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Coarse Personalization
\begin{spacing}{1.0}
\title{Coarse Personalization\thanks{Zhang: \protect[email removed],
Misra: \protect[email removed];
We thank the data science and analytics team at the company that is
the source of our data for their help and numerous insights. We are
grateful for the discussions, comments, and suggestions of conference
participants at Marketing Science 2021, TADC 2021, MLESI 2022, QME
2022, ICSDS 2023, and EC 2024.}}
\author{Walter W. Zhang\\
{\normalsize\emph{The Wharton School, University of Pennsylvania}}\and
Sanjog Misra\\
{\normalsize\emph{University of Chicago Booth School of Business}}}
\date{June 30, 2025}
\maketitle
\begin{abstract}
\begin{singlespace}
With advances in estimating heterogeneous treatment effects, firms
can personalize and target individuals at a granular level. However,
feasibility constraints limit full personalization. In practice, firms
choose segments of individuals and assign a treatment to each segment
to maximize profits: We call this the coarse personalization problem.
We propose a two-step solution that simultaneously makes segmentation
and targeting decisions. First, the firm personalizes by estimating
conditional average treatment effects. Second, the firm discretizes
using treatment effects to choose which treatments to offer and their
segments. We show that a combination of available machine learning
tools for estimating heterogeneous treatment effects and a novel application
of optimal transport methods provides a viable and efficient solution.
With data from a large-scale field experiment in promotions management,
we find our methodology outperforms extant approaches that segment
on consumer characteristics, consumer preferences, or those that only
search over a prespecified grid. Using our procedure, the firm recoups
over $99.5\%$ of its expected incremental profits under full personalization
while offering only five segments. We conclude by discussing how coarse
personalization arises in other domains.\vspace{0.5cm}
\end{singlespace}
\begin{singlespace}
\noindent\emph{Keywords}: Personalization, Targeting, Segmentation,
Optimal Transport, Machine Learning
\end{singlespace}
\end{abstract}
\noindent\end{spacing}\thispagestyle{empty}
\newpage{}
\section{Introduction\label{sec:Introduction}}
\setcounter{page}{1}
Improvements in estimating heterogeneous treatment effects enable
firms to target customers on an almost individual level. Leveraging
state-of-the-art machine learning techniques, firms can fully personalize;
individual customers with a unique set of covariates can form their
own segment.\footnote{In one-to-one marketing, a firm's marketing mix is tailored to each
individual \citep{Arora2008}. Personalization based on heterogeneous
treatment effects provides a modern implementation that approximates
one-to-one marketing using a high-dimensional vector of individual
characteristics. If the set of covariates is relevant and large enough,
the two approaches will coincide. } Full personalization involves both tailoring the marketing mix for
each customer and targeting only customers who generate positive incremental
profits for the firm. Consequently, full personalization is profit-maximizing
because individuals are targeted only when their marginal benefit
of targeting outweighs their marginal cost.
However, in practice, firms face barriers implementing these personalization
and targeting procedures. First, firms are concerned about the cost:
As an example, if a small firm sends an identical email to 25,000
customers using an email marketing platform (e.g., MailChimp), it
would need to spend tens of dollars. If that same firm were to send
fully personalized messages, it would need to spend a few thousand
dollars\textemdash that is, two orders of magnitude more. Second,
firms face the potential for unstable estimates: Firms using complex
machine learning methods may be concerned that estimated treatment
effects are noisy and their personalization models could be hindered
by variance. Finally, firms are concerned about downstream fairness
or inequity complaints: As another consequence of noisy estimates,
two similar customers may receive significantly different sets of
marketing mix variables.
More generally, full personalization is limited when there are menu
costs \citep{Sheshinski1977}, concerns about the personalization
model's stability \citep{Hitsch2024}, and issues pertaining to inequity
or fairness \citep{Kahneman1986}. The trade press often points to
the complexities of implementing personalized policies with particular
focus on the high costs of setting up the infrastructure to conduct
such personalization.\footnote{\citet{HarvardBusinessReviewAnalyticServices2018} notes that firms
are rapidly trying to fully personalize their content, but continue
to face implementation costs in doing so. Managers are selectively
choosing which products to personalize, as firms compete with one
another to provide the best personalized experiences for their customers.
\citet{Gilmore1997} observe that it is infeasible for many firms
to offer mass customization because of implementation costs and suggest
that managers should selectively choose which options to offer. }
With these constraints, it is not surprising that firms restrict the
set of unique marketing mix variables, or treatments, to offer. Consequently,
firms must decide to which customers they will assign these treatments.
In essence, firms create smaller sets, or segments, of customers and
optimally choose treatments for each set with the aim of maximizing
profits. We call this problem the \emph{coarse personalization problem}.
The coarse personalization problem is not entirely new. The marketing
literature has long understood the need to segment consumers with
the idea of personalizing the marketing mix for each segment while
keeping costs within bounds. The typical process is sequential: Marketers
first group customers into segments and then personalize marketing
mix treatments for the targeted segments. Indeed, the textbook approach
is to implement a \textquotedblleft discretize then personalize\textquotedblright{}
framework where they form customer segments based on some predefined
source of customer heterogeneity, identify which segments they wish
to target, and then tailor marketing mix variables for those segments
\citep{KotlerP.andKeller2014}. Over time, the sources of customer
heterogeneity that underpin the segmentation procedure have evolved.
Initially, marketing researchers proposed forming segments on demographic
and psychographic variables \citep{Smith1956,Wind1978,Gupta1994}.
More recently, researchers have formed segments based on estimated
consumer preferences and responses \citep{Kamakura1989a,Bucklin1992,Krieger1996,Bucklin1998a}
via either continuous or discrete mixtures \citep{Grover1987,Jain1990}.
In practice, the ``discretize then personalize'' framework is broadly
used in marketing, especially in customer relationship management
(CRM), database marketing, and direct marketing, where marketers first
form segments before making targeting decisions. While the literature
has evolved considerably in terms of the information it uses, the
structure of the framework and the sequence of steps have essentially
remained the same.
In this paper, we show that implementing these steps sequentially
is not necessarily profit maximizing: The initial segmentation procedure
is based on a distance metric that is statistical (e.g., Euclidean)
rather than economic in nature (e.g., based on profit maximization).
To address this problem, we recharacterize the coarse personalization
problem as an optimal transport problem and derive a theoretically
rigorous yet practically implementable solution. In doing so, we draw
on modern techniques from both the incrementality-based targeting
literature and the optimal transport literature. In particular, we
advocate for a \textquotedblleft personalize then discretize\textquotedblright{}
approach where we \emph{first} construct personalized policies based
on heterogeneous treatment effects obtained using now widely available
machine learning tools, and in a \emph{second }step\emph{,} we use
these policies to construct segments of consumers by minimizing profit
regret. Crucially, in the second step we \emph{simultaneously} determine
each segment\textquoteright s optimal treatment and customer assignment
to maximize profits. Put simply, this approach inverts the \textquotedblleft discretize
then personalize\textquotedblright{} approach by first searching for
what works for each customer and then deciding on a coarsened marketing
mix offering that would not significantly degrade profits relative
to full personalization.
Our two-step framework solves the coarse personalization problem precisely
because it is able to segment consumers based on a profit-relevant
distance metric. As such, we view our key contribution to be the design
of the second step that allows for the construction and implementation
of this solution. The first step's machine learning-based personalization,
done by exploiting heterogeneous treatment effects, is now relatively
straightforward from the literature. On the other hand, the issue
of coarsening these to satisfy the problem is novel and poses challenges
on multiple fronts. The combinatorial nature of the assignment problem
coupled with the optimization of the treatment policy makes the problem
especially difficult.
Our proposed solution coarsens the personalization policy to account
for the constraint that policy space is limited to a finite number
of segments. We use optimal transport methods that minimize the regret
between the profits that accrue from the full personalization solution
to those from any proposed coarsened solution. The regret minimizing
solution guarantees that we attain the highest possible profits while
incorporating the additional constraint on the number of segments
available.
To be clear, while one might be able to solve the coarse personalization
problem by brute force, the complexity of the problem is exponential
in the number of people and suffers from the curse of dimensionality.
Instead, we propose a computational solution that adapts Lloyd\textquoteright s
Algorithm \citep{Lloyd1982} and leverages the convexity of the problem.
Our algorithm is scalable and provides a transparent visualization
of the solution's procedure.
In our empirical application, we use data from a food delivery platform
experimenting with its promotions management. We find that our solution
significantly outperforms traditional marketing procedures that sequentially
segment on demographics, RFM variables, and consumer preferences in
generating profits. Further, we find that if the firm issues only
five unique optimized treatments, our solution recovers over $99.5\%$
of the full personalization's expected incremental profits. These
results illustrate the extent to which our \textquotedblleft personalize
then discretize\textquotedblright{} approach outperforms the traditional
\textquotedblleft discretize then personalize\textquotedblright{}
approach.
Substantively, our solution lets us interpolate between no personalization
to full personalization and evaluate the surplus at each level of
personalization. While full personalization is well-studied in the
literature, the gap between full personalization and blanket targeting
is not.\footnote{The literature is divided on how personalization affects surplus.
\citet{Bergemann2011} theoretically show as firms personalize more
granularly, both producer surplus and total surplus increase. \citet{Dube2017}
empirically find that consumer surplus changes nonmonotonically as
firms personalize more granularly in their application.} Using our framework, we empirically document the impact personalization
has on consumer and producer surplus. Coarser personalization reduces
producer surplus due to the extra constraint on the firm's optimization,
but it can boost consumer surplus for some individuals by assigning
them treatment levels that differ from those under full personalization.
We document that the impact of personalization on consumer welfare
is ambiguous in our application.
Lastly, we discuss how the coarse personalization problem naturally
arises in other marketing settings. In salesforce contract design,
managers need to determine which geographic blocks for salespeople
to exert effort in, how much time to allocate, and whether to call
or visit in person. Advertising designers need to choose which set
of advertisements to send out to their customer base. Pricing managers
can choose to use nudges or price subsidies to influence consumer
behavior. All of these marketing problems can be tackled with our
framework.
Our application contributes to the promotions management literature.
Traditionally, price promotions are personalized based on customer
heterogeneity \citep{Rossi1996,Shaffer1995}. With the advent of online
marketplaces, promotions are more readily personalized, customized,
and distributed to customers \citep{Ansari2003,Zhang2004}. The optimization
angle to promotions management has been explored in the literature
\citep{Duvvuri2007,Zhang2009}, but promotions are only optimized
after first forming segments. Most recently, marketers have used tools
from causal machine learning to target and personalize promotions
based on customers' heterogeneous treatment effects \citep{Ascarza2018,Simester2020,Rafieian2021,Ellickson2022,Yoganarasimhan2020,Zhang2023,Hitsch2024}.
The operations management literature has looked at optimal segmentation
strategies without fully considering the causal impact of the marketing
mix \citep{Cui2022,Aouad2023}. Our framework integrates the classical
optimization approach with modern techniques leveraging heterogeneous
treatment effects. We also provide a framework for incrementality-based
personalization for vector treatments, which, to our knowledge, has
not been explored in the marketing literature. To summarize the contributions
of this paper, we have (1) highlighted the problem of optimal coarsening
and (2) examined this problem for multiple interconnected treatments
that create methodological challenges, and (3) provided a solution
via optimal transport.
Our paper additionally provides a novel application of the optimal
transport literature to the marketing literature. Optimal transport
problems have been applied to discrete choice models under the mass
transport framework \citep{Chiong2016,Bonnet2017}, matching markets
\citep{Galichon2011}, quantile regression \citep{Carlier2016}, and
bounding regression discontinuity design estimates \citep{Daljord2019};
its general applications to economics are surveyed in \citep{Galichon2016}.
Computational methods for evaluating optimal transport problems are
detailed in \citep{Peyre2019}, and a survey of recent mathematical
developments in the field can be found in \citet{Villani2009}.
Section \ref{sec:Methodological-overview} overviews the proposed
methodology. Section \ref{sec:Model} formally presents our coarse
personalization solution: the personalization in the first step is
described in Section \ref{subsec:Treatment-effects-estimation} and
the discretization in the second step is detailed in Section \ref{subsec:Optimal-transport}.
Our computational solution that adapts Lloyd's Algorithm is described
in Section \ref{sec:Computational-solution} and its comparison to
standard brute-force grid search is provided in Section \ref{subsec:Grid-search-comparison}.
We then discuss why using optimal transport is effective for coarse
personalization in Section \ref{sec:Why-optimal-transport}. We apply
our methodology to an empirical application for promotions management
for a large food delivery firm in Section \ref{sec:Empirical-Application}.
The surplus analysis is discussed in Section \ref{sec:Surplus-Analysis}.
We discuss how our framework can be adapted to address other classic
marketing problems in Section \ref{sec:Discussion}.
\section{Methodological overview\label{sec:Methodological-overview}}
Simply put, marketers maximize profits by simultaneously forming segments
and choosing the segments' marketing mix. We first demonstrate that
our ``personalize then discretize'' approach yields higher profits
compared to the classical approaches that ``discretize then personalize,''
where segments and their assigned marketing mix, or treatments, are
selected sequentially rather than simultaneously. Then, we demonstrate
that solving the two components simultaneously is difficult in practice,
which motivates our recasting of the problem in an optimal transport
framework.
Intuitively, the simultaneous approach yields higher profits than
the sequential one because both segmentation and targeting rules are
chosen jointly. To demonstrate this idea, we work through the following
stylized example.
