suffer from the IIA property etc., it in fact does not; we note an important difference: The \(\mathbf{z}_{L}\) is a function of \(\mathbf{z}_{0}=\mathbf{S}\), i.e., the assortment effect is encoded into each \(z_{L,i}\) and subsequently in \(Y_{i}\) through the neural network layers. (We should be hence be representing \(\mathbf{z}_{L}\) as \(\mathbf{z}_{L}(S)\) but for avoiding cumbersome notation, just write as \(\mathbf{z}_{L}\).

The differences in the architectures are as follows.

**Gated-Assort-Net (GAsN)**

The GAsN model is a fully-connected neural network designed specifically for the discrete-choice model. It adopts the classic structure commonly found in traditional neural networks, making it straightforward to implement. It takes the assortment vector \(\mathbf{S}\) as its input layer and runs through a number of fully connected layers. Finally, it uses the assortment \(\mathbf{S}\) to create an output gate to ensure \(\mathbb{P}(i|\mathcal{S})=0\) if \(i\notin\mathcal{S}\). We initialize \(\mathbf{z}_{0}=\mathbf{S}\) as the input layer and for each layer \(l=1,...,L\),

\[\mathbf{z}_{l}=(\mathbf{W}_{l}\mathbf{z}_{l-1}+\mathbf{b}_{l})^{+} \tag{4}\]

where \((\cdot)^{+}=\max\{\cdot,0\}\).

**Res-Assort-Net (RAsN)**

The RAsN model incorporates the concept of residual learning, which has demonstrated a remarkable enhancement in image recognition compared to the performance of plain networks (He et al., 2016). As

Figure 1: Neural choice models. \([n]\) denotes the dimension of the corresponding layer.
in the GAsN, it is initialized by \(\mathbf{z}_{0}=\mathbf{S}\) and finally uses the assortment to create an output gate to ensure \(\mathbb{P}(i|\mathcal{S})=0\) if \(i\notin\mathcal{S}\). However, for each layer \(l=1,...,L\) in RAsN,

\[\mathbf{z}_{l}=\left(\mathbf{W}_{l}\mathbf{z}_{l-1}+\mathbf{b}_{l}\right)^{+}+\mathbf{z}_{l-1}. \tag{5}\]

Here \(\mathbf{z}_{l}=(z_{l,1},...,z_{l,n})\in\mathbb{R}^{n}\) is the output of the \(l\)-th layer and the input for the next layer, \(\mathbf{W}_{l}\in\mathbb{R}^{n\times n}\) and \(\mathbf{b}_{l}\in\mathbb{R}^{n}\) are the weight matrix and bias vector, parameters of the neural network model. Note we require the dimensions \(n_{l}=n\) for all \(l=1,...,L\) for making the operation of adding \(\mathbf{z}_{l-1}\) in (5) available.

Compared to (4), the additional (directly added) \(\mathbf{z}_{l-1}\in\mathbb{R}^{n}\) in (5) represents the residual effect. Intuitively, directly adding it in (5) can avoid forgetting/distorting the information extracted from previous layers with extreme \(\mathbf{W}_{l}\) (e.g., all entries are close to 0) during the training, which can hopefully further lead to a more robust training process and a better trained network.

#### Computational Experiment Design

We give a preview of our numerical experiment methodology and also demonstrate the model capacity of the two neural choice model architectures (full details are presented in Appendix A).

We first generate training data from four models: MNL, MCCM, NP and MMNL. We then compare the predictive performance of our two neural choice models, GAsN and RAsN, against three benchmarks and an oracle that has knowledge of the true generative model (equivalently, the negative entropy of the true distribution). Table 4 shows the results.

First, both neural choice models benefit from a larger sample size, while the performances of maximum likelihood estimation (MLE) and expectation maximization (EM) do not improve much as the sample size increases. Second, when the true model is MNL, we know the method of MLE is provably asymptotically optimal and this is unsurprisingly verified from our experiment. But when the true model becomes more complex, such as NP and MMNL, the neural network models show their advantage. Third, the learning

\begin{table}
\begin{tabular}{c|c|c c|c c|c c|c c} \hline  & \# Samples & \multicolumn{2}{c|}{\begin{tabular}{c} MNL \\ \(|\mathcal{N}|=20\) \\ \end{tabular} } & \multicolumn{2}{c|}{\begin{tabular}{c} MCCM \\ \(|\mathcal{N}|=20\) \\ \end{tabular} } & \multicolumn{2}{c|}{\begin{tabular}{c} NP \\ \(|\mathcal{N}|=20\) \\ \end{tabular} } & \multicolumn{2}{c|}{
\begin{tabular}{c} MMNL \\ \(|\mathcal{N}|=20\) \\ \end{tabular} } \\ \hline Uniform & – & 2.51 & 3.4 & 2.52 & 3.44 & 2.50 & 3.43 & 2.48 & 3.40 \\ \hline \multirow{3}{*}{MNL-MLE} & 1,000 & 1.87 & 2.65 & 1.96 & 0.50 & 1.73 & 2.50 & 1.95 & 2.41 \\  & 5,000 & 1.86 & 2.63 & 1.96 & 0.50 & 1.71 & 2.48 & 1.96 & 2.37 \\  & 100,000 & 1.86 & 2.62 & 1.95 & 0.52 & 1.71 & 2.47 & 1.96 & 2.36 \\ \hline \multirow{3}{*}{MCCM-EM} & 1,000 & 1.86 & 2.67 & 1.92 & 0.54 & 1.67 & 2.35 & 1.86 & 2.27 \\  & 5,000 & 1.86 & 2.66 & 1.88 & 0.53 & 1.60 & 2.35 & 1.88 & 2.23 \\  & 100,000 & 1.87 & 2.66 & 1.88 & 0.51 & 1.62 & 2.37 & 1.88 & 2.22 \\ \hline \multirow{3}{*}{Gated-Assort-Net} & 1,000 & 2.00 & 3.04 & 1.97 & 0.54 & 1.69 & 2.71 & 2.01 & 2.66 \\  & 5,000 & 1.89 & 2.81 & 1.88 & 0.48 & 1.55 & 2.43 & 1.87 & 2.30 \\  & 100,000 & 1.86 & 2.63 & 1.84 & 0.40 & 1.50 & 2.23 & 1.82 & 2.08 \\ \hline \multirow{3}{*}{Res-Assort-Net} & 1,000 & 2.05 & 3.05 & 2.03 & 0.61 & 1.80 & 2.79 & 2.18 & 2.96 \\  & 5,000 & 1.93 & 2.82 & 1.91 & 0.56 & 1.56 & 2.51 & 1.97 & 2.53 \\ \cline{1-1}  & 100,000 & 1.86 & 2.64 & 1.84 & 0.41 & 1.49 & 2.22 & 1.82 & 2.10 \\ \hline Oracle & – & 1.86 & 2.62 & 1.82 & 0.38 & 1.42 & 2.13 & 1.80 & 2.04 \\ \hline \end{tabular}
\end{table}
Table 4: Feature-free choice modeling: The training data are generated from the model of MNL, MCCM, NP and MMNL, with randomly generated true parameters. The number of samples refers to the number of training samples, and additional 5,000 samples are reserved to validate the training procedure. The reported numbers are the average out-of-sample cross-entropy (CE) loss over 10,000 test samples and 10 random trials. The row Uniform denotes the performance of a uniform distribution for predicting. MNL-MLE implements the standard maximum likelihood estimation to estimate a MNL model, while MCCM-EM implements the expectation maximization algorithm to fit a MCCM model. GAsN and RAsN both have one hidden fully-connected layer. The row Oracle denotes the CE loss of the true model.
of a choice model requires a large sample size when the underlying true model becomes more complex. For example, under the true model of MCCM, NP or MMNL, none of these methods give a satisfactory performance when the sample size \(m=1000\). Finally, the neural choice models perform consistently well in recovering these true models with a large amount of data, \(m=100000\). They have the model capacity to capture complex structures and they are also capable of fitting a simpler true model such as MNL. In comparison, MNL choice model suffers from too little model capacity to capture them. Meanwhile, it is widely acknowledged that the models of NP and MMNL are certainly flexible and can model behaviour realistically, but lack an effective learning/estimation algorithm given the number of variables that are unobservable in their stories, hence making them ill-suited for operational purposes.

