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Switchback Price Experiments with Forward-Looking Demand
The proliferation of online e-commerce platforms has afforded sellers an increasingly rich plethora of algorithmic approaches to optimize the pricing of their products; see, e.g., mckinsey,bain,chen2016empirical,caro2012clearance; even brick-and-mortar retail has benefited from these advances with the introduction of digital price tags nicas2015now. A basic tool in this algorithmic toolbox for the seller is {\em price experimentation}: varying the price of a product in a controlled fashion to learn about the price sensitivity of customers---i.e., the {\em demand gradient}, or derivative of the demand function around the current price. Ideally, if the seller can learn this demand gradient accurately, they can use the information to move prices in a revenue maximizing direction. Numerous modern algorithmic dynamic pricing methods are founded on the ability to precisely estimate this demand gradient, ranging from simple gradient ascent to more complex policy gradient methods using deep reinforcement learning liu2019dynamic.
In this paper, we consider a seller of a single product who uses the following simple price experimentation approach to learning the demand gradient, common in online retail. Starting with a current reference price of $p$, during the experiment the seller {\em switches} prices between $p$ and a slightly lower (discounted) price $p - \epsilon$. In particular, the experiment horizon is divided into regular intervals, and in each interval with probability $q$ (resp., $1-q$), the price is set to $p - \epsilon$ (resp., $p$). See (ref) for an example from the price tracking site \url{www.camelcamelcamel.com} that illustrates such a price experiment. (Experiments with this temporal switching behavior are called {\em switchback} experiments, and commonly used in a wide range of online platforms; see lyft-switchback, kastelman2018switchback for details.) The seller measures observed demand during the “high” price periods (when the price is $p$) with observed demand in the “low” price periods (when the price is $p-\epsilon$), and computes a discrete derivative to estimate the demand gradient. In principle, if $\epsilon$ is small, this should be a reasonably accurate estimate of the local demand gradient at $p$.
Although seemingly sound in principle, this approach can be quite problematic if {\em customers} are able to anticipate the possibility that discounts might be offered due to price experimentation. In that case, customers might {\em intertemporally substitute}, and choose to {\em wait} to purchase if they see (relatively) “high” prices, in anticipation of possible future “low” prices. Indeed, the same price trackers like camelcamelcamel that reveal price experiments are the tools that customers can use to strategize in this way, a behavior we refer to as {\em forward-looking}.
It's straightforward to understand the estimation bias that arises as a result. To be precise, we refer to the strategy where the price is fixed at $p$ (resp., $p-\epsilon$) for all time as {\em global control} (resp., {\em global treatment}). Observe that in these two regimes with fixed prices throughout the time horizon, forward-looking customers have no reason to delay. However, in the switchback experiment, customers who arrive to see the price at $p$ can wait to buy at a lower price $p-\epsilon$. This behavior inflates observed treatment demand, and depresses observed control demand, biasing the resulting estimated demand gradient.
Our paper is about understanding, quantifying, and removing this bias. We make three main contributions:
To carry out our analysis, we introduce a benchmark model of forward-looking behavior. In the model we consider, the seller faces an arrival process of forward-looking customers who form {\em rational expectations} with respect to the future price trajectory they expect to see. In our simplest model, all customers discount future rewards with a discount factor $\gamma < 1$ (we also consider a generalization with heterogeneous discount factors). Individual customers have unit demand, and a private value drawn from an unknown distribution; their only decision is whether or not to purchase the good in a given period at the current price, or wait. In global treatment and global control, forward-looking behavior has no effect: if buying is rational, they will buy immediately. But in the experiment, customers may strategically wait as discussed above.
To enable tractable analysis, we consider experiments with a random horizon length, with geometric length of mean $1/\delta$; we carry out our analysis in the limit where $\delta \to 0$. In this setting, we are able to prove a key structural result that simplifies customer behavior. Informally, we show that if a customer with value $v$ arrives in a “high” (control) price interval, then they will behave {\em as if} they were not forward-looking, but with value $v - {\cal B}$, where ${\cal B}$ is a “shading” term that captures the option value of purchasing in the future at a potentially lower price. Notably, ${\cal B}$ only depends on the buyer's patience level and $q$, as well as the expected future discount over the high price, in a multiplicatively separable manner; crucially, this shading amount does {\em not} depend on the buyer's value. This structural characterization is essential to our analysis.
Our {\em first} key result exploits this structural characterization to exactly quantify the estimation bias of the two-price switchback experiment and associated canonical estimator. In particular, we show that the resulting estimator is a multiple of the true demand gradient with a multiplicative factor that depends on the patience level $\gamma$. This factor depends on whether the seller can {\em distinguish} which sales are from new customer arrivals (“same-day” sales), and which are from prior arrivals. When the seller can distinguish these categories of customers, the factor is $1 + \frac{\gamma}{1-\gamma}q$; if it cannot, the factor increases to $\frac{1}{q}+\frac{\gamma}{1-\gamma}$.
