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Coarse Personalization
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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 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 Sheshinski1977, concerns about the personalization model's stability Hitsch2024, and issues pertaining to inequity or fairness 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{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. 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 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 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 Smith1956,Wind1978,Gupta1994. More recently, researchers have formed segments based on estimated consumer preferences and responses Kamakura1989a,Bucklin1992,Krieger1996,Bucklin1998a via either continuous or discrete mixtures 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 first construct personalized policies based on heterogeneous treatment effects obtained using now widely available machine learning tools, and in a second step, we use these policies to construct segments of consumers by minimizing profit regret. Crucially, in the second step we 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 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. Bergemann2011 theoretically show as firms personalize more granularly, both producer surplus and total surplus increase. 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 Rossi1996,Shaffer1995. With the advent of online marketplaces, promotions are more readily personalized, customized, and distributed to customers Ansari2003,Zhang2004. The optimization angle to promotions management has been explored in the literature 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 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 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 Chiong2016,Bonnet2017, matching markets Galichon2011, quantile regression Carlier2016, and bounding regression discontinuity design estimates Daljord2019; its general applications to economics are surveyed in Galichon2016. Computational methods for evaluating optimal transport problems are detailed in Peyre2019, and a survey of recent mathematical developments in the field can be found in Villani2009.
Section (ref) overviews the proposed methodology. Section (ref) formally presents our coarse personalization solution: the personalization in the first step is described in Section (ref) and the discretization in the second step is detailed in Section (ref). Our computational solution that adapts Lloyd's Algorithm is described in Section (ref) and its comparison to standard brute-force grid search is provided in Section (ref). We then discuss why using optimal transport is effective for coarse personalization in Section (ref). We apply our methodology to an empirical application for promotions management for a large food delivery firm in Section (ref). The surplus analysis is discussed in Section (ref). We discuss how our framework can be adapted to address other classic marketing problems in Section (ref).
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 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,
where we suppressed the dependence on $\boldsymbol{x}_{i}$ in our notation. The left-hand side in Equation (ref) 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 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 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.
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) and the second step is detailed in Section (ref). 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).
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,
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 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) 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
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,
In the case for feasible treatment $\boldsymbol{t}_{i}$ that is nonzero in dimension $d$, we have $E[\text{R}_{i}|\boldsymbol{x}_{i},T_{i,1}=0,\ldots,T_{i,d}=t_{i,d},\ldots,T_{i,D}=0]=E[Y_{i}|\boldsymbol{x}_{i},T_{i,1}=0,\ldots,T_{i,d}=t_{i,d},\ldots,T_{i,D}=0]-c_{d}(t_{i,d}).$
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 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 ($t_{i,d}=0$),
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,
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 Athey2016,Wager2018,Farrell2020,Farrell2021.
Full personalization represents an upper bound on profits by the principle of maximization in Equation (ref). In our empirical application, we benchmark our coarse personalization solution to full personalization to quantify lost profits due to coarsening.
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), 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) 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) 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
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) is also known as the Monge-Kantorovich problem in the optimal transport literature. A more formal discussion is in Appendix Section (ref).} 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
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)) 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)) 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 $\boldsymbol{x}_{i}$,
We then define $\bar{R}_{i}$ to be the profits for an individual with $x_{i}$ if firm fully personalizes,
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
in the optimal transport problem (Equation (ref)). The cost function in Equation (ref) 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).} 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), 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, $c_{d}(t_{i,d})$.
Assumption (ref) 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) 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)) 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) demonstrates that the optimal transport problem in Equation (ref) is a convex program in treatment values under Assumptions (ref) and (ref). We stress that any strictly concave function of CATEs will satisfy Assumption (ref), and any convex function will satisfy Assumption (ref).
Corollary (ref) 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). The proofs are in Appendix Section (ref).
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) provides a formal derivation of this intuition using the first order condition of Equation (ref).
Coarse personalization can be interpreted as a more regularized version of full personalization. Heterogeneous treatment effect estimates and CATEs are known to be noisy 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 Swaminathan-Joachims-2015,pmlr-v84-kallus18a and operations literature Zhou2022. However, these methods either cannot incorporate the concavity condition from Assumption (ref) 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.