### Theoretical Analysis

In this subsection, we briefly discuss some theoretical properties of the neural choice model in terms of its parameter estimation. Throughout this subsection, we will focus on the model of GAsN. We remark that the RAsN can be analyzed in the same way. First, we recall that the neural network model can be written as

\[\mathbf{Y}=g(\mathbf{S};\mathbf{\theta})=(g_{1}(\mathbf{S};\mathbf{\theta}),...,g_{n}(\mathbf{S};\mathbf{ \theta}))^{\top}\]

which maps the assortment \(\mathbf{S}\in\{0,1\}^{n}\) to the choice probability vector \(\mathbf{Y}\in[0,1]^{n}\). The parameters \(\mathbf{\theta}\) are estimated through _empirical risk minimization_ on a dataset

\[\mathcal{D}=\{(i_{1},\mathbf{S}_{1}),....,(i_{m},\mathbf{S}_{m})\}\]

where \((i_{k},\mathbf{S}_{k})\)'s are i.i.d. samples from some unknown distribution \(\mathcal{P}\). Specifically, we define the _risk_ as the negative log-likelihood function (also known as the cross-entropy loss)

\[r((i,\mathbf{S});\mathbf{\theta})\coloneqq-\log g_{i}(\mathbf{S};\mathbf{\theta})\]

where \(g_{i}(\mathbf{S};\mathbf{\theta})\) gives the probability that the \(i\)-th product is chosen under parameter \(\mathbf{\theta}\).

The estimated parameters \(\mathbf{\theta}\) are obtained by maximum likelihood estimation as follows

\[\hat{\mathbf{\theta}}\coloneqq\operatorname*{arg\,min}_{\mathbf{\theta}\in\Theta}\hat {R}_{m}(\mathbf{\theta})\coloneqq\frac{1}{m}\sum_{k=1}^{m}r((i_{k},\mathbf{S}_{k});\bm {\theta})\]

where the function \(\hat{R}_{m}(\cdot)\) is also known as the _empirical risk_. Here the set \(\Theta\) is defined as follows:

\[\Theta\coloneqq\big{\{}\mathbf{\theta}:\|\mathbf{W}_{l}\|_{\infty}\leq\bar{W},\|\mathbf{ b}_{l}\|_{\infty}\leq\bar{b},\text{ for }l=1,...,L\big{\}}\]

where \(\mathbf{W}_{l}\) and \(\mathbf{b}_{l}\) are the weight matrix and the bias vector of the \(l\)-th layer of the neural network. Here for a matrix \(\mathbf{W}\in\mathbb{R}^{n_{1}\times n_{2}}\) we define \(\|\mathbf{W}\|_{\infty}\coloneqq\max_{i=1,..,n_{1}}\sum_{j=1}^{n_{2}}|W_{ij}|\) and for a vector \(\mathbf{b}_{l}\in\mathbb{R}^{n_{1}}\), \(\|\mathbf{b}_{l}\|_{\infty}\coloneqq\max_{i=1,..,n_{1}}|b_{i}|\). The restriction to this bounded set usually improves both the stability of the training procedure and the generalization performance of the neural network.

Also, we define the _expected risk_ as

\[R(\mathbf{\theta})=\mathbb{E}\left[r((i,\mathbf{S});\mathbf{\theta})\right]\]

where the expectation is taken with respect to \((i,\mathbf{S})\sim\mathcal{P}.\) The expected risk can be interpreted as the
expected negative likelihood on some unseen/new data sample. We further define \(\mathbf{\theta}^{*}\) as its minimizer:

\[\mathbf{\theta}^{*}:=\operatorname*{arg\,min}_{\mathbf{\theta}\in\Theta}R(\mathbf{\theta}).\]

The theorem below shows the gap between \(R(\hat{\mathbf{\theta}})\), the expected risk of the learned parameter, and the optimal risk \(R(\mathbf{\theta}^{*})\).