This result suggests it should be easy for the seller to avoid the estimation bias: all it needs to do is correct for the multiplicative factor. But if the seller does not know $\gamma$, then a fundamental problem arises, even if the seller can distinguish same-day sales from delayed sales. In particular, given a stream of data from the two-price switchback experiment, our {\em second} key result is that the seller cannot identify the true demand gradient in a two-price switchback experiment. Informally, we can see this issue as follows: suppose the seller observes that in the data, customers are delaying purchases at a relatively high rate. The seller can't know if this is primarily because customers are highly price sensitive, and not very patient; or primarily because they are not too price sensitive, but they are very patient. In other words, a {\em range} of combinations of true demand gradient and discount factor $\gamma$ can give rise to the {\em same} distribution on experimental data.
So what is the seller left to do? Our {\em third} main contribution is to show that a simple modification of the switchback experiment to use {\em three} prices, and an associated simple estimator, suffice to completely remove the preceding estimation bias. In particular, suppose the seller randomly selects between the control price $p$ and {\em two} treatment prices $p-\epsilon, p-2\epsilon$. For simplicity suppose that both treatment prices are chosen with probability $q$ each, and the control price is chosen with probability $1 - 2q$, and also suppose the seller can distinguish between same-day and delayed sales.
Let $\Delta_1$ be the difference in average same-day sales between periods with price $p-\epsilon$ and $p-2\epsilon$; and let $\Delta_2$ be the difference in average same-day sales between periods with price $p$ and $p-\epsilon$. These are both discrete estimates of the same demand gradient, and so if buyers were not forward looking, both differences should be essentially the “same” (within $O(\epsilon^2)$ error). Any larger error must be due to forward-looking behavior of buyers.
We show the remarkable fact that the following simple estimator is asymptotically (as $\epsilon \to 0$ and $\delta \to 0$) {\em unbiased} for the true demand gradient:
This estimator uses the difference of differences $\Delta_2 - \Delta_1$ to “debias” the (naive) difference $\Delta_1$, taking advantage of the fact that if $\Delta_2 - \Delta_1$ is of $O(\epsilon)$, it must be due to a distortion introduced by forward-looking buyers. This result uses the same structural characterization of buyer behavior in the limit $\delta \to 0$: since the shading factor ${\cal B}$ depends in a multiplicatively separable way on the discount factor $\gamma$ and the expected future price discount, we can use the difference in $\Delta_2 - \Delta_1$ to remove the bias introduced by forward-looking behavior.
The resulting three-price switchback design and estimator are remarkably easy to implement and use; indeed, we encourage practitioners to employ this small change to standard practice, given the substantial robustness gained. Despite the simplicity of the estimator, the analysis is more involved than the intuition described above: in fact, we show asymptotic unbiasedness in a fairly general setting, where the seller may not be able to distinguish same-day and delayed sales, and where buyers have heterogeneous and unknown patience levels that are arbitrarily correlated with their values. When the seller can only observe total sales and not sales on the same day, the form of the estimator is slightly different, albeit equally simple, than the form in Equation (ref).
\paragraph{Forward-Looking Demand and Intertemporal Substitution} Demand estimation in the presence of forward-looking demand and intertemporal substitution has a long history in economics and in particular in industrial organization and econometrics. Some notable works are those of diewert1974intertemporal,blundell1989microeconometric,eichenbaum1990estimating,hendel2006measuring,hendel2013intertemporal,nevo2016usage,ching2020identification,gandhi2021empirical,birge2021markdown,akca2015identifying,nevo2001measuring,aguirregabiria2013recent,nevo2002manufacturers,hartmann2006intertemporal,chen2020pricing. Another line of work in operations management considers consumers' forward-looking behavior, mostly from a modeling perspective (e.g., swinney2011selling, su2007intertemporal, etc.), but also from an empirical one li2014consumers.
su2007intertemporal and li2014consumers are closest to our paper. su2007intertemporal designs revenue-maximizing pricing policy with a continuum of buyers. Unlike our work which makes no assumption on buyers' private type space, su2007intertemporal considers a binary setting - binary buyer values and binary buyer patience levels, under which a characterization of the optimal policy is possible.
The idea of using price variation to separately identify demand responses with forward-looking behavior is reminiscent of previous research on forward-looking consumers, which relied on observational rather than experimental data li2014consumers. However, due to the observational nature of the data in li2014consumers, more complex assumptions are required both on the structural model of buyer behavior as well as on the existence of particular forms of price trajectories over time; rather than solely assuming random price variation. Since our goal is only estimating demand gradient around a particular point, and since we assume random price assignment over time, we can achieve identification with fewer assumptions.
