Estimating Heterogeneous Consumer Preferences for Restaurants and Travel Time Using Mobile Location Data

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Estimating Heterogeneous Consumer Preferences for Restaurants and Travel Time Using Mobile Location Data



\title{Estimating Heterogeneous Consumer Preferences for Restaurants and Travel Time Using Mobile Location Data}

\shortTitle{Consumer Preferences for Restaurants}

\author{Susan Athey, David Blei, Robert Donnelly, Francisco Ruiz and
  Tobias Schmidt\thanks{Athey: Stanford University, 655 Knight Way,
    Stanford, CA 94305, [email removed].  Blei: Columbia
    University, Department of Computer Science, New York, NY, 10027,
    [email removed]. Donnelly: Stanford University, 655 Knight
    Way, Stanford, CA 94305, [email removed]. Ruiz: Columbia
    University, Department of Computer Science, New York, NY, 10027,
    [email removed], and University of Cambridge, Department of
    Engineering, Cambridge CB2 1PZ, UK. Schmidt: Stanford University,
    655 Knight Way, Stanford, CA 94305, [email removed]. The authors are listed in alphabetical order. We are
    grateful to SafeGraph and Yelp for providing the data, and to
    Paula Gablenz, Renee Reynolds, Tony Fan, and Arjun Parthipan for
    exceptional research assistance.  We acknowledge generous
    financial support from Microsoft Corporation, the Sloan
    Foundation, the Cyber Initiative at Stanford, and the Office of
    Naval Research. Ruiz is supported by the EU H2020 programme (Marie
    Sk\l{}odowska-Curie grant agreement 706760).}}\pubMonth{May}
\date{\today} \pubYear{2018} \pubVolume{Vol} \pubIssue{Issue}
\begin{abstract}
\small
This paper analyzes consumer choices over lunchtime restaurants using data from a sample of several thousand anonymous mobile phone users in the San Francisco Bay Area. The data is used to identify users’ approximate typical morning location, as well as their choices of lunchtime restaurants. We build a model where restaurants have latent characteristics (whose distribution may depend on restaurant observables, such as star ratings, food category, and price range), each user has preferences for these latent characteristics, and these preferences are heterogeneous across users. Similarly, each item has latent characteristics that describe users’ willingness to travel to the restaurant, and each user has individual-specific preferences for those latent characteristics. Thus, both users’ willingness to travel and their base utility for each restaurant vary across user-restaurant pairs. We use a Bayesian approach to estimation. To make the estimation computationally feasible, we rely on variational inference to approximate the posterior distribution, as well as stochastic gradient descent as a computational approach. Our model performs better than more standard competing models such as multinomial logit and nested logit models, in part due to the personalization of the estimates. We analyze how consumers re-allocate their demand after a restaurant closes to nearby restaurants versus more distant restaurants with similar characteristics, and we compare our predictions to actual outcomes. Finally, we show how the model can be used to analyze counterfactual questions such as what type of restaurant would attract the most consumers in a given location.
\end{abstract}
\maketitle


Where should a a new restaurant be located?  What type of restaurant
would be best in a given location? How close does a competitor need to
be to matter?  These are examples of questions about product
design and product choice.  While there is extensive literature on
consumer response to prices, there is relatively little attention to firm
choices about physical location and product characteristics.  Recent trends
in digitization have led to the creation of many
large panel datasets of consumers, which in turn motivates the development
of models that exploit the rich information in the data and provide precise
answers to these questions.

Answering many of these questions requires a model that incorporates individual-level heterogeneity in
preferences for product attributes and travel time, as these
characteristics might vary substantially even within a city.  More
broadly, understanding individual heterogeneity in travel preferences
is a key input for urban planning.  To this end, we develop an
empirical model of consumer choices over lunchtime restaurants, the
Travel-Time Factorization Model (TTFM).  TTFM incorporates rich
heterogeneity in user preferences for both observed and unobserved
restaurant characteristics as well as for travel time.  We apply the
model to a dataset derived from mobile phone locations for several
thousand anonymized mobile phone users in the San Francisco Bay Area; this is the
first structural model of individual travel choice based on
mobile location data.

TTFM can answer counterfactual questions.  For example, what would
happen if a restaurant with a given set of characteristics opened or
closed in a particular location?  Using data about several hundred
openings and closings of restaurants, we compare TTFM's predictions to
the real outcomes. TTFM can also make personalized
predictions for individuals and restaurants.  Its personalized predictions are more
accurate than existing methods, especially for high-activity
individuals and popular restaurants.