**Theorem 1**.: _The following inequality holds with probability \(1-\delta\),_

\[R(\hat{\mathbf{\theta}})\leq R(\mathbf{\theta}^{*})+\frac{4n}{\sqrt{m}}\left(\bar{b} \cdot\frac{(2\bar{W})^{L}-1}{2\bar{W}-1}+(2\bar{W})^{L}\sqrt{2\log(2n)}\right)+ 5C\sqrt{\frac{2\log(8/\delta)}{m}}\]

_where \(C\) is the upper bound of risk function such that \(|r((i,\mathbf{S});\mathbf{\theta})|\leq C\) holds for all \((i,\mathbf{S})\in\mathcal{D}\) and \(\mathbf{\theta}\in\Theta\)._

Theorem 1 shows the excess risk of the empirical risk minimizer \(\hat{\mathbf{\theta}}\). The proof is based on the standard analysis of neural networks (Wan et al., 2013; Neyshabur et al., 2015; Golowich et al., 2018), but we need to customize it to GAsN due to the special gated operator in the final layer. We defer the proof to Appendix C. By exploring the structure of risk function, we can further bound the distance between the true distribution of choice probability and the estimated choice probability by \(\hat{\mathbf{\theta}}\):

**Corollary 1**.: _The following inequality holds with probability \(1-\delta\),_

\[\mathbb{E}\left[\text{KL}(\mathcal{P}_{\mathbf{S}},g_{.}(\mathbf{S};\hat{\mathbf{\theta}} ))\right]\leq\mathbb{E}\left[\text{KL}(\mathcal{P}_{\mathbf{S}},g_{.}(\mathbf{S};\mathbf{ \theta}^{*}))\right]+\frac{4n}{\sqrt{m}}\left(\bar{b}\cdot\frac{(2\bar{W})^{L} -1}{2\bar{W}-1}+(2\bar{W})^{L}\sqrt{2\log(2n)}\right)+5C\sqrt{\frac{2\log(8/ \delta)}{m}}\]

_where \(C\) is defined as in Theorem 1, \(\mathcal{P}_{\mathbf{S}}\) is defined as the distribution of choice probability given assortment \(\mathbf{S}\), \(\text{KL}(\cdot,\cdot)\) is defined as the Kullback-Leibler divergence over two choice probability distributions and expectation is with respect to \(\mathbf{S}\)._

Proof.: Given an assortment \(\mathcal{S}\), the Kullback-Leibler divergence over the true choice probability \(\mathcal{P}_{\mathbf{S}}\) and \(g_{.}(\mathbf{S};\mathbf{\theta})\) is

\[\text{KL}(\mathcal{P}_{\mathbf{S}},g_{.}(\mathbf{S};\mathbf{\theta}))=\sum_{i\in\mathcal{ S}}P(i|\mathcal{S})\left(\log(P(i|\mathcal{S}))-\log(g_{i}(\mathbf{S};\mathbf{\theta})) \right),\]

where \(P(i|\mathcal{S})\) is the choice probability of \(i\)-th product given assortment \(\mathcal{S}\).

Note that

\[\mathbb{E}\left[-\sum_{i\in\mathcal{S}}P(i|\mathcal{S})\log(g_{i}(\mathbf{S};\mathbf{ \theta}))\right]=R(\theta),\]

where expectation is with respect to the \(\mathcal{S}\), with Theorem 1, we complete our proof. 

The term \(\mathbb{E}\left[\text{KL}(\mathcal{P}_{\mathbf{S}},g_{.}(\mathbf{S};\mathbf{\theta}^{*}))\right]\) is usually referred as _approximation error_, which depends on the capacity of function family \(\{g_{.}(\mathbf{S};\mathbf{\theta}):\mathbf{\theta}\in\Theta\}\), and the remaining two terms with order \(O(1/\sqrt{m})\) are usually together referred as _estimation error_, which depends on the number of training samples \(m\) and also the capacity of function family through \(W\), \(\bar{b}\) and \(L\). In general, with fixed training samples, a larger function family \(\{g_{.}(\mathbf{S};\mathbf{\theta}):\mathbf{\theta}\in\Theta\}\) (i.e., deeper and wider networks) will reduce the approximation error but will increase the estimation error.

### Feature-based Neural Choice Models

Now we extend our neural network models to the setting with features. We first make a distinction between product feature and customer feature:
* Product feature: there is a feature vector \(\mathbf{f}_{i}^{P}\in\mathbb{R}^{d}\) associated with each product \(i\in\mathcal{N}\). We also refer to these features as static features as they usually remain the unchanged over all the samples in the training data \(\mathcal{D}\).
* Customer feature: Sample in \(\mathcal{D}\) represents different customers, each of which is associated a feature vector \(\mathbf{f}_{k}^{C}\in\mathbb{R}^{d^{\prime}}\) for \(k=1,...,m\). With these features, the dataset is augmented as \[\mathcal{D}=\{(i_{k},\mathcal{S}_{k},\mathbf{f}_{k}^{C}),k=1,...,m\}.\] We refer to these features as _dynamic_ features as they may vary across different samples. Modeling-wise, if there are product features that change over time, we can simply view them as dynamic features.

Figure 3: Choice networks with features: The networks take (i) the latent utilities from feature encoder (in Figure 2) and (ii) the assortment vector as inputs.

Figure 2: Feature encoder: Product features and customer features are encoded and then taken inner product to obtain the latent utility for each product. \(d\) is the dimension of the product features, and \(d^{\prime}\) is the dimension of the customer features. Both features are encoded to \(h\)-dimensional latent features, and the latent utilities are obtained by the inner product of the corresponding latent features.
Figure 2 and Figure 3 describe our choice networks with features. The neural networks inherit the architectures in the feature-free setting, but both take an additional vector of inputs which we call latent utilities. Both feature-based networks of GAsN(f) and RAsN(f) encode product features and customer features to obtain one latent utility for each product. The product encoder is an \(L\)-layer fully-connected network shared by all the products (\(L\) is 1 or 2 in our experiments). By default, the number of nodes in the intermediate layers of the encoder is the same as the dimension of the encoder's input. When there is no available customer feature, we can simply treat the customer feature as 1 for all the samples.

**Unnecessity of Static Features**

Table 5 presents a synthetic experiment with only product (static) features to illustrate a new insight for choice modeling with features. The experiment first ignores all the product features and trains GAsN and RAsN as in the feature-free setting. Then it takes into account the product features and trains the feature-based version of the two neural networks. We find that both versions of the networks are capable of recovering the true model from the comparison against the oracle performance, and that the two versions of each neural network achieve a similar performance. This might seem counter-intuitive at first sight, because the product features may contain useful information about the product's utility, and ignoring the features may accordingly hurt the performance. However, we argue that the product utilities are essentially encoded in the choice data \((i_{k},\mathcal{S}_{k})\)'s but not in the product features. In fact, the experiment shows that such utilities can be learned by the neural networks of GAsN and RAsN without explicitly accessing the product features. An alternative interpretation is that the product utility here should be viewed as a population-level utility that reflects the preference or popularity of a product over the whole population. Things become different when customer features are also available. In that case, the choice behaviour of each data sample is determined by the personalized utilities, and thus the product and customer features become indispensable.