\paragraph{Interference in Pricing Experiments} Several recent papers study interference in market experiments and in particular as it relates to pricing experiments blake2014marketplace,wager2021experimenting,munro2021treatment,li2023experimenting,johari2022experimental,holtz2020reducing,farias2022markovian,le2023price. Several types of interference patterns have been considered, such as interference via equilibrium effects in price experiments in two-sided platform wager2021experimenting,munro2021treatment, or interference due to congestion effects li2023experimenting,johari2022experimental,farias2022markovian or interference due to substitution effects across products dhaouadi2023price. However, no prior work analyzed intertemporal interference due to forward-looking behavior and intertemporal substitution; which is the topic of our work.
From a technical perspective, the closest to our work is that of dhaouadi2023price. dhaouadi2023price present an analysis of estimation bias due to interference stemming from substitution effects across similar products in a pricing experiment. They consider a single period of experimentation, with applications to random treatment on both sides of a two-sided market. The bias in their model can be corrected through a single experiment, supplemented by pre-experimental data. In contrast, we examine a scenario where interference emerges across time periods in a switchback experiment. We offer a novel characterization of bias for the demand gradient, which takes a multiplicative factor on the true gradient. Furthermore, the method to correct bias in their paper, no longer works when the interference effect happens across different time periods. Instead, we show that this bias can be corrected via an experiment with three prices.
\paragraph{No-regret reserve price optimization with strategic buyers} Our work is also related to the line of work on dynamic pricing and regret minimization with strategic buyers mohri2014learning,mohri2014optimal,mohri2016learning,mohri2015revenue,tang2017reinforcement,chen2022bayesian,feldman2016online,liu2024contextual,mashiah2023learning, chen2018robust. These works typically assume a stationary i.i.d. stochastic demand. The key intuition of the solutions underlying most of the work in this line is that a seller should change prices infrequently, so that buyers are not patient enough to anticipate the price discount that will occur when the price gets updated. However, in a constantly changing demand system, this can lead to very infrequent adaptation and inaccurate learning. Moreover, our goal is in the analysis of a widely used practice in price experimentation, namely switchback experiments. Perhaps closest to our work is chen2018robust, which considers online revenue maximization with randomly arriving customers. Despite the differences discussed above, chen2018robust also considers selling to forward-looking buyers who discount future utility with a patience level.
\paragraph{Dynamic pricing and demand learning} Our work is also related to the line of work in operations research on dynamic pricing with an unknown demand agrawal2021dynamic,keskin2017chasing,keskin2014dynamic. Compared to agrawal2021dynamic, keskin2014dynamic which make structural assumptions about the demand and keskin2017chasing which assumes the demand variation is bounded, our work captures more general demand-generating processes only with locally approximable assumptions.
We consider an infinite discrete time horizon setting, where at each time period $t\in \{1, 2, \ldots\}$, a unit mass of buyers arrives in a market for a single product with infinite supply of units and a posted per-unit price $p_t$ set by the seller. Each buyer who arrives has a value $v$ for the product and has unit-demand. The value of an arriving buyer remains fixed throughout his stay on the market. Moreover, the distribution of values of the arriving unit mass of buyers at each period is denoted by $F_t$. The distribution of value $F_t$ is randomly chosen at each period, based on some arbitrary stochastic process.
\paragraph{Forward-Looking Buyer Behavior} At each period, each buyer forms a belief about future prices and decides between waiting or buying. The buyer discounts future utility with a discount factor $\gamma\in[0, 1)$, which we refer to as the buyer's patience level. Define $\mu_t(\gamma)$ as the marginal density of patience level $\gamma$ at period $t$. The buyer derives utility if they end up buying at period $t+\tau$ at price $p$, which is quasi-linear and takes the form:
The parameters $(v, \gamma)$ as well as the belief about the future prices can vary across buyers (subject to a consistency constraint which we will introduce in (ref)) and will be referred to as the buyer's type, which in this case is multidimensional. Moreover, the type of the buyers is arbitrarily distributed at each period, subject to the marginal distribution of values being $F_t$. Other than that the parameters that constitute the type of a buyer (e.g. patience level, value) can be arbitrarily correlated.
In a given period $t$, with posted price $p_t$, the buyer has two possible actions:
The buyer's strategy $s$ is a function of the historical prices $h_t = (p_1,\dots, p_{t-1}, p_t)$ (including current price $p_t$), with $s(h_t) = 1$ representing a purchase, and $s(h_t) = 0$ representing waiting. Moreover, the buyer leaves after he makes a purchase, i.e.\ $s(h_t) = 0$ if $s(h_i) = 1$ for some $h_i$ ($i<t$), a sub-history of $h_t$. The buyer calculates expected future utility from waiting with strategy specified by $s$:
where the expecation is taken over the buyer's beliefs about the stochastic future evolution of prices. The buyer selects a strategy that maximizes expected utility. Note that there may exist buyers who never buy the product. For example, a buyer with value $v$ lower than any of the posted prices will never buy.