We first denote individuals' covariates as $\boldsymbol{x}_{i}$,
the segmentation rule that assigns individuals to segments as $S(\boldsymbol{x}_{i})$,
and the function that maps segments to their assigned treatment as
$T(S(\boldsymbol{x}_{i}),\boldsymbol{x}_{i})$.
The classical sequential approach forms segments a priori based on
consumer characteristics and preferences \citep{KotlerP.andKeller2014}.
We denote a priori segmentation rule as $S^{0}(\boldsymbol{x}_{i})$.
After segments are formed, the marketer then solves for the optimal
treatment values $T(S^{0}(\boldsymbol{x}_{i}),\boldsymbol{x}_{i})$
by taking the a priori segmentation as given.
In contrast, our approach solves the two problems simultaneously.
We define profits as $\Pi(S(\boldsymbol{x}_{i}),T(\boldsymbol{x}_{i},S))$.
Our solution simultaneously solves for the segmentation rule $S(\mathbf{x}_{i})$
and chooses treatments by $T(S(\boldsymbol{x}_{i}),\boldsymbol{x}_{i})$
to maximize profits. We now show the simultaneous approach yields
higher profits,
\begin{equation}
\underbrace{\max_{S,T}\Pi(S,T(S))}_{\text{Simultaneous approach profits}}\geq\underbrace{\max_{T}\Pi(S^{0},T(S^{0}))}_{\text{Sequential approach profits}}\label{eq:Toy-Example}
\end{equation}
where we suppressed the dependence on $\boldsymbol{x}_{i}$ in our
notation. The left-hand side in Equation \ref{eq:Toy-Example} represents
profits from the simultaneous approach where the profits are maximized
to both arguments. The right-hand side represents the sequential approach
where treatments are chosen only after segments $S^{0}$ are formed.
The inequality holds by the property of the maximization operator
over $S$, and it will bind if and only if the a priori segmentation
is ex post optimal for profit maximization.\footnote{Specifically, only if $S^{0}(\mathbf{x}_{i})\in\operatorname*{argmax}_{S'}\max_{T}\Pi(S'(\mathbf{x}_{i}),T(S'(\mathbf{x}_{i})))$.}
Hence, the simultaneous approach will be an upper bound in profits
for \emph{any} sequential segmentation procedure. In our empirical
application, we validate this intuition by evaluating our simultaneous
approach to different sequential benchmarks.
While our simultaneous approach is analytically more profitable, in
practice solving it is quite difficult. As an example, forming three
segments for fifteen individuals, each segment receiving a promotion
from ten possible dollar-off or percentage-off values, yields a total
of ${10\times2 \choose 3}3^{15}$ combinations, which is over $16$
billion combinations. Even in this stylized example, we see that
the search space is combinatorially large. For modern digital marketing
applications with many treatment arms that are issued to millions
of customers, it only gets more complex.
Modern approaches attempt to solve the problem by testing each discrete
treatment arm in an A/B test, evaluating its heterogeneous treatment
effects, and forming segments among the extant treatment arms to maximize
profits \citep{Hitsch2024}. Policy learning methods can directly
learn the most profitable treatment to offer from the experiment.
In our running example, we have ten different promotional values tested
by the firm for both dollar and percentage off promotions. Effectively,
we have twenty treatment arms for heterogeneous treatment effects
estimation or policy learning. Segments are then formed by assigning
individuals to their best profit-maximizing treatment arm by either
using heterogeneous treatment effects or policy learning.
However, in both of these approaches, the continuity of the treatment
variables is ignored. In our application, this means the dollar off
and percentage off promotions are treated as discrete instead of as
continuous. Ignoring the continuity of the treatments has two implications:
First, the choice of which segments to offer is still combinatorially
large and suffers from the curse of dimensionality. In our running
example, to offer three segments, the marketer needs to search over
${10\times2 \choose 3}=1,140$ combinations. Second, many marketing
mix variables, such as price, promotional value, and product attributes,
are fundamentally continuous. When a firm only looks at discrete values,
it forgoes possible profits by ignoring values between those from
the experiment. We benchmark these approaches in our empirical application
and document the loss in profits by ignoring the continuity of the
treatment effects.
To mitigate the complexity of the problem, we leverage tools from
optimal transport and convex analysis to provide a scalable solution
with continuous treatments. We show that (1) the segmentation problem
can be written as a semidiscrete optimal transport problem, where
we map individuals to segments, and (2) the problem is convex in the
treatment values. These results ensures our computational solution
adequately solves the problem.
However, we make two conventional assumptions to ensure the problem
is tractable. First, we assume that individuals have diminishing sensitivity
in each type of treatment\textemdash i.e., people are less sensitive
to changes at higher promotional values. Second, we assume the firm's
cost function for issuing larger promotions has no economies of scale\textemdash i.e.,
the firm does not have lower marginal cost at higher promotional values.
These two assumptions afford us the convexity of the problem and ensure
each individual is deterministically assigned to a segment. In the
next section, we provide details of our methodology.
\section{Model\label{sec:Model}}
We solve the coarse personalization problem in two steps. The first
estimates treatment effects from a randomized controlled trial (RCT).
The second chooses which treatments to offer and their segmentation
rule using an optimal transport framework. The first step is described
in Section \ref{subsec:Treatment-effects-estimation} and the second
step is detailed in Section \ref{subsec:Optimal-transport}. We emphasize
that the second step is a novel application of optimal transport,
and the first step is presently used for full personalization. We
provide an overview of the mathematics behind the second step in Appendix
Section \ref{sec:Mathematics-of-Coarse-Personalization}.\textbf{}
At a high level, the firm chooses to offer $L$ segments that are
each assigned to a unique treatment. The assigned treatments represent
a marketing mix variable and are non-zero in only one dimension $d\in\{1,\ldots,D\}$
(e.g., only one type of promotion is given to a segment). The coarse
personalization solution solves for profit-maximizing segments by
simultaneously finding (1) which treatments to offer and (2) which
individuals $i\in\{1,\ldots,N\}$ to assign to each treatment.
We now define the notation for our setup. For individual $i$, we
denote the individual characteristics as $\boldsymbol{x}_{i}$ and
the firm's outcome measure as $Y_{i}$. Treatments are finite $D$-dimensional
and can be thought of as marketing mix variables. The treatment vector,
for individual $i$, is denoted as,
\begin{equation}
\boldsymbol{t}_{i}=(t_{i,1},\ldots,t_{i,d},\ldots,t_{i,D}),
\end{equation}
where $t_{i,d}$ is the treatment level in dimension $d$. If the
firm only sends one type of marketing mix variable, treatments are
nonzero in only one dimension and have the format $(0,\ldots,t_{i,d},\ldots,0)$
where $t_{i,d}\neq0$. We denote these treatments as \emph{feasible.}\footnote{In our application, feasible treatments implies that each segment
receives either a dollar-off coupon (e.g., \$2 off) or a percentage-off
coupon (e.g., 5\% off), but not both. However, our theoretical results
continue to hold if treatments simultaneously vary across multiple
dimensions. Section \ref{sec:Discussion} discusses applications with
this generalization.} A vector of zeros for $\boldsymbol{t}_{i}$ represents the no treatment
case. In our empirical application, there are two treatment dimensions\textemdash dollar
off and percentage off coupons.
The cost of issuing treatment $\boldsymbol{t}_{i}$ is $c(\boldsymbol{t}_{i})$.
We define the cost of issuing feasible treatment $\boldsymbol{t}_{i}$
that is nonzero in dimension $d$, or $t_{i,d}$, as
\begin{equation}
c_{d}(t_{i,d})\triangleq c(T_{1,1}=0,\ldots,T_{i,d}=t_{i,d},\ldots,T_{i,D}=0),
\end{equation}
and the cost of not targeting is normalized to be zero, $c_{d}(0)=c(\mathbf{0})=0$.\footnote{We use capital letters to denote random variables, $T_{i,d}$, and
lowercase letters to denote their realizations, $t_{i,d}$.} Then, the firm's expected return from assigning treatment $t_{i}$
to individual $i$ is the expected outcome minus the cost of treatment,
\begin{equation}
E[\text{R}_{i}|\boldsymbol{x}_{i},t_{i,1},\ldots,t_{i,d},\ldots,t_{i,D}]=E[Y_{i}|\boldsymbol{x}_{i},t_{i,1},\ldots,t_{i,d},\ldots,t_{i,D}]-c(\boldsymbol{x}_{i},t_{i,1},\ldots,t_{i,d},\ldots,t_{i,D}).
\end{equation}
In the case for \emph{feasible treatment} $\boldsymbol{t}_{i}$ that
is nonzero in dimension $d$, we have $E[\text{R}_{i}|\boldsymbol{x}_{i},T_{i,1}=0,\ldots,T_{i,d}=t_{i,d},\ldots,T_{i,D}=0]=E[Y_{i}|\boldsymbol{x}_{i},T_{i,1}=0,\ldots,T_{i,d}=t_{i,d},\ldots,T_{i,D}=0]-c_{d}(t_{i,d}).$
\subsection{Step 1: Personalize with heterogenous treatment effects\label{subsec:Treatment-effects-estimation}}
We first estimate continuous, conditional average treatment effect
(CATE) functions from the experiment. This gives us a continuous response
surface that we will coarsen in the second step. The continuous CATEs
represent the heterogeneous treatment effects and allow us to construct
the full personalization benchmark.
We consider an RCT setting where the randomization is over the treatment
values and dimensions. For example, in our empirical application,
the RCT in our study has two dimensions of treatments (dollar off
and percentage off promotions), different levels over which the promotions
are randomized, and only one dimension of promotion assigned to each
customer in the experiment.\footnote{Each customer in the experiment received a dollar off promotion, a
percentage off promotion, or was in the control group and did not
receive a promotion.} Formally, we assume that each treatment is randomized between $[0,\bar{t}_{d}]$,
where $\bar{t}_{d}\in\mathbb{R}^{+}$ represents the upper bound of
the treatment value in each dimension $d$, and the treatment vector
$\boldsymbol{t}_{i}$ has domain over $[0,\bar{t}_{1}]\times\cdots\times[0,\bar{t}_{d}]\times\cdots\times[0,\bar{t}_{D}]$.
We can compute the CATEs for each of the treatments relative to the
control (or no treatment) arm under standard assumptions \citep{Imbens2015}.
Since our treatments are feasible, we can compute the continuous CATE
separately for each dimension given $\boldsymbol{x}_{i}$. We define
the continuous CATE in dimension $d$ as $\tau_{d}(\boldsymbol{x}_{i},t_{i,d})$
for an individual with covariates $\boldsymbol{x}_{i}$ given treatment
$t_{i,d}$. Individuals are proxied by their set of covariates $\boldsymbol{x}_{i}$.
For each treatment dimension $d$ and set of covariates $\boldsymbol{x}_{i}$,
firms will choose the treatment level that yields the highest expected
profits. We can solve the firm's problem separately in each dimension
because we prohibit treatments of different dimensions to be simultaneously
administered (feasible treatments).
Without loss of generality, we consider a continuous treatment that
is nonzero in dimension $d$ or $t_{i,d}$. The firm will target the
customer $i$ with treatment $t_{i,d}>0$ if and only if the treatment
yields positive incremental expected returns over the no treatment
case ($t_{i,d}=0$),
\begin{align}
E[\text{R}_{i}|\boldsymbol{x}_{i},T_{i,1}=0,\ldots,T_{i,d}=t_{i,d},\ldots,T_{i,D}=0] & >E[\text{R}_{i}|\boldsymbol{x}_{i},T_{i,1}=0,\ldots,T_{i,d}=0,\ldots,T_{i,D}=0]\nonumber \\
E[Y_{i}|\boldsymbol{x}_{i},0,\ldots,t_{i,d},\ldots,0]-c_{d}(t_{i,d}) & >E[Y_{i}|\boldsymbol{x}_{i},0,\ldots,0]\nonumber \\
E[Y_{i}|\boldsymbol{x}_{i},0,\ldots,t_{i,d},\ldots,0]-E[Y_{i}|\boldsymbol{x}_{i},0,\ldots,0] & >c_{d}(t_{i,d})\nonumber \\
\tau_{d}(\boldsymbol{x}_{i},t_{i,d}) & >c_{d}(t_{i,d}),
\end{align}
where $\tau_{d}(\boldsymbol{x}_{i},t_{i,d})\triangleq E[Y_{i}|\boldsymbol{x}_{i},0,\ldots,t_{i,d},\ldots,0]-E[Y_{i}|\boldsymbol{x}_{i},0,\ldots,0]$
is the expected incremental outcome from issuing treatment $t_{i,d}$.
The treatment will be administered to an individual with covariates
$\boldsymbol{x}_{i}$ if $\tau_{d}(\boldsymbol{x}_{i},t_{i,d})>c_{d}(t_{i,d})$,
or, equivalently, if the continuous CATE is greater than the cost
of treatment.
In each dimension $d$ of treatment, the firm chooses the optimal
treatment level $t_{i,d}^{*}$ for individual $i$ by solving the
program,
\begin{equation}
t_{i,d}^{*}\in\operatorname*{argmax}_{t_{i,d}\in[0,\bar{t}_{d}]}~\tau_{d}(\boldsymbol{x}_{i},t_{i,d})-c_{d}(t_{i,d}),\label{eq:Firm-Program-Treatment}
\end{equation}
which optimizes the firm's expected returns over treatment $t_{i,d}$
by maximizing the difference between marginal revenue ($\tau_{d}(\boldsymbol{x}_{i},t_{i,d})$)
and marginal cost ($c_{d}(t_{i,d})$).