We emphasize that whether to include the (dynamic) customer features in choice modeling is determined not by the availability of the features, but by the downstream application. For example, the assortment optimization problem aims to find an assortment that maximizes profits under a choice model. For a brick-and-mortar store or an online retailer where the personalized assortment is not allowed, customer features should not be used to fit the choice model even if they are available, because one is more interested in the population-level choice behavior. It also underscores the potential of feature-free networks of GAsN and RAsN despite their simple architectures.

\begin{table}
\begin{tabular}{c|c c} \hline \hline  & MNL(f) & MCCM(f) \\ \hline MNL(f)-MLE & 2.903 & 2.212 \\ \hline Gated-Assort-Net & 2.921 & 2.010 \\ Gated-Assort-Net(f) & 2.918 & 2.010 \\ \hline Res-Assort-Net & 2.914 & 1.988 \\ Res-Assort-Net(f) & 2.914 & 1.984 \\ \hline Oracle & 2.900 & 1.932 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Choice modeling with only product (static) features: The training data (with \(n=50\), \(m=100,000\), and \(d=5\)) are generated from the feature-based version of MNL and MCCM. All networks have two-layer structures (for the networks with features, the encoder part has one layer). The reported numbers are the average out-of-sample cross-entropy (CE) loss over 10 random trials. The benchmark method implements MLE for the featured-based MNL model. More details about experiment setup can be found in Appendix B.1.
### Comparison on Real Datasets

In this subsection, we test the predictive performances of our neural choice model with several benchmarks on four real datasets, two private datasets from revenue management industries without features and two public ones with features. We repeat all experiments 10 times, choosing the train/validation/test sets independently and randomly, and report the averaged results below. In Section 5.2 we perform extensive numerical experiments on synthetic datasets to jointly test prediction and optimization performance.

#### 4.4.1 Feature-free Neural Choice Model

We conduct numerical experiments on two private real datasets: airline data and retailing data. The airline data contains three flight markets and the offered products are different bundles, i.e., bundle of seat, food etc. The offered assortments are decided by both the selling strategy and remaining seats. The retailing data is collected from a fashion-retailing company, where the products are clothes (from a same category and thus most customers purchase at most one item) and the assortments are decided by the inventories. Both datasets contain records of purchase transactions only, so we add no-purchases as in Simsek and Topaloglu (2018): for each purchase, we create four no-purchases with the same assortment as the original transaction but as choosing the outside no-purchase option. The summary statistics of the dataset is given in Table 6.

The testing result is summarized in Table 7, where the reported numbers are out-of-sample cross-entropy loss of two benchmark methods (with the same configurations as in Table 4) and the two neural network models with one layer.

We observe that our neural networks achieve the best performance on these datasets where the generative model is unknown and likely complex. One interesting observation is unlike other numerical results in this paper, MNL-MLE is better than MCCM-EM for the airline markets. One potential reason for the relatively poor performance of MCCM-EM might be the complicated assortment generation: in Subsection 6.1, we numerically show the performances of MCCM-EM is correlated with the assortment distributions, which are special in the airline markets due to their nested booking-limits selling strategy.

\begin{table}
\begin{tabular}{c|c|c|c|c} \hline  & Flight 1 & Flight 2 & Flight 3 & Retailing \\ \hline \# Samples & 90634 & 37075 & 70600 & 17200 \\ \# Products & 25 & 25 & 25 & 20 \\ \# Train Samples & 60000 & 30000 & 60000 & 10000 \\ \# Validate Samples & 1000 & 1000 & 1000 & 1000 \\ \# Test Samples & 10000 & 5000 & 10000 & 5000 \\ \hline \end{tabular}
\end{table}
Table 6: Description of flights and retailing data.

\begin{table}
\begin{tabular}{c|c|c|c|c} \hline  & Flight 1 & Flight 2 & Flight 3 & Retailing \\ \hline MNL-MLE & 2.798 & 2.490 & 2.547 & 1.073 \\ MCCM-EM & 3.972 & 3.470 & 3.596 & 1.063 \\ \hline Gated-Assort-Net & 2.738 & 2.436 & **2.489** & 1.063 \\ Res-Assort-Net & **2.734** & **2.434** & 2.490 & **1.060** \\ \hline \end{tabular}
\end{table}
Table 7: Performance on the airline data and retailing data. The reported numbers are out-of-sample CE loss.
#### 4.4.2 Feature-based Neural Choice Model

For the case of estimation with features, we perform numerical experiments on two public datasets - SwissMetro and Expedia Search. These two datasets contain dynamic customer/product features; that is, the features associated with each training sample may be different.

The SwissMetro dataset is a public survey dataset to analyze traveller preference among three transportation modes, and the Expedia dataset consists of hotel search records and the associated customer choices. More details of datasets, experiment setup and benchmarks can be found in Appendix B.

We implement a number of benchmark models: MNL, feature-based MNL, TasteNet (Han et al., 2020), DeepMNL (Wang et al., 2020; Sifringer et al., 2020), random forest (Chen et al., 2021). The MNL and feature-based MNL are learned by MLE. TasteNet and DeepMNL model the product utility \(u_{i}\) as a function of the product and customer features and map the utility to choice probability via MNL (3). TasteNet assumes the utility \(u_{i}\) is a linear function of product features and uses the customer feature to determine the coefficients of the linear function, while DeepMNL concatenates the customer and product features and feeds both into a fully-connected network to obtain the utility \(u_{i}\). Both models bring assortment effects only through the MNL part.

From the results in Table 8, our two neural networks give a better performance than the benchmark models. Even though the benchmark models include neural networks in this case, the key difference is that our models explicitly take the assortment as an input. This experiment result shows that even with the presence of product and customer features, incorporating assortment effects can help to better explain customer choices. Comparing the two datasets, the Gated-Assort-Net(f) performs better on SwissMetro, while Res-Assort-Net(f) performs better on Expedia. Recall that the Res-Assort-Net(f) makes a stronger usage of the assortment vector throughout the architecture than Gated-Assort-Net(f). Accordingly, one explanation can be that for the transportation setting, customer choices are less affected by the available options but more by their personal preference, but for the hotel search, the customer choices are more affected by the provided assortment, which gives Res-Assort-Net(f) more of an advantage. As for the random forest model, it is trained as a discriminative model using both features and assortment as input, so we believe it also has better potential with some further model recalibration.