Throughout the paper, we assume that the buyer's patience level is bounded away from $1$. If, instead, the buyer has patience level $\gamma = 1$ and derives the same future utility as the current period, the buyer who can afford the lowest price over an infinite time horizon will always wait, regardless of the expected waiting time.
We define the static demand $D_t(p\ifx\empty\else^{()}\fi)$ as the mass of arriving buyers who would have purchased immediately at period $t$, in a setting where the price is fixed at $p\ifx\empty\else^{()}\fi$ for all time periods -- meaning there is no price variation. Note that $D_t(p\ifx\empty\else^{()}\fi)$ is the survival function of the buyers' valuation distribution for the item, i.e. $D_t(p\ifx\empty\else^{()}\fi)=1 - F_t(p\ifx\empty\else^{()}\fi)$. We make the following benign assumptions about the stochastic process that generates the daily demand functions:
\paragraph{Demand Gradient} For conciseness, we refer to the limit function $D(p)$ as the average static demand. The main estimand that the seller is interested in identifying is the gradient of the average static demand at some reference price $p_*$, i.e.,
For most of the paper except in (ref), we refer to the average static demand as the demand, and the average static demand gradient as the demand gradient. We assume that both the demand gradient and the demand hessian is bounded. Moreover, for the construction of a uniformly asymptotically unbiased estimator, we assume that the demand function can be uniformly second-order approximated at $p_*$.
For our positive result on an asymptotically unbiased estimator, we need to make further regularity assumptions on the conditional demand gradient function, conditional on the patience level. Denote with $F_{t\mid \gamma}$ to be the CDF of the buyer values, conditional on the patience level $\gamma$ and $D_{t\mid\gamma}(p) = 1 - F_{t\mid \gamma}(p)$ to be the corresponding conditional demand function. Moreover, we let $\mu_t(\gamma)$ to be the distribution of patience levels of arriving customers on day $t$. We make the following regularity assumption on the conditional demands conditioning on the buyer having patience $\gamma$:
\paragraph{Revenue Gradient} The seller is also interested in infinitesimal adjustments to the price around $p_*$, so as to improve upon the average static price revenue function, defined as:
For this purpose, the seller is also interested in estimating the gradient of the revenue function:
Note that the revenue gradient can be derived directly from the demand and the demand gradient.
As will become evident in later sections, identifying the demand function $D(p_*)$ is easy. Thus, we will mainly focus on estimating the static demand gradient $d(p_*)$.
Trivially, the seller can identify, in the limit as $\delta\to 0$, the static demand and the static revenue at a price point $p_*$, by simply posting the price and measuring the average mass of purchases at that price point (since the stochastic demand process converges in the limit almost surely). However, to identify the demand gradient (equiv. revenue gradient), the buyer needs to experiment with prices and set prices around the price point $p_*$.
\paragraph{Switchback Experiment} A switchback experiment is characterized by a set of $K$ prices $S = \{ p^{(0)}, p^{(1)}, \dots, p^{(K)}\}$, with $p^{(i)}\in [0, \bar{p}]$ for some constant $\bar{p}$, together with associated probabilities $\ensuremath{\boldsymbol{q}} = (q_0, q_1, \ldots, q_K)$, such that $\sum_i q_i = 1$. As a convention prices are sorted in decreasing value: $p^{(0)}> p^{(1)}>\dots> p^{(K)}$. At each time $t$, the experiment period ends with probability $\delta$. If the experiment does not end, the seller chooses a price $p_t$ from the set $S$ according to the probabilities $\ensuremath{\boldsymbol{q}}$. Prices are chosen i.i.d. each time period. We will refer to the tuple $(S, \ensuremath{\boldsymbol{q}}, \delta)$ as the switchback design. Since we focus on estimating the demand gradient in the limit as $T\to \infty$, we mainly consider the limit where the experiment period is long enough, i.e.\ $\delta\to 0$.
\paragraph{Local Price Experiments} Since we are interested in identifying the demand gradient at a reference price point $p_*$, we focus mainly on {\em local price experiments} where the prices in $S$ are {\em bounded} within an $\bar{\epsilon}$ range of the reference price $p^{(0)}=p_*$. We will refer to $\bar{\epsilon}$ as the {\em price perturbation bound}.
We mainly consider the case where the reference price is $p^{(0)}$, i.e., alternative prices are “discounts” relative to the reference price $p^{(0)}$. This assumption is without loss of our main results,\footnote{Our main negative result on the insufficiency of two-price switchback experiment generalizes to the setting where the reference price is the higher price and our positive results obviously remain true if one is given more freedom in the choice of experimental price levels.} but matches the reality of price experiments, which are typically implemented via random small discounts, primarily driven by the intuition that customers should not be worse off when treated. Moreover, for all of our results we will only be analyzing two-price or three-price experiments, i.e., $K = 2$ or $3$, which also matches the small number of distinct discount levels that arise in retail practice jha2024can (e.g., 10% or 20% discount).