The results from the first step provide the firm enough information
to fully personalize: For individual $i$, the firm assigns treatment
$t_{i,d}^{*}$ that yields the highest return across possible treatments
dimensions ($\{t_{i,d}^{*}\}_{d=1}^{D}$) to do so. We can use off-the-shelf
machine learning algorithms to estimate the continuous CATEs \citep{Athey2016,Wager2018,Farrell2020,Farrell2021}.
Full personalization represents an upper bound on profits by the principle
of maximization in Equation \ref{eq:Firm-Program-Treatment}. In our
empirical application, we benchmark our coarse personalization solution
to full personalization to quantify lost profits due to coarsening.
\subsection{Step 2: Discretize via optimal transport\label{subsec:Optimal-transport}}
In the coarse personalization problem, the firm can form at most $L$
segments to personalize its marketing mix for its customers. To effectively
account for this constraint, we use tools from transport theory. In
our solution, we need to determine (1) the treatments to offer and
(2) the assignment rule of individuals to their treatment to form
segments.
We first introduce some notation to formally define the problem. We
denote $\mathscr{L}$ as the set of treatments the firm offers and
the cardinality of this set is the number of segments ($|\mathscr{L}|=L$).\footnote{The firm will generally not choose less than $L$ unique treatments
because it would weakly hinder the firm's ability to target more granularly.} We assume the number of customers ($N$) is greater than the number
of segments ($L$), i.e., $L<N$. Naturally, as $L\to N$ the firm
recovers full personalization and when $L=1$ the same treatment is
offered, or blanketed, to all individuals.
The firm then chooses which treatments to offer and which consumers
the treatment is assigned to. Under the feasible treatment constraint,
segment $l\in\mathscr{L}$ is assigned a treatment vector with structure
$\boldsymbol{t}_{l}=(0,\ldots,t_{d},\ldots,0)$ that is nonzero in
dimension $d\in\{1,\ldots,D\}$. From our first step in Section \ref{subsec:Treatment-effects-estimation},
the firm has already estimated the continuous CATEs, $\{\{\tau_{d}(\boldsymbol{x}_{i},t_{i,d})\}_{d=1}^{D}\}_{i=1}^{N}$,
and optimal treatment levels $\{\{t_{i,d}^{*}\}_{d=1}^{D}\}_{i=1}^{N}$
for each individual and dimension of treatment.
We now formulate the firm's coarse personalization problem as an optimal
transport problem that simultaneously selects customer segments and
their corresponding assigned treatments. Appendix Section \ref{sec:Mathematics-of-Coarse-Personalization}
provides an overview of the mathematics behind our optimal transport
framework.
We first lay the groundwork for the problem. We define $P$ as the
distribution over the domain $\mathcal{X}=[0,\bar{t}_{1}]\times\cdots\times[0,\bar{t}_{d}]\times\cdots\times[0,\bar{t}_{D}]\subset\mathbb{R}^{D}$.
Individuals are sufficiently represented by their continuous CATE
function, and $P$ represents the distribution of these continuous
CATEs.
We then define $\tilde{\mathbf{t}}=\{\tilde{\boldsymbol{t}}_{1},\tilde{\boldsymbol{t}}_{2},\ldots,\tilde{\boldsymbol{t}}_{L}\}$
as the $L$ assigned treatments for the $L$ segments. Each component
of $\tilde{\mathbf{t}}$, denoted as $\tilde{\boldsymbol{t}}_{l}$,
represents the treatment vector assigned to segment $l$ and has $D$
elements. Specifically, $\tilde{\mathbf{t}}$ consists of $L$ points
within the domain $[\boldsymbol{0},\bar{\boldsymbol{t}}_{1}]\times\cdots\times[\boldsymbol{0},\bar{\boldsymbol{t}}_{l}]\times\cdots\times[\boldsymbol{0},\bar{\boldsymbol{t}}_{L}]\subset\mathbb{R}^{LD}$.\footnote{We define $\bar{t}_{d},\forall d\in D$ as the upper bound that a
treatment can take in that dimension. We also define $\bar{\boldsymbol{t}}_{l}=[\bar{t}_{1}~\cdots~\bar{t}_{D}]$
to be a vector of dimension $D$ that represents treatment $l$'s
upper bound in each dimension of treatment.}
We further define $Q^{\tilde{\mathbf{t}},\mathbf{q}}$ as the discrete
distribution representing the $L$ segments. The segments are assigned
treatments $\tilde{\mathbf{t}}$ and have segment sizes given by the
vector $\mathbf{q}$. Specifically, $Q^{\tilde{\mathbf{t}},\mathbf{q}}$
is defined over the set of $L$ points that assigns probability mass
$q_{l}$ to each treatment $\tilde{t}_{l}$ and $\sum_{l=1}^{L}q_{l}=1$.
Here, $q_{l}$ represents the relative size of the segment $l$, and
$\boldsymbol{q}$ is a vector of segment sizes $q_{l}$ and has length
$L$.
Lastly, we define $\pi\in\mathcal{M}(P,Q^{\tilde{\mathbf{t}},\mathbf{q}})$
to be the assignment rule of individuals to their segment or the optimal
transport plan. Specifically, $\pi$ represents the allowable couplings
between $P$ and $Q^{\tilde{\mathbf{t}},\mathbf{q}}$ where $\mathcal{M}(P,Q^{\tilde{\mathbf{t}},\mathbf{q}})$
is the set of possible joint probability distributions. These couplings
represent the mapping between the distributions of treatment effects
$P$ and segments $Q^{\tilde{\mathbf{t}},\mathbf{q}}$. Substantively,
the couplings are the assignment rule of individuals to their segment.
Figure \ref{fig:OT-Semidiscrete} provides a stylized overview of
this problem. To recap the notation, $\tilde{\mathbf{t}}$ are the
treatments that the firm needs to choose and $\pi$ represents the
segmentation rule of individuals of $\boldsymbol{x}_{i}$ to their
segment. For segment $l$, it is assigned treatment $\tilde{\boldsymbol{t}}_{l}$
and has size $q_{l}$.
The optimal transport problem is
\begin{equation}
\mathcal{W}(\tilde{\mathbf{t}},\mathbf{q})\triangleq\min_{\pi\in\mathcal{M}(P,Q^{\tilde{\mathbf{t}},\mathbf{q}})}E_{\pi}[C(X,T)]=\min_{\pi\in\mathcal{M}(P,Q^{\tilde{\mathbf{t}},\mathbf{q}})}\sum_{l=1}^{L}q_{l}E_{\pi}[C(X,T)|T=\tilde{\boldsymbol{t}}_{l}],\label{eq:MK-Problem}
\end{equation}
where $\mathcal{W}(\tilde{\mathbf{t}},\mathbf{q})$ is the objective
function at the solution to the optimal transport problem and $C(X,T)$
is a cost function that will represent profit regret to full personalization.\footnote{Equation \ref{eq:MK-Problem} is also known as the Monge-Kantorovich
problem in the optimal transport literature. A more formal discussion
is in Appendix Section \ref{sec:Mathematics-of-Coarse-Personalization}.} This problem takes as inputs the treatment values $\tilde{\mathbf{t}}$
and segment sizes $\mathbf{q}$, and outputs an optimal assignment
rule $\pi$ that forms the segments.
The coarse personalization problem is
\begin{align}
\min_{\tilde{\boldsymbol{t}}\in\mathbb{R}^{LD},\boldsymbol{q}\in\mathbb{R}^{L}} & \mathcal{W}(\tilde{\mathbf{t}},\mathbf{q})\nonumber \\
\text{s.t.}~~ & \boldsymbol{q}\geq0 & \text{(non-negative weights)}\nonumber \\
& \sum_{l=1}^{L}q_{l}=1 & (\text{adding up})\label{eq:OT-Problem}\\
& \text{\ensuremath{\tilde{\boldsymbol{t}}_{l}} takes form \ensuremath{(0,\ldots,\tilde{t}_{d},\ldots,0)} } & (\text{feasible treatment})\nonumber
\end{align}
where the decision variables are the segments' treatment values $\tilde{\mathbf{t}}$
and their sizes $\mathbf{q}$.\footnote{The nonnegative constraint ensures the segments have one or more individuals
assigned to them. The adding-up constraint ensures that everyone is
assigned to a segment. The feasible treatment constraint implies the
segment's assigned treatment can take values in only one treatment
dimension.}
In our framework, the inner problem (Equation \ref{eq:MK-Problem})
is a semidiscrete optimal transport problem that solves for the optimal
assignment rule of individuals, described by their CATEs, to their
segment. The outer minimization problem (Equation \ref{eq:OT-Problem})
chooses the segments' treatment values and sizes, specifically the
$L$ unique $D$-dimensional treatments $\tilde{\boldsymbol{t}}$
and corresponding segment sizes $\boldsymbol{q}$.
To link the optimal transport problem to profit maximization, we choose
the cost function, $C(X,T)$, to represent profit loss from coarse
to full personalization. We first define $\mathcal{R}_{i}(\boldsymbol{x}_{i},\tilde{\boldsymbol{t}}_{l})$
to be the expected profits from assigning treatment $\tilde{\mathbf{t}}_{l}$
to an individual with covariates $\boldsymbol{x}_{i}$,
\begin{multline}
\mathcal{R}_{i}(\boldsymbol{x}_{i},\tilde{\boldsymbol{t}}_{l})\triangleq E_{\pi}[\text{R}_{i}|\boldsymbol{x}_{i},T_{i,1}=\tilde{t}_{l,1},0,\ldots,0]+\cdots+E_{\pi}[\text{R}_{i}|\boldsymbol{x}_{i},0,\ldots,T_{i,d}=\tilde{t}_{l,d},\ldots,0]+\cdots\\
+E_{\pi}[\text{R}_{i}|\boldsymbol{x}_{i},0,\ldots,T_{i,D}=\tilde{t}_{l,D}].\label{eq:Cost-Expected-Return}
\end{multline}
We then define $\bar{R}_{i}$ to be the profits for an individual
with $x_{i}$ if firm fully personalizes,
\begin{equation}
\bar{R}_{i}\triangleq\max\big\{ E_{\pi}[\text{R}_{i}|\boldsymbol{x}_{i},t_{i,1}^{*},\ldots0],\ldots,E_{\pi}[\text{R}_{i}|\boldsymbol{x}_{i},0,\ldots,t_{i,d}^{*},\ldots],\ldots,E_{\pi}[\text{R}_{i}|\boldsymbol{x}_{i},0,\ldots,t_{i,D}^{*}]\big\}.\label{eq:Cost-Max-Return}
\end{equation}
We further define $\bm{\mathcal{R}}(\boldsymbol{x},\tilde{\boldsymbol{t}}_{l})$
as the vector of $\mathcal{R}_{i}(\boldsymbol{x}_{i},\tilde{\boldsymbol{t}}_{l})$
across individuals $i\in\{1,\ldots,N\}$, $\bar{\boldsymbol{R}}$
as the vector of $\bar{R}_{i}$ across individuals, and $\boldsymbol{x}$
as the stacked covariate matrix across individuals.
The cost function is
\begin{align}
E_{\pi}[C(X,Y)|t=\tilde{\boldsymbol{t}}_{l}]= & \big|\mathcal{\bm{\mathcal{R}}}(\boldsymbol{x},\tilde{\boldsymbol{t}}_{l})-\bar{\boldsymbol{R}}\big|^{2}\label{eq:Cost-Function}
\end{align}
in the optimal transport problem (Equation \ref{eq:MK-Problem}).
The cost function in Equation \ref{eq:Cost-Function} represents
profit regret: It is the square of the difference of coarse personalization
profits ($\bm{\mathcal{R}}(\boldsymbol{x},\tilde{\boldsymbol{t}}_{l})$)
to full personalization profits ($\bar{\boldsymbol{R}}$).\footnote{We can extend the cost function to account for the number of segments
in Online Appendix Section \ref{sec:Treatments-in-Cost-Function}.} Because full personalization is profit maximizing, the firm will
receive weakly lower profits from coarse personalization as it can
only form $L$ segments to personalize. Thus, minimizing profit regret
to an upper bound corresponds to profit maximization in the coarse
personalization solution.
As we will detail in Section \ref{sec:Why-optimal-transport}, the
problem, as it stands, is computationally burdensome. To alleviate
this concern, we impose two assumptions: The first is a strict concavity
assumption on the continuous CATE function, $\tau_{d}(\boldsymbol{x}_{i},t_{i,d})$.
The second is a weak convexity assumption on the cost function, $c_{d}(t_{i,d})$.
\begin{assumption}
(Strict Concavity of the Continuous Conditional Average Treatment
Effects)\label{assu:TE-Concavity}
The continuous conditional average treatment effect $\tau_{d}(\boldsymbol{x}_{i},t_{i,d})$
is strictly concave in $t_{i,d}$. For $t,t'\in[0,\bar{t}_{d}]\subset\mathbb{R}^{+}$
and $\alpha\in[0,1]$, $\tau_{d}(\boldsymbol{x}_{i},\alpha t+(1-\alpha)t')>\alpha\tau_{d}(\boldsymbol{x}_{i},t)+(1-\alpha)\tau_{d}(\boldsymbol{x}_{i},t')$.
\end{assumption}
\begin{assumption}
(Convexity of the Cost Function)\label{assu:Cost-Convexity}
The cost function $c_{d}(t_{i,d})$ is convex in $t_{i,d}$. For $t,t'\in[0,\bar{t}_{d}]\subset\mathbb{R}^{+}$
and $\alpha\in[0,1]$, $c_{d}(\alpha t+(1-\alpha)t')\leq\alpha c_{d}(t)+(1-\alpha)c_{d}(t')$.