## 5 Assortment Optimization for Neural Choice Model

An important downstream task after the estimation of a choice model is the assortment optimization problem, where the seller aims to find an assortment that maximizes the profits

\[\max_{\mathcal{S}\subset\mathcal{N}}\mathrm{Rev}(\mathcal{S})\coloneqq\sum_{i \in\mathcal{N}}\mu_{i}\mathbb{P}(i|\mathcal{S}) \tag{6}\]

\begin{table}
\begin{tabular}{c|c|c} \hline \hline  & SwissMetro & Expedia \\ \hline MNL-MLE & 0.883 & 2.827 \\ MNL(f)-MLE & 0.810 & 2.407 \\ \hline TasteNet & 0.681 & 2.436 \\ DeepMNL & 0.741 & 2.374 \\ Random Forest & 0.633 & 2.458 \\ \hline Gated-Assort-Net(f) & **0.598** & 2.403 \\ Res-Assort-Net(f) & 0.607 & **2.325** \\ \hline \hline \end{tabular}
\end{table}
Table 8: Performance on the SwissMetro and Expedia. The reported numbers are out-of-sample CE loss.
where \(\mu_{i}\) is the profits/revenue of selling one unit of product \(i\). The problem usually has additional space or cardinality constraints.

The literature on assortment optimization fixes an underlying choice model \(\mathcal{M}\) and devises algorithms with provable performance guarantees. However, in practice, the underlying choice model is unknown a priori and a model has to be selected first and its parameters estimated from data \(\mathcal{D}\). In this case, the suboptimality of a proposed assortment may come from two sources: (i) the approximation error: the inaccuracy of the choice model estimation and (ii) the optimization error: the optimality gap induced by solving the assortment optimization problem under the estimated model. The following bound shows the intuition:

\[\text{Rev}(\mathcal{S}^{*})-\text{Rev}(\hat{\mathcal{S}})\leq \underbrace{\bar{\mu}\cdot\text{dist}(\mathcal{M},\hat{\mathcal{M}})}_{\text{ Model Approx.Error}}+\underbrace{\text{Opt. Gap}}_{\text{Opt. Error}}. \tag{7}\]

Here \(\mathcal{M}\) denotes the true choice model and \(\hat{\mathcal{M}}\) is the choice model estimated from the data \(\mathcal{D}\). \(\mathcal{S}^{*}\) is the optimal assortment obtained from (6) using the true model \(\mathcal{M}\), and \(\hat{\mathcal{S}}\) is obtained from \(\hat{\mathcal{M}}\) via some assortment optimization algorithm, where the term Opt. Gap refers to the suboptimality induced by the algorithm for assortment optimization. The function \(\text{dist}(\cdot,\cdot)\) refers to some (pseudo-)distance function between two models, and \(\bar{\mu}\coloneqq\max_{i\in\mathcal{N}}\mu_{i}\).

We use the inequality (7) to emphasize the decomposition of the revenue gap into the approximation error and the optimization error. The existing works on assortment optimization have striven to improve the optimization error, while assuming the approximation error is zero. We make the case that both errors should be taken into account when measuring the performance of an assortment algorithm as the firm is interested in the combined performance. In light of this, we measure the performance of our neural choice model illustrating the tradeoff between these two errors.

### Integer Programming Formulation of the Neural Choice-based Assortment Optimization

In this section we present a mixed integer programming (MIP) formulation for the assortment optimization problem under the neural choice model. Denote \(\boldsymbol{\mu}=(\mu_{1},...,\mu_{n})^{\top}\in\mathbb{R}^{n}\) as the revenue vector as in (6). Throughout this section, we will focus on the model of GAsN. We remark that the RAsN gives a similar MIP formulation and numerical performance. We formulate the assortment optimization problem under the GAsN as follows,

\[\max \frac{\sum_{i=1}^{n}\mu_{i}z_{0,i}\exp(z_{L,i})}{\sum_{i=1}^{n}z_ {0,i}\exp(z_{L,i})}\] (8) s.t. \[\boldsymbol{z}_{l}-\tilde{\boldsymbol{z}}_{l}=\boldsymbol{W}_{l }\boldsymbol{z}_{l-1}+\boldsymbol{b}_{l},\text{ for }l=1,...,L,\] \[\boldsymbol{0}\leq\boldsymbol{z}_{l}\leq M\boldsymbol{\zeta}_{l}, \ \boldsymbol{0}\leq\tilde{\boldsymbol{z}}_{l}\leq M(\boldsymbol{1}- \boldsymbol{\zeta}_{l}),\text{ for }l=1,...,L,\] \[\boldsymbol{\zeta}_{l}\in\{0,1\}^{n_{l}},\text{ for }l=1,...,L,\] \[\boldsymbol{z}_{0}=(z_{0,1},...,z_{0,n})^{\top}\in\{0,1\}^{n},\]

where the decision variables are \(\boldsymbol{z}_{l}\in\mathbb{R}^{n_{l}}\), \(\tilde{\boldsymbol{z}}_{l}\in\mathbb{R}^{n_{l}}\), \(\boldsymbol{\zeta}_{l}\in\{0,1\}^{n_{l}}\), all for \(l=1,...,L\), and \(\boldsymbol{z}_{0}\in\{0,1\}^{n}.\) The decision variables \(\boldsymbol{z}_{0}\) for the input layer provide a binary representation of the assortment decision, and \(\boldsymbol{z}_{l}\) represents the values of the intermediate layers of the neural network. The equality constraints describe the forward propagation of the neural network layers, and the inputs \(\boldsymbol{W}_{l}\) and \(\boldsymbol{b}_{l}\) are the parameters of the neural network learned from the data. The objective function describes the mapping from the final
layer of the network to the revenue under the assortment \(\mathbf{z}_{0}\).

The (auxiliary) decision variables \(\mathbf{z}_{l}\) and \(\mathbf{\zeta}_{l}\) implement the big-M method (Fischetti and Jo, 2017; Conforti et al., 2014) where the value of \(M\) is a large positive number obtained by some prior estimate. These two jointly ensure that

\[\mathbf{z}_{l}=\left(\mathbf{W}_{l}\mathbf{z}_{l-1}+\mathbf{b}_{l}\right)^{+}.\]

It is well known that the choice of \(M\) affects the running time of solving the MIP (Belotti et al., 2016). In our numerical experiments, we follow the methods in (Fischetti and Jo, 2017) to tighten the upper bound. Also, additional linear constraints can be added to incorporate the cardinality or capacity constraints. When there is a no-purchase option, one may introduce an additional constraint of \(z_{0,n}=1\) (\(n\) being the index of the no-purchase option).

We remark that the constraints of the MIP (8) are all linear, and the objective function resembles that of the MNL model. For a better numerical performance, we replace the exponential function in the objective function with its second-order approximation \(\exp(x)\sim 1+x+x^{2}/2\)(Banerjee et al., 2020). Though the problem (8) cannot be solved in polynomial time as is the case for some of the other models such as MNL and MCCM (when there is no capacity constraint), it does provide a formulation that allows us to take advantage of off-the-shelf integer programming solvers.