TTFM incorporates recently developed approaches from machine learning
for estimating models with a large number of latent variables.  It
uses a standard discrete choice framework to model each user's choice
over restaurants, inferring the parameters of the users' utility
functions from their choice behavior. TTFM differs from more
traditional models in the number of latent variables; it incorporates
a vector of latent characteristics for each restaurant as well as
latent user preferences for these characteristics.  In addition, it
incorporates heterogeneous user preferences for travel distance, which
vary by restaurant.  These distance preferences are represented as the
inner product of restaurant-specific factors and user willingness to
travel to restaurants with those factors.  Finally, TTFM is a
hierarchical model, where observable restaurant characteristics affect
the distribution of latent restaurant characteristics.  We use a
Bayesian approach to inference, where we estimate posterior
distributions over each user's preferences and each restaurant's
characteristics. The posterior is complex and the dataset is large.
Thus, to make the estimation computationally feasible, we rely on
stochastic variational inference to approximate the posterior
distribution with a stochastic gradient optimization algorithm.

Our approach builds on a large literature in economics and marketing
on estimating discrete choice models of consumer behavior; see
\citet{keane2013panel} for a survey. It also relates to a decades-old
literature in marketing on inferring ``product maps'' from panel data
\citep{elrod1988choice}.  Our estimation strategy is drawn from
approaches developed in \citet{athey2017counterfactual} and
\citet{ruiz2017shopper}, both of which considered the problem of
choosing items from a supermarket, and it also relates to
\citet{Wan2017}, who take a matrix factorization approach to consumer
choice.  Though less well-studied, there has also been some work on
estimating consumer preferences for travel time, e.g.,
\citet{neilsontargeted}'s study of school choice.

\section{Empirical Model and Estimation}

We model the consumer's choice of restaurant conditional on deciding
to go out to lunch.  We assume that the consumer selects the
restaurant that maximizes utility, where the utility of user $u$ for
restaurant $i$ on her $t$-th visit is
\[U_{uit} = \lambda_i + \theta_u^\top \alpha_i + \mu_i^\top \delta_{w_{ut}} - \gamma_u^\top \beta_i \cdot \text{log}(d_{ui}) + \epsilon_{uit},
\]
where $w_{ut}$ denotes the week in which trip $t$ happens, and
$d_{ui}$ is the distance from $u$ to $i$. This gives a parameterized
expression for the utility: $\lambda_i$ is an intercept term that
captures a restaurant's popularity; $\theta_u$ and $\alpha_i$ are
latent vectors that model a user's latent preferences and a
restaurant's latent attributes; $\beta_i$ is a vector that captures a
restaurant's latent factors for travel distance and $\gamma_u$ is a
user's latent preferences of willingness to travel to restaurants with
those factors; $\delta_w$ and $\mu_i$ are latent vectors of
week/restaurant time effects (this allows us to capture varying
effects for different parts of the year); and $\epsilon_{uit}$ are
error terms, which we assume to be independent and identically Gumbel
distributed.  We specify a hierarchical model where observable
characteristics of restaurants, denoted by $x_i$, affect the mean of
the distribution of latent restaurant characteristics $\alpha_i$ and
$\beta_i$. This hierarchy allows restaurants to share statistical
strength, which helps to infer the latent variables of low-frequency
restaurants. We estimate the posterior over the latent model
parameters using variational inference. Our approach is similar to
\citet{ruiz2017shopper}, but differs in a few respects.  First, we
assume that each consumer chooses only one restaurant on a purchase
occasion, so interactions among products are not important.  Second,
TTFM is hierarchical, allowing observed restaurant characteristics to
affect the prior distribution of latent variables.  (See
Appendix~\ref{sec:estdetails} for details.)

For comparison, we also consider a simpler model, a standard
multinomial logit model (MNL), which is a restricted version of our
proposed model: the term $\lambda_i$ is constant across restaurants,
$\alpha_i$ is set to be equal to the observable characteristics of
items, $\theta_u$ is constant across users, $\delta_w$ is omitted (including it created problems with convergence
of the estimation),
and $\gamma_u \cdot \beta_i$ is restricted to be constant across users
and restaurants.