\end{assumption}
Assumption \ref{assu:TE-Concavity} imposes a strict concavity requirement
on the estimated continuous CATEs. It captures the notion of diminishing
sensitivity of the treatment level\textemdash that is, how at higher
treatment levels, further changes in the treatment level do not increase
the treatment effect as much. Assumption \ref{assu:Cost-Convexity}
rules out economies of scale in the cost of issuing the treatment
as the treatment level increases.
Because the difference between a strictly concave function and a convex
function is strictly concave, the firm's program in the first step
(Equation \ref{eq:Firm-Program-Treatment}) now has a unique solution
$t_{i,d}^{*}$ that is either an interior solution or on the boundary.
After the firm solves $D$ programs for each individual $i$ to fully
personalize, it would have constructed optimal treatment levels $\{\{t_{i,d}^{*}\}_{d=1}^{D}\}_{i=1}^{N}$
and have estimated continuous CATE functions $\{\{\tau_{d}(\boldsymbol{x}_{i},t_{i,d})\}_{d=1}^{D}\}_{i=1}^{N}$.
Proposition \ref{prop:OT-Problem-Convex} demonstrates that the optimal
transport problem in Equation \ref{eq:OT-Problem} is a convex program
in treatment values under Assumptions \ref{assu:TE-Concavity} and
\ref{assu:Cost-Convexity}. We stress that \emph{any} strictly concave
function of CATEs will satisfy Assumption \ref{assu:TE-Concavity},
and \emph{any} convex function will satisfy Assumption \ref{assu:Cost-Convexity}.
Corollary \ref{cor:Deterministic-mappings} shows that the convex
program ensures that the assignment rule (optimal transport plan)
deterministically maps individuals to segments. Having a deterministic
segmentation rule is beneficial because it ensures the segments are
easily implementable, and we further discuss its downstream practical
implications in Section \ref{sec:Why-optimal-transport}. The proofs
are in Appendix Section \ref{sec:OT-Proofs}.
\begin{prop}
Under Assumptions \ref{assu:TE-Concavity} and \ref{assu:Cost-Convexity},
$E_{\pi}[C(X,T)]$ is strictly convex in $\tilde{\boldsymbol{t}}$.
\label{prop:OT-Problem-Convex}
\end{prop}
\begin{cor}
Under Assumptions \ref{assu:TE-Concavity} and \ref{assu:Cost-Convexity},
the optimal transport plan $\pi$ that solves Equation \ref{eq:OT-Problem}
deterministically maps individuals to their segment. \label{cor:Deterministic-mappings}
\end{cor}
\subsubsection*{Discussion }
The coarse personalization solution maximizes profits by simultaneously
choosing (1) the segmentation rule of individuals and (2) the assigned
treatment for each segment. At the solution, the assigned treatment
level is chosen such that its average marginal effect is equal to
its average marginal cost in each segment. With full personalization,
the marginal effect of treatment equals the marginal cost of treatment
at the individual level. Intuitively, our solution shifts the profit-maximizing
personalization principle from equating marginal effect and marginal
cost individually to equating them at the segment level. Online Appendix
Section \ref{sec:First-order-condition-discussion} provides a formal
derivation of this intuition using the first order condition of Equation
\ref{eq:OT-Problem}.
Coarse personalization can be interpreted as a more regularized version
of full personalization. Heterogeneous treatment effect estimates
and CATEs are known to be noisy \citep{Wager2018,Rafieian2021,Hitsch2024},
and by coarsely assigning segments or groups of customers, firms are
able to explicitly choose the level of regularization when using heterogeneous
treatment effects to personalize. At one extreme, full regularization
implies everyone receives the same treatment ($L=1$) and at the other,
full personalization implies everyone gets their own fully personalized
treatment ($L=N$). The coarse personalization framework enables navigation
between two extremes while ensuring the profit maximizing groups and
treatments are formed at each level of regularization.
We also highlight that nonparametric approaches to estimate continuous
CATEs have been developed in the machine learning literature \citep{Swaminathan-Joachims-2015,pmlr-v84-kallus18a}
and operations literature \citep{Zhou2022}. However, these methods
either cannot incorporate the concavity condition from Assumption
\ref{assu:TE-Concavity} or they require that treatment values for
each segment be specified ex ante for computational tractability.
Thus, without additional adaptions and theoretical adjustments, these
methods cannot be directly applied to our setting.
\section{Computational solution\label{sec:Computational-solution}}
To computationally solve our coarse personalization problem, we adapt
a version of Lloyd's Algorithm (Algorithm \ref{alg:Adapted-Lloyd's-Algorithm}),
from \citet{Lloyd1982} to maximize profits while accounting for all
the constraints. Our algorithm provides an iterative method to efficiently
solve the problem. Various versions of Lloyd's Algorithm have been
used in the literature to solve optimal transport problems \citep{Pollard1982,NIPS2012_c54e7837}.
Figure \ref{fig:Adapted-Lloyd's-Algorithm} illustrates the algorithm's
optimization procedure for two-dimensional treatments. Our formulation
of the optimal transport problem falls under the class of mixed-integer
linear programs; other methods to solve semidiscrete optimal transport
problems can be found in \citet{Peyre2019}. We discuss the benefits
of Algorithm \ref{alg:Adapted-Lloyd's-Algorithm} over the brute force
solution via grid search in Section \ref{subsec:Grid-search-comparison}.
Using the promotions-management setting from our empirical example,
we outline the procedure in Online Appendix Section \ref{subsec:Outline-of-Algorithm}.
Given continuous CATE estimates $\{\{\tau_{d}(\boldsymbol{x}_{i},t_{i,d})\}_{d=1}^{D}\}_{i=1}^{N}$
and $\{\{t_{i,d}^{*}\}_{d=1}^{D}\}_{i=1}^{N}$ from the first step,
the firm can solve the coarse personalization problem by using Algorithm
\ref{alg:Adapted-Lloyd's-Algorithm}. The firm chooses the number
of segments ($L$) to offer and our algorithm will supply the profit-maximizing
segments and their assigned treatments.
As Figure \ref{fig:Adapted-Lloyd's-Algorithm} demonstrates, in each
iteration, our algorithm cycles through updating the segments and
their assigned treatment values. The feasible treatment constraint
is enforced by choosing the dimension of the treatment that yields
the highest expected return to the firm for each cell. The algorithm
terminates when the change to the treatment values after each iteration
is smaller than a prespecified tolerance value. The final profits
for the firm is the sum of the expected profits from the terminal
step's segmentation rule.
\begin{spacing}{1.0}
\begin{algorithm}
\caption{Adapted Lloyd's Algorithm\label{alg:Adapted-Lloyd's-Algorithm}}
\textbf{Notation: }There are $L$ segments that need to each be assigned
a treatment and $D$ treatment dimensions. The candidate treatment
for $\boldsymbol{t}_{l}$ in step $k$ of the algorithm is denoted
$\boldsymbol{t}_{l}^{k}$. The set of individuals assigned to treatment
$l$ is denoted $\ell$. The candidate treatments have form $\boldsymbol{t}_{l}=(0,\ldots,t_{l,d},\ldots,0)$
and are nonzero only in dimension $d$ with value $t_{l,d}$. $\delta$
represents a prespecified tolerance value.
\textbf{Step }$\boldsymbol{0}$: Guess the initial set of $L$ treatment
vectors $\tilde{\boldsymbol{t}}^{0}=\{\tilde{\boldsymbol{t}}_{1}^{0},\ldots,\tilde{\boldsymbol{t}}_{l}^{0},\ldots,\tilde{\boldsymbol{t}}_{L}^{0}\}$.
\textbf{Step $\boldsymbol{k}$}:
\begin{itemize}
\item (\emph{Form segments}) Compute the Voronoi Cells for each proposed
treatment $\tilde{\boldsymbol{t}}_{l}^{k},\forall l\in\{1,\ldots,L\}$
\[
v_{l}^{k}=\{x\in X:|C(X,\tilde{\boldsymbol{t}}_{l}^{k})|\leq|C(X,\tilde{\boldsymbol{t}}_{j})|,\forall j\neq l\}.
\]
\item (\emph{Choose new candidate treatments for each segment}) Compute
the Barycenter values in each dimension $d\in\{1,\ldots,D\}$ and
for each Voronoi Cell $v_{l}^{k},\forall l\in\{1,\ldots,L\}$
\[
\tilde{\boldsymbol{t}}_{l}'=(\tilde{t}_{l,1}',\ldots,\tilde{t}_{l,D}')=\frac{1}{P(v_{l}^{k})}\int_{v_{l}^{k}}wdP(w).
\]
\item (\emph{Evaluate profits for candidate treatments for each segment})
Compute profits for each feasible treatment generated from $\tilde{\boldsymbol{t}}_{l}'$,
which are of form $\{(0,\ldots,\tilde{t}_{l,d},\ldots0)\}_{l=1}^{L}$,
and have expected profits
\begin{align*}
\sum_{i\in\ell}E[\text{R}_{l,d}|x_{i},t_{i,1}=0,\ldots,t_{i,d}=\tilde{t}_{l,d},\ldots,t_{i,D}=0] & =\sum_{i\in\ell}\tau_{d}(\boldsymbol{x}_{i},\tilde{t}_{l,d})-c_{d}(\tilde{t}_{l,d}).
\end{align*}
\item (\emph{Update candidate treatments to ensure feasibility}) Update
$\tilde{\boldsymbol{t}}$ by forcing the feasibility constraint. We
choose the feasible treatment that yields highest profits across each
dimension $d\in D$ {\footnotesize
\[
\tilde{\boldsymbol{t}}_{l}^{t+1}=\begin{cases}
(\tilde{t}_{l,1}',\ldots,0) & \text{if \ensuremath{\sum_{i\in\ell}E[\text{R}_{l,1}|\boldsymbol{x}_{i},\tilde{t}_{i,1},\ldots,0]>\sum_{i\in\ell}E[\text{R}_{l,d'}|\boldsymbol{x}_{i},0,\ldots,\tilde{t}_{i,d'},\ldots,0],\forall d'\neq1}}\\
\hspace{1cm}\vdots & \hspace{1cm}\vdots\\
(0,\ldots,\tilde{t}_{l,d}',\ldots0) & \text{if \ensuremath{\sum_{i\in\ell}E[\text{R}_{l,d}|\boldsymbol{x}_{i},0,\ldots,\tilde{t}_{i,d},\ldots,0]>\sum_{i\in\ell}E[\text{R}_{l,d'}|\boldsymbol{x}_{i},0,\ldots,\tilde{t}_{i,d'},\ldots,0],\forall d'\neq d}}\\
\hspace{1cm}\vdots & \hspace{1cm}\vdots\\
(0,\ldots,\tilde{t}_{l,D}') & \text{otherwise}.
\end{cases}
\]
}{\small}{\small\par}
\end{itemize}
\textbf{Terminate} the algorithm when $\tilde{\boldsymbol{t}}^{k+1}$
is close enough to $\tilde{\boldsymbol{t}}^{k}$, or $||\tilde{\boldsymbol{t}}^{k+1}-\tilde{\boldsymbol{t}}^{k}||<\delta$.
\end{algorithm}
\end{spacing}
\subsection{Discussion of the Lloyd's Algorithm's adaptation}
In Algorithm \ref{alg:Adapted-Lloyd's-Algorithm}, we make two adaptations
to Lloyd's Algorithm to directly maximize profits. First, the optimization
metric is different. Rather than forming segments (Voronoi Cells)
and their assigned treatments (Barycenters) to maximize Euclidean
distance between treatments, we construct both simultaneously to maximize
profits. Second, since each feasible treatment is only nonzero in
one dimension, we include a dimension-reduction step where the Barycenter
of the Voronoi cell is projected down to a treatment value in one
dimension.\footnote{These are in the last two substeps in Step $k$ of Algorithm \ref{alg:Adapted-Lloyd's-Algorithm}.}
For each segment, we choose the treatment dimension that maximizes
profits.
Even though the coarse personalization problem is strictly convex
in the treatment values by Proposition \ref{prop:OT-Problem-Convex},
the requirement that treatments are feasible, or are nonzero in one
dimension, adds a combinatorial constraint to the problem.\footnote{The dimension-reduction step in Algorithm \ref{alg:Adapted-Lloyd's-Algorithm}
leads to discontinuities and possible sudden jumps in the assigned
treatment values when the assigned treatment dimension changes. These
discontinuous updates arise due to the combinatorial constraint.} As a result, the optimization problem becomes NP-hard and a global
optimum is not guaranteed to be found with our algorithm or even with
an exhaustive grid search.
To mitigate this concern, we (1) run Algorithm \ref{alg:Adapted-Lloyd's-Algorithm}
five times with different starting values and choose the best performing
run and (2) compare our results from our adapted Lloyd's Algorithm
with a more exhaustive search using BFGS. We find that the best performing
run of our algorithm is effectively identical to that of the exhaustive
search. We discuss the difference between our algorithm and the grid
search in the next subsection.
If we relax the treatment feasibility constraint in our coarse personalization
problem, or let the assigned treatments take values in each dimension
of the treatment, then we can avoid the combinatorial constraint.
In our application, we require treatment feasibility because individuals
are only sent one type of promotion. However, this constraint can
be relaxed in a product design application when we optimize all attributes
of the product simultaneously.\footnote{In Algorithm \ref{alg:Adapted-Lloyd's-Algorithm}, we would then omit
the dimension-reduction step and only run the first two substeps under
step $k$. The coarse personalization problem then collapses to a
standard convex optimization problem over a compact set. Since Lloyd's
Algorithm is a greedy algorithm, it converges to a local optimum \citep{Lu2016},
which is the global optimum by the convexity of the problem.}
Lastly, our results in Section \ref{subsec:Optimal-transport} show
that the solution under our assumptions leads to a deterministic transport
map: Individuals are assigned to one segment and not probabilistically
assigned to different segments. This allows us to form discrete segments
that are more practically implementable than probabilistic segments.