### Numerical Study of Joint Estimation and Optimization

In this subsection, we provide a comparative study of the four models (MNL, MCCM, 1 and 2-layer GAsN) along with their corresponding estimation methods and assortment optimization algorithms by running them on a total of 9600 different synthetic problem instances generated by a mix of ground-truth strategies. Since there are no computationally efficient estimation methods for NP and MMNL choice models, we do not include them here. Each problem instance is a combination of historical sales and revenue data, and additional capacity constraints (if any), and the algorithm(s) output a recommended assortment following this data-to-decision pipeline:

\[\mathcal{D}=\{(i_{k},\mathcal{S}_{k}),k=1,...,m\},(\mu_{1},...,\mu_{n}),\text{ Constraints}\ \rightarrow\ \text{Recommended Assortment}\ \hat{\mathcal{S}}.\]

The numerical experiment is to test performance where the true model is unknown and has to be estimated from data.

We generate the training data \(\mathcal{D}\) from four different choice models as ground truth, MNL, MCCM, NP, and MMNL with the number of training samples fixed at \(m=30,000\) for all the trials. The exact optimal assortment \(\mathcal{S}^{*}\) is computed with the knowledge of the generative model, and then the recommended assortment \(\hat{\mathcal{S}}\) is evaluated by the optimality ratio

\[\text{Opt. Ratio}\coloneqq\frac{\text{Rev}(\hat{\mathcal{S}})}{\text{Rev}( \mathcal{S}^{*})}.\]

For each problem instance, we consider both unconstrained and constrained settings and implement several benchmark methods as follows. For the estimation part, we implement the maximum-likelihood estimation to estimate the parameters of the MNL model, and the expectation-maximization algorithm to estimate those of MCCM. We provide more details of the experiments in Appendix D. Below are the assortment optimization algorithms:

* Unconstrained: The estimated choice model is used to solve for a recommended assortment through the assortment optimization specific for the method.
* Constrained:
* Revenue-ordered policy (RO): All the \(n\) products are ordered in an decreasing order by their revenues to form a nested set of assortments and the feasible assortment with largest expected revenue is picked from that nested set (Talluri and Van Ryzin, 2004).
* MNL-MIP: A mixed integer programming is solved based on the estimated MNL model (Mendez-Diaz et al., 2014).
* ADXOpt: The ADXOpt algorithm proposed by Jagabathula (2014) is a greedy algorithm to solve assortment optimization under a general choice model. We implement the algorithm on the estimated Markov chain choice model. In our implementation, we increase the removal limit (the maximal number of times for a product to be removed from the assortment) from 1 in (Jagabathula, 2014) to 5 so as to boost the algorithm performance.
* Markov Chain Capacity-Assort (MC-CA): We implement the algorithm given in Desir et al. (2020) on the estimated Markov chain choice model.
* Neural Network Optimization (NN): We implement the MIP (8) on the fitted GAsN with 1 or 2 layers (denoted as NN(1) and NN(2)).

The results of our numerical experimets are given in Table 9 and Table 10. In general, the neural network with MIP formulation performs well in all underlying choice models achieving the best performance in several cases (averaged over different numbers of products), with and without a capacity constraint; and under MNL choice model, the (averaged) approximation ratios of NN(1) are above 95%, both with and without constraint.

The advantage of a neural network is more apparent when the generative models are complicated: with data generated by NP and MMNL and with constraints, the averaged approximation ratios of NN are 92.55% and 84.79%, where the best averaged performance of the other methods are 88.28% and 67.14%. The relatively poor performances of the other methods can be due to the limited expressive power or the hardness of the subsequent optimization. While the structural limitations of MNL are well known, and hence is not surprising that it cannot approximate NP and MMNL generated data. For the MCCM, Blanchet et al. (2016) show that the worst case error bound of MCCM to approximate a choice probability of an assortment in MMNL is negatively correlated to "the maximum probability that the most preferable product for a random customer from any segment belongs to that assortment" (Theorem 4.2). Thus, when the offered assortment includes many "preferable products", the estimated MCCM may not approximate the underlying MMNL choice model well. Also, constraints make approximation ratios worse by making the optimization harder; this can be observed by comparing Table 9 and Table 10.

Comparing NN(1) with NN(2), NN(1)'s average approximation ratios are better than NN(2)'s in all cases except MCCM with constraint. In fact, we observe the overfitting of NN(2) during the training in some cases, which is a plausible explanation for its worse relative performance.

In summary, when trying to model and optimize the assortment optimization problems, our finding is that, when evaluating joint estimation and assortment optimization, a shallow network may be preferable to a deep neural network, as it reduces overfitting and leads to a more manageable optimization problem. Of course, if the its estimation error is large, one can try adding more layers to enlarge the capacity and expressive power of the neural network.

## 6 Extensions and Discussions

In this section, we discuss several properties of neural choice models gained from our experience with the experiments, including robustness under the assortment distribution's shift in the testing data and the effects on the depth and width of the networks. We also list some tricks for deploying neural choice
models when we need to add new products or the training samples are not sufficient. We defer the details of all experiments in this section to Appendix E.

### Assortment Distribution Shift in the Training Data

A very practical but often neglected aspect of learning a choice model is the assortment distribution in the training data \(\mathcal{D}\), i.e., the distribution of \(\mathcal{S}_{k}\)'s. Figure 4 gives empirical evidence that the learning of an MCCM using the EM algorithm (implementing (Simsek and Topaloglu, 2018)) is affected by the assortment size of the training samples. Intuitively, a smaller assortment size means it will take longer for the Markov chain to hit the exiting state, and thus the E-step needs to impute larger unobserved quantities which may result in larger variance. Alternatively, we find the performance of the neural network model is quite robust in terms of the generalization on the assortment domain. Specifically, in

\begin{table}
\begin{tabular}{c c|c c c|c c c|c} \hline \hline \multicolumn{2}{c|}{Model} & \multicolumn{4}{c|}{MNL} & \multicolumn{4}{c}{MCCM} \\ \(|\mathcal{N}|\) & & 20 & 40 & 60 & Avg. & 20 & 40 & 60 & Avg. \\ \hline \multirow{2}{*}{MNL (MLE)} & RO & 77.82 & 78.03 & 74.47 & 76.77 & 59.81 & 47.37 & 45.46 & 50.88 \\  & MIP & 86.86 & 90.37 & 87.20 & 88.14 & 75.39 & 65.44 & 60.60 & 67.14 \\ \hline \multirow{2}{*}{MCCM (EM)} & MC-CA & 84.96 & 85.06 & 84.49 & 84.84 & 70.54 & 55.26 & 56.17 & 60.66 \\  & ADXOpt & 89.02 & 88.81 & 87.02 & 88.28 & 73.60 & 61.70 & 59.26 & 64.85 \\ \hline \multirow{2}{*}{NN (1-layer)} & MIP & 93.90 & 93.28 & 90.47 & **92.55** & 92.76 & 84.99 & 76.63 & **84.79** \\ NN (2-layer) & MIP & 94.80 & 91.27 & 87.53 & 91.20 & 91.58 & 77.59 & 68.28 & 79.15 \\ \hline \hline \end{tabular}
\end{table}
Table 10: The approximation ratios (\(\times 100\%\)) of heuristics for assortment optimization with capacity constraint. Each reported value is averaged over 100 trials.