\section{The Data and Summary Statistics}

The dataset is from SafeGraph, a company that collects anonymous, aggregates locational
information from consumers who have opted into sharing their location
through mobile applications.  The data consists of ``pings'' from
consumer phones; each observation includes a unique device identifier that we
associate with a single anonymous consumer, the time and date of the ping, and
the latitude, longitude and accuracy of the ping over a sample period from January
through October 2017.

From this data, we construct the key variables for our
analysis. First, we construct the approximate ``typical'' morning location of the
consumer, defined as the most common place the consumer is found from
9:00 to 11:15 a.m.\ on weekdays.  We restrict attention to consumers
whose morning locations are consistent over the sample period, and for
which these locations are in the Peninsula of the San Francisco Bay
Area (roughly, South San Francisco to San Jos\'{e}, excluding the
mountains and coast). We determine that the consumer visited a
restaurant for lunch if we observed at least two pings more than 3
minutes apart during the hours of 11:30 a.m.\ to 1:30 p.m.\ in a
location that we identify as a restaurant. Restaurants are identified
using data from Yelp that includes geo-coordinates, star ratings,
price range, restaurant categories (e.g., Pizza or Chinese), and we
also use Yelp to infer approximate dates of restaurant openings and
closings.  Last, we narrow the dataset to consumer choices over a
subset of restaurants that appear sufficiently often in the data, and
to consumers who visit a sufficient number of restaurants.  This
process results in a final dataset of 106,889 lunch visits by 9,188
users to 4,924 locations.
Table~\ref{tab:summary} provides summary
statistics on the users and restaurants included in the dataset.
(Appendix~\ref{sec:sampleselect} gives all details about the dataset
processing pipeline.)


\begin{table}
  \caption{Summary Statistics.\label{tab:summary}}

\end{table}
\afterpage{\clearpage}
\section{Estimation and Model Fit}

We divide the dataset into three parts, 70.6 percent training, 5.0 percent
validation, and 24.4 percent testing.
We use the validation dataset to
select parameters such as the length of the latent vectors $\alpha_i$
and $\beta_i$ ($k_1$ and $k_2$, respectively), while we compare models
and evaluate performance in the test dataset.  (See
Section~\ref{sec:fit} for details.) We
select $k_1 = 80$ and $k_2 = 16$.  In the hierarchical prior,
the distribution of a restaurant's components depends on price range,
star ratings, and restaurant
category.

Across several measures evaluated on the test set, TTFM is a better model than
MNL.  For example, precision@5 is the percentage of times that a user's
chosen restaurant is in the set of the top five predicted restaurants.
It is 35\% for TFMM and 11\% for MNL. Further, as shown in
Figures~\ref{fig:fitbyuserdecile} and~\ref{fig:fitbyrestdecile}, TTFM
predictions improve significantly for high-frequency users and
restaurants, while MNL does not exhibit that improvement.  This
highlights the benefits of personalization: When given enough data,
TTFM learns user-specific preferences.

\begin{table}[!h]
  \centering
  \caption{Goodness of Fit of Alternative Models}\label{fig:modelfitsummary}


  \begin{tablenotes}
  Precision measures the share of visits in the set of the top \{1,5,10\} restaurants predicted by the model.
  \end{tablenotes}
\end{table}
\afterpage{\clearpage}
Figure~\ref{fig:demandbydistance} illustrates that both TTFM and MNL
fit well the empirical probability of visiting restaurants at varying
distances from the consumer's morning location.  But
Figure~\ref{fig:predvactual-itemdec} shows that TTFM outperforms MNL
at fitting the actual visit rates of different restaurants; here
restaurants are grouped by their visit-frequency deciles.  The rich
heterogeneity of TTFM allows personalized predictions for restaurants.

\begin{figure}
  \begin{minipage}{\linewidth} \centering
    \includegraphics[width = \textwidth]{./demand-by-distance.pdf}
    \caption{Predicted Versus Actual Shares By Distance}\label{fig:demandbydistance}
  \end{minipage}
  \begin{minipage}{\linewidth} \centering
    \includegraphics[width = \textwidth]{./predvactual-itemdec.pdf}
    \caption{Predicted Versus Actual Shares by Restaurant Visit Decile}\label{fig:predvactual-itemdec}
  \end{minipage}
\end{figure}
\afterpage{\clearpage}
\section{Parameter Estimates}