\subsection{Grid search comparison\label{subsec:Grid-search-comparison}}
A common alternative approach to solving the optimal transport problem
is using a brute-force grid search, and we now discuss why our solution
is advantageous over a grid search. To implement a grid search, the
firm can discretize over the treatment domain $\mathcal{X}$ ex ante,
thereby converting the semidiscrete optimal transport problem into
a fully discrete optimal transport problem.\footnote{The use of discretization to approximate a continuous domain underlies
the mass transport approach to solving continuous-continuous optimal
transport problems in \citet{Chiong2016}.} Then, over the finite grid of possible assignments and treatment
levels, the firm can combinatorially search for segments and their
assigned treatments. While we can relax Assumptions \ref{assu:TE-Concavity}
and \ref{assu:Cost-Convexity} in the grid search, the combinatorial
space will grow exponentially in the number of individuals and as
it suffers from the curse of dimensionality. Specifically, for $N$
individuals, $L$ treatments, and $D$ dimensions with $G$ discrete
points in each dimension, the combinatorial space for the grid search
is ${GD \choose L}L^{N}$.\footnote{For example, if the firm chooses to assign three different promotions
($L=3$) across dollar off and percentage off promotion types ($D=2$)
that take on ten possible values $(G=10)$ for fifteen people $(N=15)$,
there are over $16$ billion combinations to search over.}
The grid search algorithm's run time is $\mathcal{O}((GD)^{GD/2}LN)$.
In contrast, the run time of $I$ iterations of our adapted Lloyd's
Algorithm is $\mathcal{O}(IDLN)$ and will be substantially faster
as it removes the exponential factor. For many high-dimensional treatments,
the standard grid search approach is computationally infeasible. Instead,
using adapted Lloyd's Algorithm or any convex optimizer that leverages
Assumptions \ref{assu:TE-Concavity} and \ref{assu:Cost-Convexity}
is recommended.
Algorithm \ref{alg:Adapted-Lloyd's-Algorithm} and the brute force
approach yield similar profits. In Online Appendix Section \ref{sec:Grid-search-details},
we describe our grid search implementation and discuss why Algorithm
\ref{alg:Adapted-Lloyd's-Algorithm} runs $37$ times faster than
the grid search using an example from our empirical application.
We also provide intuition why our approach is faster. In short, our
algorithm uses a more computationally efficient update rule and avoids
the curse of dimensionality.
\section{Why optimal transport?\label{sec:Why-optimal-transport}}
Given the constraint that only $L$ segments can be offered, our coarse
personalization solution solves which segments to offer and their
corresponding treatment to maximize profits. In essence, the distribution
of CATEs estimated in the first step is mapped to a discrete distribution
of assigned treatments. Figure \ref{fig:OT-Semidiscrete} visualizes
this procedure in a stylized example.
In essence, full personalization is profit maximizing for the firm
because it optimally chooses profit maximizing treatments at the individual
level. Coarse personalization accounts for the extra constraint on
the number of segments and uses optimal transport to solve the segmentation
problem. Our coarse personalization solution generates the maximum
possible expected profits given the limitation on the number of segments.
Unlike the traditional marketing segmentation framework, we have solved
both segmentation and targeting steps simultaneously. The former are
the assignment of individuals to each segment and the latter are the
unique treatment values chosen for each segment. Because we are discretizing
full personalization's estimates into a few segments in the second
step, our ``personalize then discretize'' approach inverts the ``discretize
then personalize'' approach used in classical marketing. The optimal
transport problem minimizes profit regret relative to full personalization,
so its segmentation and targeting rules should yield higher profits
than those sequentially constructed by first segmenting on covariates
or preferences and then forming targeting rules. We additionally
recover full personalization profits when we let the number of segments
equal to the number of individuals ($L=N$). The final choice of the
number of segments $L$ to offer is a free parameter for the firm
to tune.
More generally, we espouse the optimal transport approach because
it (1) allows us to interpret the coarse personalization problem in
an economic manner, (2) suggests a simple computational solution in
Algorithm \ref{alg:Adapted-Lloyd's-Algorithm}, and (3) links the
problem to the broader transport theory literature. Naturally, marketing
researchers can use black-box optimizers to solve the optimization
program via brute-force instead of using our optimal transport framework.
However, this approach forgoes the benefits from the optimal transport
approach and can be computationally slower as discussed in Section
\ref{subsec:Grid-search-comparison} and in Appendix Section \ref{sec:Grid-search-details}.
Economically, the optimal transport solution finds segmentation plans
such that the average marginal benefit is equal to the marginal cost
of treatment across assigned segments (Appendix Section \ref{sec:First-order-condition-discussion}).
Algorithm \ref{alg:Adapted-Lloyd's-Algorithm} provides a computationally
simple and scalable solution, and its link would not be made clear
without viewing the problem through an optimal transport lens. Our
algorithm is especially efficient in higher dimensions as it searches
over treatments that are close to the individual optimal values: Compared
to a black-box optimizer, our algorithm embeds a ``warm start''
in the optimization process. Lastly, recasting our coarse personalization
problem as a semidiscrete optimal transport problem allows us to connect
our marketing problem to the broader transportation theory literature.
Exploring additional links between the two may prove fruitful for
future research.
We emphasize that the link to the optimal transport framework is still
beneficial even if Assumptions \ref{assu:TE-Concavity} and/or \ref{assu:Cost-Convexity}
are relaxed. In that case, the segmentation rule shifts from deterministically
assigning each individual to a single segment to probabilistically
assigning each individual to multiple segments.\footnote{In the optimal transport literature, the solution would then be the
Kantorovich relaxation to our Monge-Kantorovich problem and a pure
Monge mapping is not guaranteed to exist. Additional details can
be found in Appendix Section \ref{sec:Mathematics-of-Coarse-Personalization}.} This probabilistic assignment to segments also appears in the latent
class and finite mixture models in the marketing literature \citep{Kamakura1989a}.
Thus, our recasting of the classical marketing segmentation problem
as an optimal transport framework is beneficial because (1) the correct
target metric of profit is used when forming segments, (2) it lends
a link to Lloyd's Algorithm for a computationally efficient implementation,
and (3) under Assumptions \ref{assu:TE-Concavity} and \ref{assu:Cost-Convexity}
we show the solution's optimal segments and their assigned treatments
are unique and each individual is deterministically assigned to a
segment.
\section{Empirical application\label{sec:Empirical-Application}}
In this section, we illustrate our methodology with data from a large-scale
field experiment conducted by a food delivery platform. Marketing
mix treatments are two dimensional: dollar amount off and percentage
off promotions. Customers will be assigned to a promotion of one of
the two treatment dimensions.
We show that with our coarse personalization framework, the company
can achieve profits close to full personalization while using only
a handful of optimized segments. Further, we demonstrate that the
coarse personalization solution significantly outperforms classic
sequential segmentation procedures in generating expected profits.
\subsection{Setting}
We use a large-scale RCT dataset for a food delivery platform across
three core-based statistical areas in the United States. There are
over 1.2 million unique customers in the data with around $100$ pretreatment
variables that describe customers\textquoteright{} recency, frequency,
and monetary (RFM) characteristics. Across these customers, the following
promotional treatments were randomly assigned: dollar amount off ($\$2,\$3,\$4,\$5$),
percentage off ($5\%,10\%,15\%,20\%$), and no treatment. The assigned
promotion can only be used on the next order by the customer, the
promotions are issued by email, and the no treatment group will act
as a control group. We assume customers do not engage in coupon gaming
or stockpiling.. Following Section \ref{sec:Model}'s framework, we
have a two-dimensional treatment vector that consists of dollar off
and percentage off promotions. We set the dollar off promotion to
be the first dimension and the percentage off promotion as the second
dimension. We conduct randomization checks in Online Appendix Section
\ref{sec:Randomization-check}.
The firm aims to maximize profits by personalizing its dollar-off
and percentage-off promotions for its customer base. The outcome
variable is the total profits generated from the customer $28$ days
after the promotion was sent. This represents the firm's net profits
per order, after subtracting payments to delivery drivers and restaurants.
The firm's profits from consumer $i$ after $28$ days is defined
as
\begin{equation}
\text{Profits after 28 days}_{i}=\text{Spending after 28 days}_{i}-\text{Promotion cost}_{i}\label{eq:Profit-Equation-Setting}
\end{equation}
where the promotion cost depends on the type of promotion issued.
Figure \ref{fig:Profits-by-Treatment} plots the profits by treatment
value for dollar off and percentage off promotions. In both treatment
dimensions, profits generally increase with higher promotion values
and have a concave shape which suggests diminishing sensitivities
at higher levels of treatment.
The ATE estimates for the firm's profits are presented in Table \ref{tab:Experiment-ATEs}.
They represent the difference in profits of comparing the treatment
arms in Figure \ref{fig:Profits-by-Treatment} to the control group
of no treatment. Using a two-sided $t$-test, we find that the ATE
estimates for four dollar off, fifteen percentage off, and the twenty
percentage off promotions are statistically significant from zero
at the $\alpha=0.05$ level. Further, the point estimates of the ATEs
are all positive, which suggests the promotions have a positive effect
on profits.
Following Section \ref{sec:Model}'s notation, profits are $\text{R}_{i}$,
spending levels are $Y_{i}$, and promotion costs are $c_{d}(t_{i,d})$,
where $t_{i,d}$ is the treatment issued to consumer $i$ that is
nonzero in dimension $d$. We only observe profits after $28$ days
for each consumer but we can recover spending levels by imposing structure
on the firm's costs. The firm's expected returns, or expected profits,
for individual $i$ assigned treatment $t_{i,d}$ that is nonzero
in dimension $d$ is
\begin{equation}
E[R_{i}|\boldsymbol{x}_{i},T_{i,1}=0,\ldots,T_{i,d}=t_{i,d},\ldots,T_{i,D}=0]=E[Y_{i}|\boldsymbol{x}_{i},T_{i,1}=0,\ldots,T_{i,d}=t_{i,d},\ldots,T_{i,D}=0]-c_{d}(t_{i,d}).
\end{equation}
We define the cost of the dollar off promotion as the dollar amount
itself. For the percentage off promotion, the costs for consumer $i$
are the percentage off times the customer's average promotional spending
on an order over the last year. The cost of the promotions are
\begin{align}
c_{d}(t_{i,1}) & =t_{i,1}\\
c_{d}(t_{i,2}) & =t_{i,2}\times\text{ average past promo spend}_{i}
\end{align}
for dollar off promotion $\ensuremath{t_{i,1}}$ and for percentage
off promotion $\ensuremath{t_{i,2}}$. For customers who have never
received a promotion before the field experiment, their past promotional
spend for an order was median imputed. The treatment vector $(t_{i,1},t_{i,2})=(0,0)$
represents the no promotion case; the firm incurs no cost for withholding
a promotion.
We also assume that the continuous CATEs, $\tau_{d}(\boldsymbol{x}_{i},t_{i,d})$,
are strictly concave in the treatment value, $t_{i,d}$, to satisfy
Assumption \ref{assu:TE-Concavity}. The strict concavity assumption
then implies that customers have diminishing sensitivities to promotion
levels. By construction, the promotions' costs are weakly convex
in $t_{i,1}$ and $t_{i,2}$ satisfying Assumption \ref{assu:Cost-Convexity}.
We interpret the continuous CATEs, $\tau_{d}(\boldsymbol{x}_{i},t_{i,d})$,
as a behavioral sensitivity to dollar off or percentage off treatments
\citep{Briesch1997}. We find that even when the two yield the same
expected value to the customers, the continuous treatment effect is
different; customers have differing sensitivity when presented with
two promotions that yield the same expected monetary value. Otherwise,
if consumers had the same sensitivity to the expected value of both
promotions, then we would be able to collapse the two dimensions of
treatments to a one-dimensional, expected value of the promotion for
each customer.
\subsection{Estimation procedure}
To estimate the continuous CATEs, $\hat{\tau}_{d}(\boldsymbol{x}_{i},t_{i,d})$,
we first implement the Causal Forest from \citet{Wager2018} in each
of the two treatment dimensions. To adjust for randomization concerns
in the field experiment (see Appendix Section \ref{sec:Randomization-check}),
we use generalized regression forests that use propensity score estimates
to adjust for possible stratification between the treatment and holdout
samples \citep{Athey2019}.\footnote{We also estimate the continuous CATEs directly using Deep Neural Networks
\citep{Farrell2020,Farrell2021}. We find similar results which suggest
that deviations from perfect randomization do not heavily influence
our results. To account for imperfect randomization in this approach,
we can implement the DR-Learner procedure from \citet{Kennedy2020}
while using DNNs for the propensity score and regression function
estimators.}
To force strict concavity of the continuous CATEs, we then impose
a logarithmic functional form that is parameterized by $\alpha_{i,d}$
and $\beta_{i,d}$,
\begin{equation}
\hat{\tau}_{d}(\boldsymbol{x}_{i},t_{i,d})=\alpha_{i,d}+\beta_{i,d}\log(1+t_{i,d})+\epsilon_{i,d},\label{eq:CATE-Log-Form}
\end{equation}
along with a shape restriction that $\beta_{i,d}>0$ and we let $\epsilon_{i,d}$
represent the approximation error.\footnote{Any strictly concave functions can be used in this procedure. The
stylized logarithmic functional form of the marketing treatment response
to capture diminishing sensitivity has also been used in the CRM literature
\citep{Rust2005}.} The individual continuous CATEs are parameterized by the estimates
$\{\{\hat{\alpha}_{i,d},\hat{\beta}_{i,d}\}_{d=1}^{2}\}_{i=1}^{N}$,
and we use these to solve for the optimal treatment levels $\{t_{i,1}^{*},t_{i,2}^{*}\}_{i=1}^{N}$
following Equation \ref{eq:Firm-Program-Treatment}. We then choose
the number of segments ($L$) and run Algorithm \ref{alg:Adapted-Lloyd's-Algorithm}
to attain the segments and their assigned treatments.