\begin{table}
\begin{tabular}{c|c c c|c|c c c|c} \hline \hline \multicolumn{2}{c|}{Model} & \multicolumn{4}{c|}{MNL} & \multicolumn{4}{c}{MCCM} \\ \(|\mathcal{N}|\) & & 20 & 40 & 60 & Avg. & 20 & 40 & 60 & Avg. \\ \hline MNL (MLE) & 99.94 & 99.36 & 99.11 & 99.47 & 93.54 & 94.26 & 94.59 & 94.13 \\ MCCM (EM) & 99.70 & 99.44 & 99.65 & **99.60** & 93.88 & 96.80 & 97.77 & 96.15 \\ NN (1-layer) & 99.71 & 98.49 & 97.80 & 98.67 & 96.93 & 95.78 & 97.68 & **96.80** \\ NN (2-layer) & 98.03 & 91.14 & 86.60 & 91.92 & 96.60 & 90.46 & 91.83 & 92.96 \\ \hline \hline \multicolumn{2}{c|}{Model} & \multicolumn{4}{c|}{NP} & \multicolumn{4}{c}{MNNL} \\ \(|\mathcal{N}|\) & & 20 & 40 & 60 & Avg. & 20 & 40 & 60 & Avg. \\ \hline MNL (MLE) & 99.94 & 99.36 & 99.11 & 99.47 & 93.54 & 94.26 & 94.59 & 94.13 \\ MCCM (EM) & 99.70 & 99.44 & 99.65 & **99.60** & 93.88 & 96.80 & 97.77 & 96.15 \\ NN (1-layer) & 99.71 & 98.49 & 97.80 & 98.67 & 96.93 & 95.78 & 97.68 & **96.80** \\ NN (2-layer) & 98.03 & 91.14 & 86.60 & 91.92 & 96.60 & 90.46 & 91.83 & 92.96 \\ \hline \hline \end{tabular} \begin{tabular}{c|c c c|c|c c c|c} \hline \hline \multicolumn{2}{c|}{Model} & \multicolumn{4}{c|}{NP} & \multicolumn{4}{c}{MNNL} \\ \(|\mathcal{N}|\) & & 20 & 40 & 60 & Avg. & 20 & 40 & 60 & Avg. \\ \hline MNL (MLE) & 99.94 & 99.36 & 99.11 & 99.47 & 93.54 & 94.26 & 94.59 & 94.13 \\ MCCM (EM) & 99.70 & 99.44 & 99.65 & **99.60** & 93.88 & 96.80 & 97.77 & 96.15 \\ NN (1-layer) & 99.71 & 98.49 & 97.80 & 98.67 & 96.93 & 95.78 & 97.68 & **96.80** \\ NN (2-layer) & 98.03 & 91.14 & 86.60 & 91.92 & 96.60 & 90.46 & 91.83 & 92.96 \\ \hline \hline \end{tabular} 
\begin{tabular}{c|c c c|c|c c c|c} \hline \hline \multicolumn{2}{c|}{Model} & \multicolumn{4}{c|}{NP} & \multicolumn{4}{c}{MNNL} \\ \(|\mathcal{N}|\) & & 20 & 40 & 60 & Avg. & 20 & 40 & 60 & Avg. \\ \hline MNL (MLE) & 99.94 & 99.36 & 99.11 & 99.47 & 93.54 & 94.26 & 94.59 & 94.13 \\ MCCM (EM) & 99.70 & 99.44 & 99.65 & **99.60** & 93.88 & 96.80 & 97.77 & 96.15 \\ NN (1-layer) & 99.71 & 98.49 & 97.80 & 98.67 & 96.93 & 95.78 & 97.68 & **96.80** \\ NN (2-layer) & 98.03 & 91.14 & 86.60 & 91.92 & 96.60 & 90.46 & 91.83 & 92.96 \\ \hline \hline \end{tabular} \begin{c|c c|c|c c c|c} \hline \hline \multicolumn{2}{c|}{Model} & \multicolumn{4}{c|}{NP} & \multicolumn{4}{c}{MNNL} \\ \(|\mathcal{N}|\) & & 20 & 40 & 60 & Avg. & 20 & 40 & 60 & Avg. \\ \hline MNL (MLE) & 99.94 & 99.36 & 99.11 & 99.47 & 93.54 & 94.26 & 94.59 & 94.13 \\ MCCM (EM) & 99.70 & 99.44 & 99.65 & **99.60** & 93.88 & 96.80 & 97.77 & 96.15 \\ NN (1-layer) & 99.71 & 98.49 & 97.80 & 98.67 & 96.93 & 95.78 & 97.68 & **96.80** \\ NN (2-layer) & 98.03 & 91.14 & 86.60 & 91.92 & 96.60 & 90.46 & 91.83 & 92.96 \\ \hline \hline \end{tabular}
\end{table}
Table 9: The approximation ratios (\(\times 100\%\)) of heuristics for assortment optimization without capacity constraint. Each reported value is averaged over 100 trials.
Appendix E.1, we train the neural network model with one distribution to generate the assortments \(\mathcal{S}_{k}\)'s (in training data \(\mathcal{D}\)) and find that it performs surprisingly well on the test data where the assortments are generated from a different assortment generating distribution (an aspect of out-of-domain performance).

### Depth and Width of the Neural Networks

Throughout the paper, we use one-layer or two-layer neural networks for our neural choice models, i.e., \(L=1\) or \(2\). We find this provides a sufficiently good fit for both synthetic and real data. For the number of neurons/nodes in the intermediate layers, we set \(n_{l}=n\) for all \(l\)'s by default. In Appendix E.2, we provide more numerical results that the performance of the neural choice models remain stable with more number of layers and wider width for the intermediate layers. Generally, it is not recommended to have many layers for the neural network, as it results in a bad landscape for the loss function and consequently bad local minima that do not generalize well (Keskar et al., 2016).