\paragraph{Rational Expectations} After a switchback experiment starts, we assume the buyers form rational expectations about the experiment setup, including the prices in $S$ and associated probabilities $\ensuremath{\boldsymbol{q}}$, and the experiment ending probability $\delta$.
Note that we do not assume buyers form rational expectations about post-experimental prices, since the seller may adjust post-experimental prices according to the experimental results in a manner that is hard for the buyer's to accurately anticipate. Therefore, at period $t$, each buyer is also endowed with a private type $\phi_t$, the belief about the post-experimental prices at period $t$. This belief $\phi_t$ describes an infinite sequence of conditional probabilities: for each $\tau\in \mathbb{N}_{+}$, conditioning on the experiment ends at $t+\tau$, the distribution $\phi_{t+\tau|t}$ is over post-experiment prices. In the limit where the experiment horizon is long enough (i.e.\ $\delta\to 0$), the buyers' beliefs about post-experimental prices has a vanishing effect on their strategy within the experiment.
\paragraph{Data} At the end of each day, the seller observes the realized demand $C_t$, that is, the mass of sales realized in period $t$. In some of our analysis, we will also assume that the seller observes the {\em same-day} sales $N_t$, i.e., the part of the mass of the realized demand $C_t$ that arises from buyers who arrived at period $t$.
\paragraph{Estimators} An {\em estimator} $\hat{d}(\cdot)$ for the demand gradient $d(p_*)$ is a function of the realized sales mass $\ensuremath{\boldsymbol{C}}$, and, when observed, of the same-day sales mass $\ensuremath{\boldsymbol{N}}$. An estimator can also depend on the setup parameters of the local experiment, i.e. $S, \ensuremath{\boldsymbol{q}}, \delta$, but we will suppress this dependence. Moreover, where obvious from context, we will also suppress the functional dependence on $\ensuremath{\boldsymbol{C}}$ and $\ensuremath{\boldsymbol{N}}$, and simply refer to $\hat{d}(p_*)$ as “the estimator” at reference price $p_*$.
Throughout, we focus on estimators that are unbiased in the limit of a vanishingly small price perturbation bound and ending probability of the experiment, defined as follows.
Note that the seller does not know the distribution of the values and patience levels or the post-experimental beliefs of the buyers. Hence, the estimator cannot be a function of these quantities.
In this section, we analyze the waiting behavior of buyers for any switchback price experiment under the discounted quasilinear utility model defined in Section (ref) and the rational experimental price expectations (Assumption (ref)).
In order to define the strategy of a buyer, it is useful to introduce the following quantities related to the switchback design. We denote with $P$ a random price drawn from the distribution of experimental prices. Then $\ensuremath{\mathcal Q}(p) := \Pr(P\leq p)$ is the probability of receiving a smaller price than $p$, and $\ensuremath{\mathcal E}(p)=\mathbb{E}\ifthenelse{\not\equal{}{}}{_{}}{}\!\left[{\def\middle|{\middle|}p - P\mid P\leq p}\right]$ is the expected discount conditional on receiving a smaller price than $p$. Note that $\ensuremath{\mathcal Q}(p)\ensuremath{\mathcal E}(p) = \mathbb{E}\ifthenelse{\not\equal{}{}}{_{}}{}\!\left[{\def\middle|{\middle|}(p - P)_+}\right]$.
Lemma (ref) provides an intuitive, albeit crucial result in our analysis, characterizing buyer behavior. In the limit as $\delta\to 0$, essentially, a buyer with value $v$, when faced with a price $p_t$, will be as if their value is slightly lower $v - \mathcal{B}$, where the option value of waiting term $\mathcal{B}$ is a function of the price level and the buyer's patience level. Now, for all patience and price levels, the behavior will always be so that buyers seem to have an additive shaded value given by the option value of waiting when deciding whether to accept or not the current price. We formalize this in the following corollary.
An important aspect of this characterization is that the amount of value-shading $\mathcal{B}$ introduced by the buyer's forward-looking behavior does not depend on the buyer's value $v$.
In this section, we analyze two-price switchback designs, with $S=\left\{p^{(0)}, p^{(1)}\right\}=\{p_*, p_*-\epsilon\}$ and probabilities $\ensuremath{\boldsymbol{q}} = (q_0, q_1)=(1 - q, q)$. This is a frequently used experimental design in practice and arguably the simplest type of a switchback price experiment one would consider, where on each day the seller will offer a discount $\epsilon$ over the reference price of the product, with some probability $q$. Specialized to this switchback design, the characterization in Corollary (ref) shows that as $\delta\to 0$, a buyer with value $v$ and patience level $\gamma$, will purchase when the reference price $p_*$ is posted if:
The buyer will purchase when the discounted price $p_*-\epsilon$ is posted if $v\geq p_*-\epsilon$, because there is no benefit of waiting given that posted price.