We bootstrap to produce standard errors for our estimates. In our
estimation results, we focus on the implementation uncertainty of
the second step, the optimal transport step, which is of main practical
interest to the firm.\footnote{See \citet{Hitsch2024} for a discussion around uncertainty in decision-making
for personalization strategies.} In practice, firms treat the first step CATE estimates as given and
focus on the implementation uncertainty of the different methods in
generating profits. Our focus on implementation uncertainty reflects
situations where the firm is concerned with how the procedure performs
across different markets, as well as uncertainty around the roll out
of the personalization policy.
To do so, we take the first stage treatments as the ground truth and
bootstrap only the second step of our procedure.\footnote{To fully account for both model uncertainty in the first step and
the implementation uncertainty in the second step, we can use the
confidence intervals from the Causal Forest's CATE estimates. We then
can sample from the CATE confidence intervals and then implement the
optimal transport step.} More specifically, we only bootstrap the segmentation and treatment
assignment steps from Section \ref{subsec:Optimal-transport} to quantify
the implementation error. For our other segmentation benchmarks, we
similarly only bootstrap the segmentation and treatment assignment
steps.
As a result, the comparisons of interest are (1) how close coarse
personalization can get to the full personalization in generating
profits and (2) how our coarse personalization solution compares to
sequential segmentation procedures.
\subsection{Results\label{subsec:Empirical-Results}}
In our application, we find that coarse personalization both (1)
outperforms sequential segmentation procedures and (2) almost recovers
full personalization profits after using five segments. Figure \ref{fig:Profit-to-granular}
summarizes the main results. We now discuss these results in more
detail.
The first step's CATE estimates were estimated using the entire set
of $126$ covariates and the entire dataset of 1,213,390 individuals.\footnote{We preprocess the CATE estimates. From our CATE parameterization,
individuals whose $\hat{\beta}_{i,d}$ are close to $0$ are not responsive
to promotions. After accounting for the promotions' cost, these individuals
would not receive a promotion. Thus, dropping these individuals does
not have an effect on incremental profits. Consequently, we exclude
individuals with values less than or equal to $10^{-6}$. The second
step was run on the remaining $1,203,379$ individuals. } The continuous CATE parameterization produced an average $R^{2}$
of $0.925$ and $0.770$ respectively for the dollar off and percentage
off dimensions. We implement the honest validation procedure \citep{Misra2021}
to compare the continuous CATE parameterization in Equation \ref{eq:CATE-Log-Form}
to the nonparametric estimates in a holdout treatment and find that
our parameterization performs well out of sample (Online Appendix
Section \ref{subsec:Honest-validation}).
To visualize the continuous CATEs, we can turn to Figure \ref{fig:Continuous-CATE-Densities}
to see the estimates across individuals evaluated at the treatment
arms. We see that the treatment effects generally rise as the promotion
value increases in each dimension. On average, customers are more
receptive to the dollar off promotions than to the percentage off
promotions since the CATE levels are higher for the former. The treatment
effect of the percentage off promotion has more variance than that
of the dollar off promotion.
We now examine the performance of our coarse personalization solution
by quantifying profits across different numbers of available segments.
Table \ref{tab:Profits-table-by-treatments} shows the percentage
of expected incremental profits recouped by coarse personalization
compared to full personalization.\footnote{Expected incremental profits are defined as additional expected profits
over the expected baseline profits of not targeting anyone.} Figure \ref{fig:Profit-to-granular} plots the coarse personalization's
expected incremental profits across a hundred bootstrap iterations.
The lines represent the bootstrap means and the bands represent one
bootstrap standard deviation. Full personalization corresponds to
issuing $2,206$ unique treatments in our scenario when rounding to
three significant figures.
We find that with three unique treatments, we recoup around $99\%$
of the full personalization's expected profits and with five unique
treatments, we recoup over $99.5\%$ of the full personalization's
expected profits. These results suggest the firm is able to match
almost all of the full personalization's expected profits by using
only a handful of unique segments with our coarse personalization
solution.
To visualize our solution, we can turn to Figure \ref{fig:Treatment-assignments-10},
which plots the optimal treatment values, $\{t_{i,1}^{*},t_{i,2}^{*}\}_{i=1}^{N}$,
across individuals and segment assignments for ten segments ($L=10$).
The optimal treatment values represent the treatment values in each
treatment dimension for the individuals under full personalization.\footnote{For example, the rightmost yellow circular point represents an individual
whose optimal treatment values, $(t_{i,1}^{*},t_{i,2}^{*})$, are
close to five dollars off and twenty percent off. That individual
is in segment six and is assigned a $2.79$ dollar off promotion.
This assignment generates around five dollars in expected incremental
profits for the firm.} Table \ref{tab:Treatment-assignments-10} provides the treatment
values for the segments' assigned treatments and their segment sizes.
The expected incremental profits for each individual under the assigned
treatment is provided on the vertical axis and a plane at zero expected
profits is provided for reference. The points represent the optimal
treatment values $\{t_{i,1}^{*},t_{i,2}^{*}\}_{i=1}^{N}$ across individuals
and the colors represent their assigned treatment. The shape of the
points represents whether the assigned treatment is a dollar off (circle)
or a percentage off (triangle) promotion.
The takeaways from Figure \ref{fig:Treatment-assignments-10} are
threefold. First, across the two dimensions of treatment, the dollar
and percentage off, we see explicit quantization of the assignments
in each dimension and individual assignment to only one segment.
The quantization structure is an artifact from our optimal transport
problem which ensures individuals that generate similar profits within
a treatment dimension are grouped together.
Second, we see substantial heterogeneity in individual-level expected
incremental profits. Most of the individual expected profits are close
to zero and profits are generally larger for those with higher optimal
treatment levels; the most receptive customers to the promotions
produce the largest profits. On the flip side, a handful of individuals
assigned the lowest dollar off and percentage off promotions generate
negative profits. If the firm could use more segments, these individuals
would be given no promotion or an even smaller promotional value.
Third, there is significant bunching of individuals at the upper bound
in Figure \ref{fig:Treatment-assignments-10} for the percentage off
treatment. This implies that we are leaving profits on the table by
enforcing this upper bound. While we can expand the upper bound of
allowed treatments using our parameterization of the continuous CATE
estimates, we retain the upper bound at $\$5$ and $20\%$ in the
optimal transport problem because this was the largest promotion from
RCT. In future experiments, the firm may consider testing higher promotional
values.
Breaking down the segments, we see from Table \ref{tab:Treatment-assignments-10}
that most of the customers are assigned to the second, seventh, and
tenth treatments and these are all dollar off promotions. Many of
the offered treatments are tightly packed between the one to two dollar
off promotion in Figure \ref{fig:Treatment-assignments-10} which
suggests that a majority of individuals are most receptive to the
firm using one dollar off to two dollars off promotions.
\subsubsection{Comparison to sequential segmentation procedures \label{subsec:Classical-segmentation-procedures}}
We now compare our coarse personalization solution to sequential segmentation
procedures that first segment on covariates, individual preferences,
and optimal treatment levels and then form targeting rules. These
comparisons (1) shed light on why our coarse personalization procedure
performs better by targeting profits when forming segments and (2)
quantify how much our proposed procedure outperforms these sequential
techniques. Figure \ref{fig:Profit-to-granular} and Table \ref{tab:Profits-table-by-treatments}
summarize our key results.
We find that our coarse personalization solution (``personalize then
discretize'') significantly outperforms the sequential segmentation
methods (``discretize then personalize''). From our methodology,
coarse personalization generates the maximum expected profits possible
given the limits on the number of segments. We now document how we
implemented each sequential segmentation procedure and then discuss
how they compare to our coarse personalization solution.
\subsubsection*{Sequential benchmark procedures}
\emph{Covariate segmentation}. We segment on customer covariates,
which contain customers' platform behavior and RFM variables. We reduce
the covariate size to $59$ covariates by retaining variables that
are relevant either for predicting subsequent purchase incidence or
purchase amount with the Lasso.\footnote{The $59$ RFM covariates include variables that capture customers'
past spending, tipping, delivery vs. pickup, and order cancellation
behavior, promotional spending, their location, and the device they
use to order.} For this reduced set of covariates, we implement $k$-means to segment
the data. We then select the best treatment value in generating profits
for each segment as the assigned treatment. This segmentation procedure
represents \emph{classical ex ante segmentation} since neither the
outcome nor treatment variables from the data are used in forming
the segments themselves after the variable reduction step.
\emph{Preference segmentation}. To segment on consumer preferences,
we form segments on the individuals' treatment sensitivity. We implement
$k$-means on $\{\hat{\beta}_{i,1},\hat{\beta}_{i,2}\}_{i=1}^{N}$
estimates from our parameterization of the CATEs (Equation \ref{eq:CATE-Log-Form})
to form segments. The $\hat{\beta}_{i,1},\hat{\beta}_{i,2}$ parameters
capture the sensitivity of the individuals to the dollar and percentage
off promotions, which in turn are proxies for the individuals' price
sensitivity. After running $k$-means to form segments, we then find
the profit-maximizing treatment for each segment.
\emph{Optimal treatment levels segmentation}. We can segment on the
optimal treatment levels across individuals by running $k$-means
directly on $\{t_{i,1}^{*},t_{i,2}^{*}\}_{i=1}^{N}$. The optimal
treatment levels are the individually optimal promotion values for
each individual under full personalization. Here, forming segments
using $k$-means allocates individuals with the same optimal treatments
(for both dollar off and percentage off) in the same segment. After
segments are constructed, we find the optimal treatment level and
treatment dimension that maximizes profits.
\subsubsection*{Discussion}
From Figure \ref{fig:Profit-to-granular} and Table \ref{tab:Profits-table-by-treatments},
we see that segmenting on covariates barely helps the firm in maximizing
profits. Segments formed only using consumer characteristics do not
correspond to the optimal segments for maximizing profits; profits
barely increase as we increase the number of segments. Segmenting
on consumer preferences and optimal treatment levels does better than
segmenting on covariates. These results corroborate the conclusions
found in the segmentation literature: Segmenting on consumer preferences
performs better than segmenting on demographics in generating profits
\citep{Gupta1994,Rossi1996}.
Since optimal treatment levels combine information on preferences
with treatment costs, segmentation on preferences and segmentation
on optimal treatment levels embed similar information and thus should
perform similarly. However, segmenting on optimal treatment levels
is less noisy because it accounts for additional information in the
treatments' costs as we see in Figure \ref{fig:Profit-to-granular}.
Coarse personalization outperforms segmenting on optimal treatment
levels because coarse personalization uses the \emph{correct distance
metric of expected profits for segmentation} instead of Euclidean
distance between the optimal treatment levels ($t_{i,1}^{*},t_{i,2}^{*}$).
This specific difference is exemplified when comparing the quantization
of the segments in each treatment dimension. Appendix Figure \ref{fig:Treatment-assignments-10-2D}
provides a top-down view of Figure \ref{fig:Treatment-assignments-10}
and shows the segmentation structure of the coarse personalization
solution. Appendix Figure \ref{fig:Treatment-assignments-10-2D-opt-t-levels}
shows segmentation structure when forming segments by optimal treatment
levels.
In the coarse personalization solution, the nearest treatment in Appendix
Figure \ref{fig:Treatment-assignments-10-2D} is not necessarily assigned
to each individual because the distance metric used for segmentation
is in expected profits (Equation \ref{eq:Cost-Function}); it could
be more profitable to give the individual a treatment in another treatment
dimension. However, within each dimension of treatment, the nearest
treatment will be assigned to the individual. In contrast, when segmenting
on optimal treatment levels, the nearest treatment is always assigned
to the individual as seen in Appendix Figure \ref{fig:Treatment-assignments-10-2D-opt-t-levels}.
As a result, the constructed segments from coarse personalization
will outperform those constructed from using the optimal treatment
level when the firm's goal is to maximize profits because the objective
function \emph{directly} targets profits.
The lack of convexity guarantees for the three sequential benchmark
procedures implies running the $k$-means solution is unstable; running
it again will yield very different segments. In contrast, rerunning
Algorithm \ref{alg:Adapted-Lloyd's-Algorithm} yields almost identical
results and the solution is quite stable.
Finally, we note that capturing $99.5\%$ of the fully personalized
profit with only five segments is unique to this empirical application.
In other contexts, the gap between coarse and full personalization
could be wider or narrower.
\subsubsection{Discrete treatments benchmark\label{subsec:A/B-testing-comparison}}
In this section, we evaluate the cost of restricting the treatment
variables (dollar off and percentage off) to the discrete levels used
in the experiment. First, we examine the optimal blanket promotion
obtained when treatments are constrained. Next, we focus on policy-learning
approaches that only assign the discrete treatments observed in the
experimental data, thus ignoring treatment continuity.\footnote{\citet{pmlr-v84-kallus18a} adapt policy learning for continuous treatments,
but their reliance on kernel estimation makes it computationally prohibitive
for high-dimensional treatments and large datasets.} Through both exercises, we demonstrate that neglecting the continuous
nature of treatments results in substantial unrealized profits.
