ISSN (Print) 0473-453X

Discussion Paper No. 1203

ISSN (Online) 2435-0982

SOCIAL LEARNINGAND STRATEGIC PRICINGWITH RATING SYSTEMS

Chia-Hui Chen
Kong-Pin Chen
Junichiro Ishida

March 2023

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Social Learning and Strategic Pricing with Rating
Systems
Chia-Hui Chen∗

Kong-Pin Chen†

Junichiro Ishida‡

March 7, 2023

Abstract
Despite widespread use in online transactions, rating systems only provide summary
statistics of buyers’ diverse opinions at best. To investigate the consequences of thiscoarse form of information aggregation, we consider a dynamic lemons market in whichbuyers share their evaluations anonymously through a rating system. When the buyers
have diverse preferences, the value of a good rating depends endogenously on the seller’s
pricing strategy, which in turn creates complicated dynamic interactions and results instochastic price fluctuations. Occasional flash sales induced by the rating system yielda non-trivial welfare effect that stands in sharp contrast to standard adverse selection
models: all buyers are weakly better off with information asymmetry than without.
Incentivizing buyers to leave ratings may backfire by exacerbating the seller’s strategic
pricing incentives.

Keywords: online platform; rating system; anonymity; preference diversity; pricefluctuation; dynamic adverse selection
JEL Classification: D82; D83; L11∗

Institute of Economic Research, Kyoto University, Japan. E-mail: chchen@kier.kyoto-u.ac.jp.
Institute of Economics, Academia Sinica, Taiwan. E-mail: kongpin@gate.sinica.edu.tw.
‡
Institute of Social and Economic Research, Osaka University, Japan. E-mail: jishida@iser.osaka-u.ac.jp.
†

1

Introduction

The development of platform transactions has provided a channel for buyers to learn thequality of a seller that is very different from traditional modes of learning such as word-of-mouthcommunication and observational learning. Since transactions are centralized on a platform,a standardized reputation mechanism, which we broadly refer to as the “rating system,” canbe designed to streamline the transmission of information. Under a rating system, buyers canpost their “ratings” for the products or sellers based on their experiences. Potential buyers canobserve the ratings, or a summary of them, left by previous buyers before they make purchasing
decisions. This trading environment constitutes a special form of social learning, which enables
potential buyers to collect information on a much wider scale than they could through wordof-mouth communication and also to gain access to more detailed information—past buyers’
evaluations rather than their actions—than through observational learning.
Rating systems are, however, subject to substantial communication frictions. There areseveral possible reasons for this,1 but the one that stands out is the fact that under a rating
system, buyers share their evaluations anonymously without knowing the identities, much lessthe preferences, of the raters. This inherently anonymous nature of rating systems creates acomplication when the buyers have diverse preferences, as they may be satisfied for different
reasons, and hence what a “good rating” implies could differ considerably across them. Particularly relevant in this regard is the difference in the way buyers evaluate the tradeoff between
price and quality, where some are more sensitive to quality while others are more sensitive to
price. This structure suggests a possibility that a seller—even a low-quality one—can induce
good ratings from those who are more concerned about price by deliberately underpricing andtemporarily boost her reputation. Under this structure, the pricing strategy of a seller plays
a dual role: on one hand, it plays the conventional role of rent extraction; on the other hand,
it also influences the buyer’s rating directly and hence the beliefs of subsequent buyers andfuture revenues indirectly. The seller’s pricing decisions must strategically balance the tradeoff
between current and future revenues as dictated by the rating system.
In this paper, we develop a dynamic adverse selection model to examine this strategic
incentive and its consequences on trading dynamics and welfare. The trading environment we
1

As we will later argue in Section 6.1, it is also important that platforms must in principle rely on thevoluntary participation of buyers: although some platforms offer small rewards for providing a rating, the size ofthe rewards is typically quite negligible, and it is in general difficult for platforms to induce buyers to exchangedetailed information in an incentive-compatible manner. There are actually two sides to this limitation. On
one hand, it is highly time-consuming for any buyer to provide a detailed account of how and in what sense
he is satisfied. On the other hand, even if some previous buyers are somehow willing to provide such detailed
reports, current buyers often do not bother to go through a pile of reports to extract relevant information.