Our first main result is an impossibility result that uncovers the limitation of such fixed discount experiments in dealing with forward-looking buyer behavior. We show that there cannot exist any asymptotically unbiased estimator of the demand gradient when faced with data that stem from a two-price switchback design. This impossibility result holds, even under the simple case where the buyers share a homogenous patience level that is unknown to the seller and even when the seller can observe the same-day sales masses $N_t$.
To prove this result, we show that any estimator necessarily depends only on the following two summary statistics of the data generating process, $D(p_*+\frac{\gamma}{1-\gamma}q\epsilon)$ and $D(p_*-\epsilon)$, since the observed data can be uniquely described by these two quantities of the underlying process. Moreover, we further show that the limit of the expected value of any asymptotically unbiased estimator must necessarily essentially only depend on the difference of these two summary statistics, i.e. only on $\Delta = D(p_*+\frac{\gamma}{1-\gamma}q\epsilon)-D(p_*-\epsilon)$. Finally, we show that one can construct two distinct demand-generating processes with different buyer patience levels, such that the demand gradients $d_1(p_*),d_2(p_*)$ are different but the observed demand differences $\Delta_1, \Delta_2$ are the same. Intuitively, these demonstrate that it is not possible to separately identify demand gradients with patience levels; hence, there cannot be an unbiased estimator in the limit. With a two-price switchback, the seller is not able to distinguish the following two buyer behaviors: 1) the buyers who could afford to buy at the higher price but select to wait for a discount, and 2) the buyers who could not afford at the higher price and select to wait.
Next, we take this impossibility result one step further and characterize the form of the bias for the natural naive estimator that one would consider (if forward-looking behavior was ignored), under the assumption that all buyers have the same patience level. We show that the bias of this estimator admits a multiplicative form, essentially yielding a multiple of the true demand gradient, by a constant that depends on the patience level of the buyers and the probability of the discounted price.
We first prove this result in the case where the seller only observes purchase masses $\{C_t\}_{t\in[T]}$ on each day. When faced with data from such a two-price experiment, a natural estimator is the difference in the time-averaged sales mass between the days where a price $p_*$ was posted vs. the days where $p_*-\epsilon$ was posted. We define the average number of purchases, within each price level as:
and consider the natural estimator $\hat{d}(p_*)$ of demand gradient $d(p_*)$ as:
As the experiment ending probability $\delta\to 0$, (ref) characterizes the bias of the naive estimator in two-price experiments.
Hence, the naive estimator multiplicatively overestimates the demand gradient due to the forward-looking behavior of customers. This result implies that removing this bias can be achieved under a very stringent set of assumptions where all buyers have the same patience and this patience level is known to the seller. When the patience level is not known, the overestimation bias can be reduced, but not totally eliminated with a two-price switchback experiment, as we argue below. The overestimation bias can be decomposed into two sources:
The second source of bias can be removed if the seller has the ability to monitor the buyers' arrival times. In that case, the seller can obtain an unbiased estimator of $D(p_*-\epsilon)$ by removing the mass of buyers who wait and buy and only considering the same-day sales mass $N_t$, instead of $C_t$. More specifically, suppose that the seller observes the mass of purchases $\{N\ifx\empty\else^{()}\fi_t\}_{t\in[T]}$ from buyers who arrive on the same day. The purchases are aggregated into same-day purchase masses within periods that had the same price level
and we can consider the natural analogue of the naive estimator $\hat{d}(p_*)$ of demand gradient $d(p_*)$, where we ignore buyer's that arrived earlier and waited for the discount:
In this case, the expected estimation of demand gradient can be shown to be $\left(1+\frac{\gamma}{1-\gamma}q\right)\cdot d(p_*)$.
The proof follows an identical argument as in the proof of (ref) and is omitted. Even though this estimator is still multiplicatively biased, the factor in the bias is a smaller constant, especially for small values of $q$.
In this section, we show that the seller can debias the estimation of the demand gradient by conducting a switchback experiment with three prices. We consider in particular an experiment with equal discounts, i.e., $S=\{p^{(0)}, p^{(1)}, p^{(2)}\}=\{p_*, p_*-\epsilon, p_*-2\epsilon\}$ and associated probabilities $\ensuremath{\boldsymbol{q}} = (q_0, q_1, q_2) = (1 - q_1 - q_2, q_1, q_2)$.
For any such switchback design, we will construct an asymptotically unbiased estimator, even in the setting where the seller only observes the total sales mass every day $C_t$, but cannot track the buyer's arrival time and hence cannot observe which part of these sales were same-day sales $N_t$, or waiting-buyer sales $C_t-N_t$. Moreover, we will allow for the buyer patience levels to be arbitrarilty heterogeneous and correlated with their values (unlike the negative result in (ref), which assumed a homogeneous patience level).