First, we calculate the difference in profits between the firm's historical
blanket promotion of ten percent off and an optimized blanket promotion.
For the blanket promotion optimization, we consider the cases where
(1) the firm only uses the experiment's discrete treatment values
and (2) the firm leverages the continuity of the treatment values.
This comparison demonstrates how much money the firm leaves on the
table by not completely optimizing the blanket promotion.
If the firm chose the best treatment arm from the RCT to blanket target,
the firm would blanket a two dollar off promotion and attain expected
profits of $789,868$ dollars. Allowing for continuous treatments,
the firm would blanket $1.70$ dollars off, which generates $808,920$
dollars in expected profit (Table \ref{tab:Profits-table-by-treatments}).
Before running the experiment, the firm historically used a blanketed
$10\%$ off promotion. This yields only $330,278$ in profits which
is $39.21\%$ of the full personalization benchmark. Thus, optimizing
the blanket treatment itself does a sizable amount of work of getting
to full personalization profits. Further allowing for continuous treatments
rather than discrete treatments additionally increases profits.
Second, we then consider the case where the firm offers up to all
eight treatments used in the experiment as possible assigned treatments
for segments, and we report it as the discrete treatments benchmark
in Figure \ref{fig:Profit-to-granular}. This benchmark with one segment
is just the optimized blanket treatment among the discrete treatment
arms at two dollars off. Substantively, this benchmark represents
the best we can do with policy learning with discrete treatment values
because policy learning only leverages extant treatment arms from
the RCT.
For example, to implement this procedure for three segments, we combinatorially
search through the $\binom{8}{3}=56$ possible combinations of treatments.
For each of these treatment combinations, we form segments by assigning
the individual to the treatment that produces the highest expected
profits. We calculate total expected profits by summing the profits
across the segments. We then choose the most profitable combination
of treatments as the discrete treatments benchmark.
From Figure \ref{fig:Profit-to-granular}, we find that the discrete
treatments benchmark does better than segmenting on covariates, but
performs significantly worse than segmenting on consumers' preferences,
optimal treatment levels, and our coarse personalization solution.
These differences demonstrate the benefit of allowing the treatments
to be continuous instead of preselecting the discrete treatment values.
Our coarse personalization solution allows for optimization over the
treatment values themselves; this generates additional profits for
the firm and demonstrates the advantage of our solution over standard
policy learning techniques.
\subsubsection{Rounded treatment levels\label{subsec:Rounded-treatment-levels}}
Decimal treatment levels may be impractical in certain domains. To
address this, we examine our coarse personalization solution with
rounded treatments. We round at the end of each step in Algorithm
\ref{alg:Adapted-Lloyd's-Algorithm} so it only considers rounded
treatments when updating. This procedure ensures that constructed
segments are still geared towards profit maximization.
We find that our coarse personalization solution still does well with
rounded treatments. Table \ref{tab:Profits-table-by-treatments-discrete}
demonstrates the percentage of expected incremental profits with coarse
personalization with the two rounding procedures (at the integer level).
Full personalization with integer treatments only allows for $26$
unique treatments in this setting. Comparing Table \ref{tab:Profits-table-by-treatments}
to Table \ref{tab:Profits-table-by-treatments-discrete}, we see that
we leave some profits on the table with the rounding procedure.
Figure \ref{fig:Profit-to-granular-rounded} demonstrates the performance
of Algorithm \ref{alg:Adapted-Lloyd's-Algorithm} and provides the
profit comparisons for the coarse personalization solution with rounded
treatments at the $0.25$, $0.5$, and $1$ (integer rounding) levels.
Our algorithm continues to perform well despite treatment rounding,
with profits rising as more segments are included.
The optimal rounding level of the treatments will ultimately depend
on the firm's application and how much it is willing to leave profit
on the table for more rounded treatments. Thus, even if the firm only
commits to offering rounded treatments, our solution still performs
well in forming segments.
\section{Surplus analysis\label{sec:Surplus-Analysis}}
We now use our coarse personalization methodology to study how the
personalization level affects consumer and producer surplus. While
it is clear that producer surplus will monotonically increase as the
firm improves its personalization, its effect on consumer surplus
is not clear. Depending on whether their assigned treatment is above
or below their fully personalized treatment, some consumers may experience
higher or lower surplus compared to that of full personalization.
Due to the uncertain effect on consumer surplus, the total surplus
may be higher or lower under coarse personalization than under full
personalization. This section provides an empirical surplus analysis
for our application.
To complete the surplus calculation, we need to impose assumptions
on the individual valuation of the treatments $v_{i}(t_{i,d})$. We
assume that the consumer values the dollar off coupon at face value
($v_{i}(t_{i,1})=t_{i,1}$) and values the percentage off coupon at
the percentage off rate times the average past promotion spend for
an order ($v_{i}(t_{i,2})=t_{i,2}\times\text{ average past promo spend}_{i}$).
These valuations match the cost of issuing the promotion for the firm
and we can treat the promotion as a transfer from the firm to the
consumer that further stimulates consumer demand. We can now compute
the total surplus (Equation \ref{eq:Total-Surplus}) and estimate
the change in surplus for each individual under different levels of
coarse personalization compared to the full personalization benchmark.
We defer the complete surplus calculation to Online Appendix Section
\ref{sec:Surplus-Appendix}.
From Figure \ref{fig:Welfare-decomposition-bootstrap}, we observe
that there is a nonmonotonic relationship of consumer surplus to the
number of segments. Figure \ref{fig:Welfare-decomposition-bootstrap}
plots the change in surplus from coarse personalization to full personalization
and Online Appendix Table \ref{tab:Welfare-decomposition-treatments}
provides the decomposition. The figure's lines represent the bootstrap
means and the bands represent one bootstrap standard deviation. Not
surprisingly, producer surplus is always reduced under coarse personalization
when compared to full personalization.
However, consumer surplus is highest when there is one unique treatment
($L=1$) and decreases as the firm can use more treatments. Consumer
surplus drops significantly after four segments and then drops significantly
again after eight segments. The nonmonotonic shifts in consumer surplus
arise because the firm optimizes its segments without explicitly considering
consumer surplus. Given the firm's control over these assignments,
consumers typically cannot gain substantially at the firm's expense.
Online Appendix Table \ref{tab:Welfare-decomposition-individuals}
demonstrates the surplus decomposition of the change in total surplus
into the change in producer surplus and consumer surplus in ten segments
($L=10$) setting. Each row in the table represents the respective
unique treatment in Table \ref{tab:Treatment-assignments-10}.
From Tables \ref{tab:Treatment-assignments-10} and \ref{tab:Welfare-decomposition-individuals},
we see that most of the customers are assigned to three unique treatments
or segments. In turn, the three segments capture a sizable portion
of the expected profit gain from coarse personalization. As a result,
when the firm moves from three to four segments, it can assign individuals
previously unprofitable under the three-segment scenario to the new,
fourth segment. These individuals likely benefited under the three-segment
arrangement, which explains the substantial decline in consumer surplus
observed in Figure \ref{fig:Welfare-decomposition-bootstrap} upon
transitioning from three to four segments.
Our empirical findings corroborate the nonmonotonic relationship between
the level personalization and consumer surplus in \citet{Dube2017}.
We further see that from Figure \ref{fig:Welfare-decomposition-bootstrap}
and Table \ref{tab:Welfare-decomposition-treatments}, the consumer
surplus gain when assigned a suboptimal treatment dwarfs the loss
in producer surplus. As a result, we see that total surplus is higher
with coarser targeting. These findings differ from the theoretical
equilibrium results in \citet{Bergemann2011}, which finds that total
surplus increases as advertisers are better able to personalize and
meet the needs of the consumers. Even though our surplus calculation
uses a simplified model of the consumer valuation, our results still
recover a complicated nonmonotonic relationship between consumer surplus
and the level of personalization. Future research should utilize
panel data for a more rigorous surplus calculation as well as examine
the long-term effect of personalized promotions on societal surplus.
Lastly, our surplus framework offers a convenient rule for choosing
the number of segments from a societal perspective. From Figure \ref{fig:Welfare-decomposition-bootstrap}
we see that offering three segments maximizes the total societal surplus
while allowing the firm to recover $99\%$ of full personalization
profits. Thus, the firm may be inclined to offer only three unique
treatments and leave $1\%$ of the full personalization profits on
the table.
\section{Discussion\label{sec:Discussion}}
Beyond our empirical application, where we revisited the traditional
segmentation for promotions management, our coarse personalization
procedure can be adapted to various other marketing applications.
In this section, we provide a handful of settings where our framework
can be used. In some of the examples, the assumption that the treatment
vector is nonzero in one treatment dimension can be relaxed and interactions
across treatments can be allowed. However, in practice, this requires
significantly more data to estimate interactions.
In the context of salesforce compensation, managers need to decide
which geographic blocks salespeople should focus on, how much time
the salespeople should spend on each visit, and whether the visits
should be in person or over the phone \citep{Misra2019}. The dimensions
of treatment are the salespeople's time spent and the type of visit.
Further, often only a handful of geographic blocks are assigned to
each salesperson \citep{Zoltners1983}. Managers can use the coarse
personalization procedure to design the optimal geographic blocks
and visits types to maximize expected sales profits.
In the ``contract externality'' setting of \citet{Daljord2016},
salesforce managers select a uniform commission rate and assign salespeople
with heterogeneous performance levels to the contract. ``Contract
externalities'' arise because the composition of the salespeoples
will affect the optimal contract commission rate. \citet{Daljord2016}
focus on a one-dimensional contract rate while our framework allows
for a finite dimensional number of contract components.
We can also apply our methodology to advertising content design. For
instance, \citet{Bertrand2010} find that modifying advertising elements
such as imagery and content significantly increases loan demand, comparable
in effect to changing the loan's interest rate. If the firm chooses
a few optimized advertisements to coarsely personalize and send out,
then the findings from \citet{Bertrand2010} provide the first step's
results in our procedure. The firm can then use our coarse personalization
solution to determine which optimal advertisements to offer and to
whom they should be sent to.
Our framework can also be used to guide pricing decisions. \citet{Chandar2019}
find that across 40 million Uber rides, riders have heterogeneous
preferences over private tipping behavior. If firm managers aim to
maximize tip and overall spending from customers but cannot fully
personalize due to fairness concerns, they can assign different geographic
blocks of riders going to the same location the same baseline price
and suggested default tip rate. By coarsely personalizing, the managers
can both increase profits and address fairness concerns: two people
in the same area searching for a ride to the nearest airport will
face the same optimized price.
\section{Conclusion \label{sec:Conclusion}}
Recent advances in estimating heterogeneous treatment effects have
made granular, incrementality-based targeting and personalization
feasible. However, firms rarely deploy full personalization in practice.
Implementation costs, fairness considerations, and other operational
constraints restrict personalization to a limited number of segments.
We refer to this as the coarse personalization problem, where firms
need to choose what marketing mix to offer and which segments of customers
to offer it to.
In this paper, we utilize methods from the optimal transport literature
to effectively solve the coarse personalization problem. We propose
a two-step approach where the first step estimates the conditional
average treatment effects and the second step solves the constrained
assignment problem. The first step itself is already used in the literature
for full personalization. The second step is a novel application of
optimal transport and maps the distribution of conditional average
treatment effects to a set of discrete segments. We advocate for
a ``personalize then discretize'' approach that forms segments and
chooses their assigned marketing mix variables simultaneously, instead
of the traditional ``discretize then personalize'' approach that
sequentially forms segments and then chooses their assigned marketing
mix variables.
To illustrate the practical relevance of our methodology, we present
an empirical application for promotions management using field experiment
data from a food delivery platform. We find that after forming five
optimized segments, the company is able to recoup over $99.5\%$ of
its expected profits under full personalization. Our approach outperforms
classical sequential procedures that first segment on covariates or
preferences and then form targeting rules. We show our algorithmic
solution is both scalable and easily implementable. These results
have direct managerial implications: We offer a practical way for
firms and marketers to optimally segment and coarsely personalize
their marketing mix.
Our methodology is quite general and can be applied for constrained
assignment problems in many settings even outside marketing. With
clinical trial data, pharmaceutical companies can coarsely personalize
medicine by finding the best drug and dosage combination. The optimal
transport solution can be adapted to address problems such as teacher
assignments with peer effects or worker allocations to collaborative
projects.
Our paper bridges optimal transport theory to practical problems in
the social sciences. Optimal transport methods have been utilized
in economics for many applications and are surveyed in \citet{Galichon2016}.
In computer science, optimal transport has been used for a variety
of problems ranging from training high-dimensional GANs to computer
vision \citep{Peyre2019}. We use optimal transport as a novel way
to solve a practical problem and future research in applying optimal
transport to other problems in social science is already underway.\newpage\begin{spacing}{1.0}{\footnotesize\bibliographystyle{ecta}
\bibliography{./Bibliography/bib}
}\end{spacing}\newpage{}
\section*{Figures}
\begin{spacing}{1.0}
\begin{figure}[H]
\begin{centering}
\includegraphics[width=0.6\textwidth]{Figures/OT-semidiscrete-detailed}\caption{Semidiscrete optimal transport \label{fig:OT-Semidiscrete}}
\medskip{}
\par\end{centering}
{\footnotesize Note: This figure provides a stylized visualization
of the semidiscrete optimal transport problem introduced in Section
\ref{subsec:Optimal-transport} (Equation \ref{eq:MK-Problem}). In
our setup, the allowable couplings $\pi$ are the assignment rule
from the distribution of treatment effects ($P$) to the discrete
distribution of segments ($Q^{\tilde{\mathbf{t}},\mathbf{q}}$) with
assigned treatments $\tilde{\mathbf{t}}=\{\mathbf{t}_{1},\ldots,\mathbf{t}_{5}\}$.