To computationally solve our coarse personalization problem, we adapt a version of Lloyd's Algorithm (Algorithm (ref)), from 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 Pollard1982,NIPS2012_c54e7837.
Figure (ref) 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 Peyre2019. We discuss the benefits of Algorithm (ref) over the brute force solution via grid search in Section (ref). Using the promotions-management setting from our empirical example, we outline the procedure in Online Appendix Section (ref).
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). 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) 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.
In Algorithm (ref), 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).} 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), 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) 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) 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), 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 Lu2016, which is the global optimum by the convexity of the problem.}
Lastly, our results in Section (ref) 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.
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 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) and (ref) 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) and (ref) is recommended.
Algorithm (ref) and the brute force approach yield similar profits. In Online Appendix Section (ref), we describe our grid search implementation and discuss why Algorithm (ref) 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.
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) 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), 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) and in Appendix Section (ref). 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)). Algorithm (ref) 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) and/or (ref) 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).} This probabilistic assignment to segments also appears in the latent class and finite mixture models in the marketing literature 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) and (ref) we show the solution's optimal segments and their assigned treatments are unique and each individual is deterministically assigned to a segment.
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.
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)'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).
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
where the promotion cost depends on the type of promotion issued. Figure (ref) 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). They represent the difference in profits of comparing the treatment arms in Figure (ref) 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)'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
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
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). 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).
We interpret the continuous CATEs, $\tau_{d}(\boldsymbol{x}_{i},t_{i,d})$, as a behavioral sensitivity to dollar off or percentage off treatments 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.
To estimate the continuous CATEs, $\hat{\tau}_{d}(\boldsymbol{x}_{i},t_{i,d})$, we first implement the Causal Forest from Wager2018 in each of the two treatment dimensions. To adjust for randomization concerns in the field experiment (see Appendix Section (ref)), we use generalized regression forests that use propensity score estimates to adjust for possible stratification between the treatment and holdout samples Athey2019.\footnote{We also estimate the continuous CATEs directly using Deep Neural Networks 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 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 $\beta_{i,d}$,
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 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). We then choose the number of segments ($L$) and run Algorithm (ref) 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 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) 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.
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) 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 Misra2021 to compare the continuous CATE parameterization in Equation (ref) to the nonparametric estimates in a holdout treatment and find that our parameterization performs well out of sample (Online Appendix Section (ref)).
To visualize the continuous CATEs, we can turn to Figure (ref) 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) 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) 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), 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) 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) 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) 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) 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) which suggests that a majority of individuals are most receptive to the firm using one dollar off to two dollars off promotions.
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) and Table (ref) 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.
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 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.
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)) 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.
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.
From Figure (ref) and Table (ref), 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 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).
Coarse personalization outperforms segmenting on optimal treatment levels because coarse personalization uses the 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) provides a top-down view of Figure (ref) and shows the segmentation structure of the coarse personalization solution. Appendix Figure (ref) shows segmentation structure when forming segments by optimal treatment levels.
In the coarse personalization solution, the nearest treatment in Appendix Figure (ref) is not necessarily assigned to each individual because the distance metric used for segmentation is in expected profits (Equation (ref)); 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). 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 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) 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.
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{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)).
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). 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), 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.
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) 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) 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) to Table (ref), we see that we leave some profits on the table with the rounding procedure.
Figure (ref) demonstrates the performance of Algorithm (ref) 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.
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)) 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).
From Figure (ref), we observe that there is a nonmonotonic relationship of consumer surplus to the number of segments. Figure (ref) plots the change in surplus from coarse personalization to full personalization and Online Appendix Table (ref) 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) 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).
From Tables (ref) and (ref), 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) upon transitioning from three to four segments.
Our empirical findings corroborate the nonmonotonic relationship between the level personalization and consumer surplus in Dube2017. We further see that from Figure (ref) and Table (ref), 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 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) 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.
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 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 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 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. 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, 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 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. 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.
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 Galichon2016. In computer science, optimal transport has been used for a variety of problems ranging from training high-dimensional GANs to computer vision 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.