2

consider consists of a long-lived seller and a sequence of short-lived buyers who arrive randomly
over time. The quality of the good supplied by the seller is either high or low and is her private
information. A crucial assumption of our analysis is that the buyers are also heterogeneousand have diverse preferences in that they are either price shoppers (who only care about price)or value shoppers (who also appreciate good quality). Upon arrival, a buyer observes thehistory of ratings left by previous buyers and the current price offer and decides whether topurchase the good given his belief.2 If he chooses to purchase the good, he learns its quality
after consumption and leaves a good rating (with some positive probability) if he is “satisfied.”
Within this setup, we argue that any rating system, which aggregates diverse opinions ofprevious buyers into some summary statistics, entails an unintended consequence of inducing
strategic behaviors from the sellers. The driving force behind our argument is the combinationof anonymity and preference diversity (buyer heterogeneity) that is inevitably present in platform transactions. In the presence of these two factors, the rating system can only provide acrude indicator of quality because there are no cohesive standards among buyers for when to
give a good rating: specifically, while value shoppers are satisfied only by high quality, there
are also price shoppers who are satisfied by low price; as such, good ratings provided by priceshoppers do not necessarily indicate high quality. The buyers can observe ratings scatteredover time and the current price but have no means to differentiate informative signals (goodratings from value shoppers) and uninformative ones (those from price shoppers) due to theanonymous nature of rating systems. This implies that the value of a good rating is endogenous
in our model, depending on the seller’s pricing strategy at the time of purchase, which marksa crucial point of departure from dynamic adverse selection models with exogenous news suchas Daley and Green (2012) and Kaya and Kim (2018).
Throughout the analysis, we focus on a natural class of equilibria, which we call monotone
equilibria, and derive properties that hold robustly within this class. This class of equilibria
exhibits an intuitive structure where both types of seller charge a higher price to extract surplus
when the belief is relatively high, while they hold a flash sale to attract a broader customer
base and garner good ratings when it is low. The seller’s pricing strategy thus repeatedly
and stochastically fluctuates between a high and a low price, giving rise to trading dynamics
induced by the rating system. When the seller charges a low price, price shoppers are satisfied
and leave good ratings which convey no information about the seller’s quality. This strategic
pricing introduces noise into the buyers’ inference process and consequently slows down social
learning. This finding is closely in line with Fan et al. (2016) who find that sellers tend to2

We therefore consider a posted price platform. Although eBay has made auctions with rating systems
popular in the C2C environment, nowadays most of the items listed in eBay (or any other transaction platforms)
are in posted prices (Einav et al., 2018).

3

underprice to acquire good ratings when their reputation is low, which incidentally indicatesthat price shoppers indeed exist and play a major role in rating systems.
All monotone equilibria of our model exhibit two absorbing states. At the upper absorbing
state, the high-quality seller is identified as such, and the belief stays constant thereafter.
Moreover, by belief consistency, only the high-quality seller can reach the upper absorbing stateon the equilibrium path, meaning that the rating system we consider displays no false negative.
In contrast, at the lower absorbing state, value shoppers do not have enough confidence in theseller and never purchase. Once the belief reaches this point, therefore, any good rating could
only come from price shoppers and its reputation value is entirely lost. Since the low-quality
seller cannot reach the upper absorbing state, she will eventually be trapped into the lower
absorbing state. The problem is that the high-quality seller also has a positive probability ofreaching the lower absorbing state, implying that the rating system may exhibit a false positive.
This result suggests that in addition to transaction delays, noisy social learning entails a social
cost that remains strictly positive even as the discount rate vanishes. To measure the extent
of this efficiency loss, we explicitly compute the asymptotic probability of identifying the highquality seller as such. Remarkably, the asymptotic probability is constant across all monotone
equilibria of our model and depends only on two factors: the seller’s initial reputation and thesensitivity of the value shoppers to the good’s quality.3
It is worth noting, however, that despite this efficiency loss, distributional consequences ofnoisy social learning are not uniformly negative across all parties involved. In fact, the seller’s
strategic pricing incentive yields a non-trivial welfare effect that stands in sharp contrast to
standard adverse selection models: the value shoppers are strictly better off with information
asymmetry than without. This result stems from a crucial feature of our model that theseller’s information manipulation occurs through her pricing decisions. To see this, if the seller
is known to be of high quality for sure, there is no point in offering a low price as the seller
only loses her current revenue without gaining any reputation. In this case, the seller always
charges a price that equals the expected quality, and the value shoppers can capture no surplus.
The situation changes substantially once there is uncertainty about the seller’s type: when the
belief is close to the lower absorbing state, even the high-quality seller is forced to salvage her
reputation, which allows some value shoppers to capture positive surplus. Interestingly, an
important qualifier of this result is that there are some price shoppers around, because thefrequency of receiving a good rating would not depend on the current price otherwise, i.e.,
preference diversity benefits the value shoppers.
We also identify the seller-optimal equilibrium and examine how its efficiency is affected by
3

To be more precise, the asymptotic probability is constant across all pooling equilibria of our model whichconstitute a larger set of equilibria including some non-monotone ones.