We will accomplish this result in two steps. First, in (ref) we will construct such an estimator for the simpler setting where the seller observes both $N_t$ and $C_t$ as a stepping stone. Then, in (ref), we will extend the result to the case when the seller has less information and does not monitor the buyer's arrival time. We construct the unbiased estimator by reducing the problem to the former setting in (ref).
Suppose that the seller observes the mass of purchases $\{N\ifx\empty\else^{()}\fi_t\}_{t\in[T]}$ from buyers who arrive on the same day. The purchases are aggregated into same-day purchase masses within periods that had the same price level $N\ifxi\empty\else^{(i)}\fi$, as defined in Equation (ref). The asymptotically unbiased estimator of the demand gradient is constructed by a weighted average of same-day purchase masses:
The construction of our unbiased estimator relies mainly on exploiting different waiting biases at different prices. With three price levels we note that there are many ways that we can attempt to identify the demand gradient via pairwise comparisons of demand at different pairs of price levels. Even though all these ways of identify the demand gradient are asymptotically biased, they have different limit biases. However, the difference in these pairwise differences contains helpful information about that bias. In particular, we can show that this difference of differences is a factor times the bias of each individual estimate, where the factor only depends in the experiments randomization probabilities and not on the patience level of the buyers.
To gain more intuition, consider the case when the experiment offers both discount levels with equal probability, i.e. $q_1=q_2$. In this case, the unbiased estimator simplifies to the following intuitive formula:
In words, we estimate the gradient by using the difference in observed same-day demands between prices $p_*-\epsilon$ and $p_*-2\epsilon$ and then correct it by subtracting the difference of differences, when measuring the gradient by using the difference between demands from $p_*$ and $p_*-\epsilon$ and correspondingly when measuring it by using the difference between demands from $p_*-\epsilon$ and $p_*-2\epsilon$.
Intuitively, with three price experiments, we have two possible ways of measuring the demand gradient, that, ideally, in the absence of forward looking behavior, should converge to the same thing. The first is $\Delta_1 = (N^{(1)}-N^{(2)})/\epsilon$ and the second is $\Delta_2=(N^{(0)} - N^{(1)})/\epsilon$. Both of these two quantities are asymptotically biased. For instance, consider the case when all buyers have the same patience level $\gamma$. Then an argument similar to the one we used in Theorem (ref) shows that both of these quantities are asymptotically biased, albeit with the following multiplicative form of asymptotic bias:
Importantly, the bias of the first estimate is smaller than the bias of the second estimate and in a manner that scales only with the probability of seeing a smaller price. In particular, buyers who face the top price $p_*$ are more likely to wait, than buyers that face $p_*-\epsilon$, as there are higher chances of a discount in the future. From the above argument, we see that the difference of differences, reveals exactly the bias of $\Delta_1$!
Thus by subtracting this difference from the estimate that is solely based on $\Delta_1$, we are completely removing its asymptotic first order bias. See also Figure (ref), for a pictorial representation of this argument. The two differences in demands capture forward-looking behavior with different intensities, and by taking the difference of such differences one can isolate the bias term. Our proof formalizes this argument and also shows that it can be extended to the case where buyers have heterogeneous patience levels that are arbitrarily correlated with their values.
When the seller does not monitor buyer arrival time, we showed that the seller can estimate the demand gradient with unbiased estimators of same-day purchase masses $N\ifx0\empty\else^{(0)}\fi, N\ifx1\empty\else^{(1)}\fi, N\ifx2\empty\else^{(2)}\fi$. The seller only observes the total sales mass $C_t$ in each period $t$. Using the total sales in each period, we show that the seller can construct unbiased estimators $\widehat{N\ifxi\empty\else^{(i)}\fi}$ of the aggregate same day sales $N\ifxi\empty\else^{(i)}\fi$ for $i=0, 1, 2$ as follows:
The proof of (ref) is based on the following high-level arguments. First, it is easy to see that only same-day buyers will purchase at a price of $p\ifx0\empty\else^{(0)}\fi$ and therefore using total sales vs. same-day sales does not change the calculation for the high price level. The second argument is the observation that as the length of the experiment goes to infinity, every buyer who has a value above $p\ifx2\empty\else^{(2)}\fi$, will eventually buy some day. Thus, the aggregate total sales, regardless of price level, converge to the average limit demand at the price point $p\ifx2\empty\else^{(2)}\fi$. As we showed earlier, this is also the limit of the aggregate same-day sales demand at this lowest price point, since everyone that arrives at this price point will immediately buy. The harder part of the argument is identifying an un-biased estimate for the aggregate same-day sales at the medium price level $N^{(1)}$. The total sales at this medium price point is a mixture of same-day sales and sales from prior periods from buyers that rejected price $p\ifx0\empty\else^{(0)}\fi$, but are willing to accept $p\ifx1\empty\else^{(1)}\fi$. Based on this reasoning, it can be shown that the agggregate total sales from a price point of $p\ifx1\empty\else^{(1)}\fi$ converges to the aggregate same-day sales at that price point, plus a term that converges to a scaled difference in same day sales between the high and medium price points $\frac{q_0}{q_1+q_2}(\mathbb{E}\ifthenelse{\not\equal{}{}}{_{}}{}\!\left[{\def\middle|{\middle|}N\ifx1\empty\else^{(1)}\fi}\right]-\mathbb{E}\ifthenelse{\not\equal{}{}}{_{}}{}\!\left[{\def\middle|{\middle|}N\ifx0\empty\else^{(0)}\fi}\right])$. Thus total aggregate sales at the medium price point is a linear combination of aggregate same-day sales at the medium price point and aggregate same-day sales at the highest price point. From this observation and using the fact that we already have an un-biased estimate of aggregate same-day sales at the highest price point, we can derive the un-biased estimate for the medium price point.