This figure represents the one-dimensional case where the distribution
of treatment effects is discretized into five different segments ($L=5$),
and the discrete distribution represents a coarsened version of the
continuous distribution. The optimal transport solution ($\pi$) ``pushes
forward'' the treatment effect distribution to their assigned segments.
The mathematics behind optimal transport are detailed in Appendix
Section \ref{sec:Mathematics-of-Coarse-Personalization}.}{\footnotesize\par}
\end{figure}
\begin{figure}[H]
\begin{centering}
\includegraphics[width=0.65\textwidth]{Figures/Llyods-Algo-2D-Rectangle}\caption{Adapted Lloyd's Algorithm ($D=2$)\label{fig:Adapted-Lloyd's-Algorithm}}
\medskip{}
\par\end{centering}
{\footnotesize Note: This figure provides a stylized visualization
of Algorithm \ref{alg:Adapted-Lloyd's-Algorithm} with two-dimensional
treatments $(D=2)$ and three segments ($L=3)$. The horizontal axis
represents the treatment values for the first dimension and the vertical
axis represents that of the second dimension. The points on the axes
represent the three segments' assigned treatment values $\boldsymbol{t}_{l}=\{\mathbf{t}_{1},\mathbf{t}_{2},\mathbf{t}_{3}\}$.
The interior points represent the individuals' optimal treatment values
$(t_{i,d}^{*},t_{i,d}^{*})$. The lines represent the Voronoi Cells'
(or segments') boundaries. The square points represent the Barycenter
(or offered treatment) of each cell. An outline of the algorithm is
provided in Appendix Section \ref{subsec:Outline-of-Algorithm}.}{\footnotesize\par}
\end{figure}
\begin{figure}[H]
\begin{centering}
\includegraphics[width=0.75\textwidth]{Figures/Sales-Average-by-Treatment.PDF}
\par\end{centering}
{\footnotesize Note: The bars represent one standard deviation. Profits
(28 days) are the average profits per customer 28 days after the promotion
was issued. The profits are defined as sales minus promotional costs
in Equation \ref{eq:Profit-Equation-Setting}.}\caption{Profits by treatment\label{fig:Profits-by-Treatment}}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{Figures/Concave-HTE-Densities-Log.PDF}\caption{Continuous CATE densities for sales \label{fig:Continuous-CATE-Densities}}
\end{figure}
\begin{figure}[H]
\begin{centering}
\includegraphics[width=0.72\textwidth]{Figures/Segmentation-Procedure-Profit-Comparison-Bootstrap-Presentation-Relabeled-1.PDF}
\par\end{centering}
{\footnotesize Note: Lines show bootstrap means and shaded bands show
one bootstrap standard deviation. The full personalization profits
band reflects the sampling variation across the $100$ bootstrap iterations.
The full personalization benchmark represents issuing $2,206$ unique
treatments when rounded to three significant figures. }\caption{Profit comparisons\label{fig:Profit-to-granular}}
\end{figure}
\begin{figure}[H]
\begin{centering}
\includegraphics[width=0.8\textwidth]{Figures/Sample-Run-Clusters-Assignments-Full-3D-Assigned.JPEG}
\par\end{centering}
\begin{raggedright}
{\footnotesize Note: The optimal treatment values ($t_{i,1}^{*},t_{i,2}^{*}$)
for each individual is plotted for the dollar off and percentage off
promotions. The colors represent each individual's assigned treatment.
The triangular points indicate those individuals who are assigned
a percentage off treatment and the circular points represents the
those who are assigned a dollar off promotion. The assigned treatments'
values are in the legend and in Table \ref{tab:Treatment-assignments-10}.
The incremental expected profits generated from the treatment assignments
for each individual is plotted on the vertical axis. A horizontal
plane plotted at zero expected profits is provided as guide.}{\footnotesize\par}
\par\end{raggedright}
\raggedright{}{\footnotesize{} }\caption{Treatment assignments ($L=10$ treatments)\label{fig:Treatment-assignments-10}}
\end{figure}
\begin{figure}[H]
\begin{centering}
\includegraphics[width=0.7\textwidth]{Figures/Segmentation-Procedure-Profit-Comparison-Bootstrap-Rounded-CT-Relabeled.PDF}
\par\end{centering}
{\footnotesize Note: Lines show bootstrap means and shaded bands show
one bootstrap standard deviation. The full personalization profits
band reflects the sampling variation across the $100$ bootstrap iterations.
The $0.25,0.5,1$ benchmarks represent the coarse personalization
solution with treatments rounded at those respective levels. The coarse
personalization benchmark is rounded to three significant figures.
The full personalization benchmark represents issuing $2,206$ unique
treatments when rounded to three significant figures. }\caption{Profit comparisons for coarse personalization with rounded treatments
\label{fig:Profit-to-granular-rounded}}
\end{figure}
\begin{figure}[H]
\begin{centering}
\includegraphics[width=0.8\textwidth]{Figures/Surplus-Plot-Bootstrap-Relabeled.PDF}
\par\end{centering}
\begin{singlespace}
{\footnotesize Note: Lines show bootstrap means and shaded bands show
one bootstrap standard deviation.}\caption{Surplus decomposition across treatments \label{fig:Welfare-decomposition-bootstrap}}
\end{singlespace}
\end{figure}
\end{spacing}
\newpage{}
\section*{Tables}
\begin{spacing}{1.0}
\begin{table}[H]
\begin{centering}
\caption{Average treatment effects on profits \label{tab:Experiment-ATEs}}
\centering\begin{tabular}{rrr|rr}
Treatment value & ATE (\$) & SE & $t$-stat & $p$-value\\
\midrule
\addlinespace[0.3em]
\multicolumn{3}{l}{{Dollar off}}\\
\hspace{1em}2 & 0.034 & 0.046 & 0.730 & 0.465\\
\hspace{1em}3 & 0.061 & 0.046 & 1.326 & 0.185\\
\hspace{1em}4 & 0.122 & 0.047 & 2.605 & 0.009\\
\hspace{1em}5 & 0.085 & 0.049 & 1.722 & 0.085\\
\addlinespace[0.3em]
\multicolumn{3}{l}{{Percentage off}}\\
\hspace{1em}5 & 0.056 & 0.045 & 1.225 & 0.220\\
\hspace{1em}10 & 0.078 & 0.047 & 1.659 & 0.097\\
\hspace{1em}15 & 0.153 & 0.047 & 3.220 & 0.001\\
\hspace{1em}20 & 0.096 & 0.045 & 2.101 & 0.036\\
\end{tabular}{\small\medskip{}
}{\small\par}
\par\end{centering}
\noindent\raggedright{}{\footnotesize Note: The outcome variable is
profits 28 days after sending out the treatment. The average treatment
effects (ATEs) are estimated from the Causal Forest comparing each
treatment arm to the holdout set of issuing no promotion and represents
the treatment effect on profits. The reported $t$-statistic and the
$p$-value are for a two-sided Welch's $t$-test against the null
hypothesis that the ATE for each promotion is 0.}{\footnotesize\par}
\end{table}
\begin{table}[H]
\begin{centering}
\caption{Treatment assignments ($L=10$) \label{tab:Treatment-assignments-10}}
\centering
\begin{tabular}{r|rr|r}
Treatment ($l$) & Dimension ($d$) & Value ($t_{i,d}$) & $N$\tabularnewline
\hline
1 & Percentage & 12.45 & 7,090\tabularnewline
2 & Dollar & 1.67 & 556,544\tabularnewline
3 & Percentage & 20.00 & 7,375\tabularnewline
4 & Dollar & 2.15 & 54,253\tabularnewline
5 & Percentage & 8.42 & 12,816\tabularnewline
6 & Dollar & 2.79 & 15,532\tabularnewline
7 & Dollar & 1.83 & 214,631\tabularnewline
8 & Dollar & 1.24 & 30,425\tabularnewline
9 & Percentage & 5.17 & 9,234\tabularnewline
10 & Dollar & 1.54 & 295,479\tabularnewline
\end{tabular}{\small\medskip{}
}{\small\par}
\par\end{centering}
\noindent\raggedright{}{\footnotesize Note: The table lists the ten
segments in Figure \ref{fig:Treatment-assignments-10}. The segments'
assigned treatment and size are provided.}{\footnotesize\par}
\end{table}
\begin{table}[H]
\begin{centering}
\caption{Expected incremental profits by segmentation method \label{tab:Profits-table-by-treatments}}
\centering\vspace{-0.5cm}
\par\end{centering}
\begin{centering}
{\small\subfloat[Profit by segmentation method and number of segments]{\begin{centering}
{\footnotesize{}
\begin{tabular}{r|cccc}
& \multicolumn{4}{c}{{\small Profits by segmentation method (\$)}}\tabularnewline
{\footnotesize Number of segments ($L$)} & {\footnotesize Covariates} & {\footnotesize Preferences} & {\footnotesize Optimal treatment levels} & {\footnotesize Coarse personalization}\tabularnewline
\hline
{\footnotesize 1} & {\footnotesize 808,920} & {\footnotesize 808,920} & {\footnotesize 808,920} & {\footnotesize 808,920}\tabularnewline
{\footnotesize 2} & {\footnotesize 808,940} & {\footnotesize 814,377} & {\footnotesize 812,609} & {\footnotesize 829,300}\tabularnewline
{\footnotesize 3} & {\footnotesize 808,969} & {\footnotesize 815,851} & {\footnotesize 824,710} & {\footnotesize 834,085}\tabularnewline
{\footnotesize 4} & {\footnotesize 809,021} & {\footnotesize 826,215} & {\footnotesize 826,581} & {\footnotesize 836,669}\tabularnewline
{\footnotesize 5} & {\footnotesize 809,114} & {\footnotesize 827,432} & {\footnotesize 828,205} & {\footnotesize 838,450}\tabularnewline
{\footnotesize 6} & {\footnotesize 809,156} & {\footnotesize 828,159} & {\footnotesize 828,766} & {\footnotesize 839,417}\tabularnewline
{\footnotesize 7} & {\footnotesize 809,190} & {\footnotesize 828,568} & {\footnotesize 828,970} & {\footnotesize 839,954}\tabularnewline
{\footnotesize 8} & {\footnotesize 809,226} & {\footnotesize 828,817} & {\footnotesize 829,159} & {\footnotesize 840,412}\tabularnewline
{\footnotesize 9} & {\footnotesize 809,247} & {\footnotesize 828,929} & {\footnotesize 829,243} & {\footnotesize 840,761}\tabularnewline
{\footnotesize 10} & {\footnotesize 809,277} & {\footnotesize 829,429} & {\footnotesize 829,315} & {\footnotesize 840,985}\tabularnewline
\end{tabular}}{\footnotesize\par}
\par\end{centering}
{\small}{\small\par}}}{\small\par}
\par\end{centering}
\begin{centering}
{\small}{\small\subfloat[{\small Benchmark profit comparisons}]{\begin{centering}
{\footnotesize{}
\begin{tabular}{c|c}
{\footnotesize Benchmark} & {\footnotesize Profits (\$)}\tabularnewline
\hline
{\footnotesize Firm's blanket promotion (10\% off)} & {\footnotesize 330,278}\tabularnewline
{\footnotesize Optimal experimented blanket promotion (\$2)} & {\footnotesize 789,868}\tabularnewline
{\footnotesize Optimal blanket promotion (\$1.70)} & {\footnotesize 808,920}\tabularnewline
{\footnotesize Full personalization} & {\footnotesize 842,366}\tabularnewline
\end{tabular}}{\footnotesize\par}
\par\end{centering}
{\small}{\small\par}}}{\small\medskip{}
}{\small\par}
\par\end{centering}
\noindent\raggedright{}{\footnotesize Note: The firm's most recent
blanket promotion was $10\%$ off. The optimal experimented blanket
promotion is a $\$2$ off promotion. The optimal blanket promotion
allowing for continuous treatments is a $\$1.70$ promotion. When
rounded to three significant figures, full personalization issues
2,206 unique promotions. }{\footnotesize\par}
\end{table}
\begin{table}[H]
\begin{centering}
\caption{Expected incremental profits by the number of segments (rounded integer
treatments) \label{tab:Profits-table-by-treatments-discrete}}
\centering\begin{tabular}{r|rr}
Number of segments ($L$) & Profits ($\$$) & \begin{cellvarwidth}[t]
\raggedleft
Percent to integer-rounded \\
full personalization $(\%)$
\end{cellvarwidth}\tabularnewline
\hline
1 & 789,868 & 96.50\tabularnewline
2 & 809,849 & 98.95\tabularnewline
3 & 812,964 & 99.33\tabularnewline
4 & 814,335 & 99.49\tabularnewline
5 & 816,677 & 99.78\tabularnewline
6 & 817,753 & 99.91\tabularnewline
7 & 818,116 & 99.96\tabularnewline
8 & 818,265 & 99.97\tabularnewline
9 & 818,394 & 99.99\tabularnewline
10 & 818,471 & 100.0\tabularnewline
\hline
Full personalization (26) & 818,481 & \textendash{}\tabularnewline
\end{tabular}{\footnotesize}{\small\medskip{}
}{\small\par}
\par\end{centering}
\noindent\raggedright{}{\footnotesize Note: Full personalization with
rounded integer treatments involves issuing $26$ unique treatments.}{\footnotesize\par}
\end{table}
\end{spacing}
\pagebreak\newpage{}