The distributions of estimated elasticities from TTFM are summarized in Table \ref{tab:distance_elasticities} and
Figure~\ref{fig:elasticity}.  Note that the elasticities in the MNL
vary only because the baseline visit probabilities vary across
consumers and restaurants.  TTFM elasticities are more dispersed,
reflecting the personalization capabilities of the TTFM model.  The
average elasticity across consumers and restaurants (weighted by
trip frequency) is $-1.41$.
Thus, distance matters substantially for lunch, which is consistent with the fact that
roughly 60 percent of visits are within two miles of the consumer's
morning location.
Furthermore, there is substantial heterogeneity in that willingness to
travel. Across users and restaurants, the standard deviation of elasticities in the TTFM model is 0.68, while the average within-user standard deviation of elasticities is 0.30 and the average within-restaurant standard deviation of elasticities is 0.60. Elasticities are substantially less dispersed in the MNL model.



\begin{table}
  \caption{Average Within-Item Elasticities by Restaurant Characteristics, TTFM model.\label{tab:prefdistancebytype_ttfm}}

\end{table}


\begin{table}
  \caption{Average Within-Item Elasticities by City, TTFM model.}\label{tab:prefdistancebycity_ttfm}

\end{table}

\begin{figure}
  \includegraphics[width=.8 \textwidth]{./item_distance_elasticity_by_geo_hash_map.png}
  \caption{Average Within-Item Elasticities by geohash6, TTFM model.} \label{fig:elasticityprefbygeo}
\end{figure}
\afterpage{\clearpage}
Tables~\ref{tab:prefdistancebytype_ttfm} and~\ref{tab:prefdistancebycity_ttfm} and Figure \ref{fig:elasticityprefbygeo} illustrate how elasticities vary across restaurant types and cities. Willingness to travel
is lower for low-priced restaurants (elasticity $-1.45$
for price range \$ (under \$10) versus $-1.37$ for price range \$\$
(\$11$-$\$30)); lower
for Mexican restaurants and Pizza places than for Chinese and Japanese
restaurants (elasticities of $-1.50$ and $-1.50$ versus $-1.35$ and $-1.32$,
respectively). Cities with many work locations nearby retail districts, including San Jos\'{e}, Sunnyvale, and Mountain
View have a lower willingness to travel than cities that are more spread out like Daly City, Burlingame,
San Bruno, and San Mateo.  Appendix
Section~\ref{sec:addlresults} provides further descriptive statistics
about latent factors and model results, illustrating for example how to model can be used to find restaurants that are intrinsically similar
(without regard to location) as well as which restaurants are similar in terms of user utilities.

\section{Analyzing Restaurant Opening and Closing}\label{sec:openclose}

The TTFM model can make predictions about how market share will be redistributed among restaurants when restaurants open or close, and these predictions can be compared to the actual changes that occur in practice.  For this exercise, we focus on 221 openings and 190 closings where, both before and after the change, there
were at least 500 restaurant visits by users with morning locations within a 3 mile radius of the relevant restaurant.  Figure~\ref{fig:openclose} illustrates that restaurant openings and closings are fairly evenly distributed over the time period.

One challenge of analyzing market share redistribution is that for any given target restaurant that opens or closes, we would expect some baseline level of market share changes of competing restaurants due to changes in the open status of neighboring restaurants.  We address this in an initial exercise where we hold the environment fixed in the following way. For each target restaurant that changed status, we first construct the predicted difference in market shares for each other restaurant between the ``closed'' and ``open'' regime (irrespective of which came first in time), and then subtract out the predicted change in market share that would have occurred for each restaurant if the target restaurant had been closed in both periods. We then sum the changes across restaurants in different groups defined by their distance from the target restaurant.   Table~\ref{tab:opcl_distance_redistribution_shares} shows TTFM model predictions for how the opening/closing restaurant's market share is redistributed over other restaurants within certain distances after the restaurant becomes unavailable (i.e.~before the opening or after the closing).  The TTFM model estimates imply that just over 50 percent of the market share impact of a closure accrues restaurants within 2 miles of the target restaurant.

\begin{table}
  \caption{Share of demand redistributed by distance, TTFM model relative to benchmark}\label{tab:opcl_distance_redistribution_shares}

\end{table}


Figure~\ref{fig:OPCL2} compares the actual changes in market share that occured against the predictions of the TTFM model.  It should be noted that baseline changes unrelated to the opening and closing of the target restaurants seem to dominate both the actual and predicted market share changes in the figure.  The figure shows that our model's predictions match well the actual changes that occurred, but it there is substantial variation in the changes that occured in the actual data, making it difficult to evaluate model performance using this exercise.