Having access to these unbiased estimates $\{\hat{N\ifxi\empty\else^{(i)}\fi}\}_{i=0, 1, 2}$, the seller can construct the analogue of the unbiased estimator of $d(p_*)$ as in (ref) in a plug-in manner:
The following theorem is an immediate corollary of (ref) and (ref), hence we omit its proof.
The estimates in Equation (ref) can be expressed in terms of aggregate total sales mass quantities $C^{(i)}$ as defined in Equation (ref).
Using this equivalence, in essence, (ref) shows that the difference $N^{(1)}-N^{(2)}$ can be replaced by the un-biased estimate $q_2 (C^{(1)} - C^{(2)})$ and the difference $N^{(0)}-N^{(1)}$ can be replaced by the un-biased estimate $(q_1 + q_2) (C^{(0)} - C^{(1)})$. In other words, the aggregate difference in total sales between two price points can be shown to be in expectation a scaled multiple of the aggregate difference of same-day sales of these two price points, where the factor only depends on the discount probabilities. Using the aggregate total sales expressions, the estimate in Equation (ref) can be simplified as follows (see Appendix (ref) for the derivation):
Thus, we derived that the unbiased estimator is simply the difference of differences in total sales, appropriately scaled. When both discounts are given with equal probability $q$, then the scaling factor is $2q$, i.e. the total probability of seeing a discounted price.
If the seller is constantly experimenting so as to adjust prices, then one wonders if static demand and its gradient are relevant quantities that the seller should pay attention to when adjusting prices. Under constant experimentation, maybe the relevant quantity is the gradient of the forward-looking demand that accounts for the price randomization. The latter could potentially be first-order different from the gradient of the static demand. In other words, if prices are never constant, then why would we care about the hypothetical demand under constant pricing?
We show that even in a platform that constantly runs local pricing experiments around some reference price, the static demand gradient is the quantity that one needs to estimate to decide how to adjust the reference price $p_*$ after one local price experiment finishes and another begins!
\paragraph{Dynamic Demand Gradient} Consider two switchback experiments with different reference prices $p_*, p_*'$ and the same price perturbation bounds $\epsilon = p_* - p\ifxK\empty\else^{(K)}\fi = p_*' - p^{(K)_2}$. The reference price adjustment $\alpha = p_*'-p_*$ is small. Instead of the static demand gradient fixing the price constant, the seller is interested in the demand gradient under experiments. Formally, for the first experiment with reference price $p_*$, define the dynamic demand under experiment:
where the expectation is taken over $T$, the day on which the experiment ends and realized purchase mass $C_t$'s. For the second experiment with price $p_*'$, define the dynamic demand under experiment the same as $D_{s_1}$:
The seller is interested in the following dynamic demand gradient in the limit
where $\lim_{\alpha\to 0} \frac{D_{s_2} - D_{s_1}}{\alpha}$ is the dynamic gradient under experiments. This gradient is the change in total demand if we first run the switchback experiment $s_1$ and then run switchback experiment $s_2$, by slightly adjusting the reference price of $s_1$ by a small $\alpha$. Identifying the demand gradient allows the seller to decide whether she should increase or decrease the reference price.
(ref) shows that the demand gradient under experiment is consistent with the static demand gradient.
We can similarly define the revenue gradient under experiments as:
for $i\in \{1, 2\}$. The dynamic revenue gradient is
\paragraph{Dynamic Revenue Gradient} (ref) shows that the dynamic revenue gradient is consistent with the static revenue gradient, if the platform only adjusts the reference price of the switchback experiment. We write $\text{Rev}_{t|\gamma}$ as the static conditional daily revenue, from the sub-population of buyers with patience level $\gamma$. We also define as $D_{t|\gamma}(p\ifx\empty\else^{()}\fi)$ the daily demand that comes from the sub-population of buyers with patience level $\gamma$, that is, it is the total mass of buyers in period $t$ who have patience level $\gamma$ and will purchase if the price is fixed to $p\ifx\empty\else^{()}\fi$ over the horizon.