### Network Augmentation with Warm Start

One drawback of the models proposed in this paper is that they all require the universe of available products to be fixed, i.e., \(\mathcal{N}=\{1,...,n\}\) is fixed. When an additional set of products \(\mathcal{N}^{\prime}=\{n+1,...,n^{\prime}\}\) is available, the model has to be retrained on the new product set of \(\mathcal{N}\cup\mathcal{N}^{\prime}\). One option is to initialize the weights of the new network according to the previously well-trained network of \(\mathcal{N}\) as a warm start. Figure 5 shows that the well-trained network of \(\mathcal{N}\) can provide a good warm start for retraining the new network.

### Neural Choice Model as a Meta Choice Model

From our numerical results in Table 4, we gain a new perspective, namely, to use neural choice model as a meta choice model. Table 4 shows how the neural choice model can benefit from large data size but can also be outperformed when the data size is small. In this case, we can use the estimated choice model with the smallest validation error to generate synthetic data to re-train another neural choice model and fine-tune it by the real training data. The pseudo code is given in Algorithm 1. Intuitively, the strong representation ability of the neural choice model can make its performance arbitrarily close to the well-fitted generative model in the re-training step and can be further improved by real data in the fine-tuning step. Since the re-training step is indeed learning the trained choice model \(\hat{\mathcal{M}}_{k^{*}}\) with the _best_

Figure 4: Smaller assortment size leads to larger estimation error. In the experiment, the total number of products \(n=20\), and the training data consists of \(m=10,000\) samples with a fixed-size assortment \(\|S_{k}\|\). The x-axis gives the assortment size and the y-axis represents the mean error. The plotted curves are based on an average over 10 independent trials.
validation error from all other trained models \(\hat{\mathcal{M}}_{k}\), which has the _best_ training error of the corresponding choice model class \(\mathcal{M}_{k}\), we name this algorithm _Learning the Best of the Bests_. In this way, the neural choice model is not trained as a competitor to the existing choice models, but more as a complementary meta choice model. In Appendix E.4, we implement Algorithm 1 on a public Hotel data (Bodea et al., 2009) as an example.

```
1:Input: Training data \(\mathcal{D}_{\text{train}}\), validation data \(\mathcal{D}_{\text{val}}\), a set of choice models \(\{(\mathcal{M}_{k},\mathcal{L}_{k})\}_{k=1}^{K}\) where each choice model \(\mathcal{M}_{k}\) is equipped with a learning algorithm \(\mathcal{L}_{k}\). \((\mathcal{M}_{1},\mathcal{L}_{1})\) corresponds to neural choice model.
2:%% Training phase
3:for\(k=1,...,K\)do
4:%% We omit the training details such as cross-validation
5: Fit the \(k\)-th candidate choice model using dataset \(\mathcal{D}_{\text{train}}\) and algorithm \(\mathcal{L}_{k}\)
6: Denote the trained model as \(\hat{\mathcal{M}}_{k}\)
7:endfor
8:%% Comparison phase
9: Compute the test performance of all \(\hat{\mathcal{M}}_{k}\)'s using \(\mathcal{D}_{\text{val}}\)
10: Let \(k^{*}\) be the index of the model with best test performance
11:if\(k^{*}=1\)then
12: Let \(\mathcal{M}_{\text{final}}=\hat{\mathcal{M}}_{1}\)
13:else
14:%% Re-training and fine-tuning
15: Generate \(m^{\prime}(=100,000)\) new training samples as \(\mathcal{D}^{\prime}_{\text{train}}\) from the model \(\hat{\mathcal{M}}_{k^{*}}\)
16: Use \(\mathcal{D}^{\prime}_{\text{train}}\) to train (re-train) a neural choice model \(\hat{\mathcal{M}}_{\text{tmp}}\) from scratch
17: Use \(\mathcal{D}_{\text{train}}\) to further train (fine-tune) \(\hat{\mathcal{M}}_{\text{tmp}}\) and obtain \(\hat{\mathcal{M}}^{\prime}_{1}\)
18: Let \(\mathcal{M}_{\text{final}}\) be the better one among \(\hat{\mathcal{M}}^{\prime}_{1}\) and \(\hat{\mathcal{M}}_{k^{*}}\)
19:endif
20:Output: \(\mathcal{M}_{\text{final}}\)
```

**Algorithm 1** Learning the Best of the Bests

## 7 Conclusion and Future Directions

The existing literature of choice modeling in the feature-free and feature-based settings have been largely separate from each other. In this paper, we propose a neural choice model, a unified neural network framework that applies to both settings through a binary representation of the assortment and feature encoders. In addition, we provide a MIP formulation for the assortment optimization for a trained neural

Figure 5: Validation losses using cold start and warm start: The left panel uses \(2,000\) samples for retraining and the right panel uses \(100,000\) samples.
choice model.

From extensive numerical experiments, we gain insights into the performance of the various methods and summarize into the following three points: First, a single neural choice architecture is capable of recovering all the existing choice models, with a standard learning procedure. The neural choice model becomes particularly effective when the underlying model/training data is too complex to be described by a simple model such as MNL and when there is a sufficient amount of training data. Second, the neural choice models are robust under the assortment's distribution shift and can serve as a meta choice model. Third, the combined calibration and assortment optimization regime based on the neural network can outperform other optimization heuristics, especially when the underlying model is complex and when we have capacity constraints; the drawback nevertheless is that as it is based on a MIP, the size of the optimization cannot scale to more than around a hundred products; we need further research and new optimization algorithms to scale beyond this.

Moreover, we conclude with the following two future directions:

* **Multiple-purchase choice model.** The existing choice models focus exclusively on modeling the single-choice behavior where only one product is chosen from the offered assortment. The multiple-purchase choice model is usually studied under an inverse optimization framework in the literature of revealed preference (Zadimoghaddam and Roth, 2012; Amin et al., 2015; Birge et al., 2022). The deep learning-based choice models provide a natural framework to study the multiple-choice behavior, and they can fit to such training data by modifying the loss function.
* **Pricing optimization.** Although we equip our neural networks with an assortment optimization MIP, in a retailing context, the product price is another important factor that affects customer choice. Our current models do not account for this factor but assume prices as fixed. It is an interesting research problem to study optimal pricing under the neural choice model.

### Acknowledgment

The authors thank Antoine Desir for helpful discussions and pointing to the two public datasets for numerical experiments.