\begin{figure}
  \includegraphics[width=.8 \textwidth]{./opcl_change_in_mean_market_share_by_distance_final.pdf}
  \caption{Model Predictions Compared to Actual Outcomes for Restaurant Openings and Closings.}\label{fig:OPCL2}
  \begin{figurenotes}
  The figure shows the average of the predicted difference in the market share of each restaurant in the group between the period where the target restaurant is closed and when it is open. The user base for the calculated market shares includes all users whose morning location is within three miles of the target restaurant and who visit at least one restaurant in both periods. We consider only restaurants that appear in the consideration sets of these users at least 500 times in both periods. User-item market shares under each regime (target restaurant open and target restaurant closed) are averaged using weights proportional to each user's share of visits in the group to any location during the open period. The bars in the figure show the point estimates plus or minus two times the standard error of the estimate, which is calculated as the standard deviation of the estimates across the different opening or closing events divided by the square root of the number of events.
  \end{figurenotes}
\end{figure}

Our final exercise considers the best choice of restaurant type for a location.
For the set of restaurants that open or close, we look at how the demand for the restaurant
that changed status (the ``target restaurant'') compares to the
counterfactual demand the model predicts in the scenario where a different restaurant
in our sample (as described by its mean latent characteristics)
is placed in the location of the target restaurant.
For each target, we consider a set of 200 alternative restaurants, 100 from the
same category as the target restaurant and 100 from a different category.
\footnote{These alternatives are sampled with equal probabilities from the set of restaurants in our
  sample.}
We then compare the target restaurant's estimated market share to
the mean demand across the set of alternatives.
In Table~\ref{tab:CF}, we see that both the restaurants that opened and those
that closed on average have higher predicted demand than either group of
alternatives. However, the restaurants that opened appear to be in more
valuable locations, since for the 200 alternative
restaurants, we predict higher average demand if they were (counterfactually) placed
at the opening locations than at the locations of closing restaurants. As a
further comparison, we split the set of alternatives into groups based on whether or not
they are in the same broad category as the restaurant that opened or closed.
We find that alternative restaurants from the same category as the target would
perform better on average than alternatives from a different category.

\begin{table}
  \caption{Alternative Restaurant Characteristics for Opening and Closing Restaurants}\label{tab:CF}

\end{table}

\section{Ideal Locations and Ideal Restaurant Types}\label{sec:marketfit}
In this section, we consider the match between restaurant characteristics and locations.
In each geohash6, we select one restaurant location at
random and use the TTFM model to predict what the total demand would have been
if a different restaurant had been located in its place. The set of alternative
restaurants was chosen to include one restaurant from each of the major
categories in the sample.\footnote{From each category, we randomly selected one
  restaurant whose market share is within $0.1$ standard deviation of the
  mean market share in the full sample.}

In Figure \ref{fig:bestlocation}, we examine
which locations are predicted to provide the largest demand
in the lunch market for each restaurant category. We can see for example that Vietnamese restaurants are
predicted to have the highest demand in a dense region in the southeastern
portion of the map. The demand for Filipino restaurants is relatively diffuse,
whereas the demand for sandwiches is characterized by small but dense pockets of relatively high demand.

In Figure \ref{fig:bestcategory}, we group the restaurant categories into coarse groups
based on the price range and the type of cuisine.  We examine within each group which
category would have the highest total demand in each location. There is
considerable spatial heterogeneity in which restaurant category is predicted to
perform best in each location.

\section{Conclusions}

This paper makes use of a novel dataset to analyze consumer choice: mobile location data.  We propose the TTFM model, a rich model that allows heterogeneity in user preferences for restaurant characteristics as well as for travel time, where preferences for travel time vary across restaurants as well.  We show that this model fits the data substantially better than traditional alternatives, and by incorporating recent advances in Bayesian inference, the estimation becomes tractable.  We use the model to conduct counterfactual analysis about the impact of restaurants opening and closing, as well as to evaluate how the choice of restaurant characteristics affects market share.  More broadly, we believe that with the advent of digitization, panel datasets about consumer location can be combined with rich structural models to answer questions about firm strategy as well as urban policy, and models such as TTFM can be used to accomplish these goals.
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