4changes in some underlying parameters of the model. We find that as the seller becomes lesspatient, the efficiency of social learning improves. This is because when the seller discounts the
future payoffs more heavily, there is less incentive to offer a low price to garner good ratings.
This result suggests a counterintuitive possibility that the high-quality seller is often madebetter off, while the value shoppers are worse off, as the market environment becomes morecompetitive with higher market exit rates. Also, when satisfied buyers leave ratings at a higher
frequency, the seller’s strategic incentive might intensify and slow down social learning. Thismeans that although many platforms adopt various measures to encourage buyers to leave
ratings, such an attempt might backfire and actually impede social learning.
A distinctive feature of transactions mediated by platforms is that the platforms typicallyhave some control over the flow of information. In the baseline model, we consider a rating
system with minimum information content, where a buyer can only leave a good rating when
he is satisfied. To examine the impact of providing more information on trading dynamics, we
extend this rating system in two different directions. In the first, we make the history of prices
publicly observable, on top of the history of ratings. In the second, we allow the buyers toleave bad ratings when they are unsatisfied. We argue that the first extension may drastically
improve the efficiency of social learning while the second has little qualitative impact.

2

Literature

Our analysis belongs to the vast literature on dynamic adverse selection and reputationbuilding. Although the literature has developed in various directions, there are very few works
which incorporate strategic pricing into this framework.4 As we will detail below, the key
departures of our analysis from the previous literature are summarized by the following twoelements: (i) both types of seller are strategic and attempt to maximize their payoffs; and (ii)they strategically set prices to manipulate the flow of information to their advantage.
Starting with Milgrom and Roberts (1982), Kreps and Wilson (1982), and Fudenberg andLevine (1989), many reputation models build on the framework where one long-run playerinteracts with a sequence of myopic short-run players, as in our trading environment.5 This
strand of literature assumes that the long-run player is either a strategic type, who maximizes
his own payoffs, or a commitment type, who simply follows a pre-specified plan of action.
4

An obvious technical challenge is that if one allows the seller to set a price, it works as a signal and hencechanges the nature of the problem. For this reason, adverse selection models in trading environments typicallyconsider buyer-offer games, where the buyer makes an offer and the seller either accepts or rejects.
5
See Liu (2011), Liu and Skrzypacz (2014), and Pei (2020, 2022) for more recent examples of this approach.
A continuous-time version of this strand of literature is developed by Faingold and Sannikov (2011).

5

While earlier models of this strand of literature look at the situation where all the past actions
are perfectly observable to the current players, subsequent works extend this setting and incorporate imperfect observability of past actions (Fudenberg and Levine, 1992; Cripps et al.,2004). Among them, Ekmekci (2011) introduces a “rating system” which aggregates all theobservable variables into a message in an arbitrary way and shows that a finite rating systemwith information censoring allows the long-run player to perpetually sustain a reputation.6
The focus of this literature is different from ours, as it is generally concerned with whether the
strategic type can approximately achieve his best feasible payoff, often called the Stackelberg
payoff in the literature, when there are some commitment types around.
In our model, the seller strategically sets prices to interfere with the buyers’ inference
process—a strategy that may be broadly referred to as “signal jamming” (Holmstrom, 1999;
Fudenberg and Tirole, 1986).7 Related in this respect are Ekmekci et al. (2022), who analyze
a dynamic stopping problem with manipulable signals. They consider a situation where aprincipal decides whether to terminate the game when an opportunity arrives stochasticallywhile continuously receiving a signal about an agent’s type. The agent in their model is either
strategic (called noninvestible in their model) or committed (investible), where the strategic
type chooses how much effort to exert at each instance to manipulate the signal-generatingprocess. As noted above, a crucial point of departure is that both types of seller in our model are
strategic. Aside from this, it is also important to note that the seller’s information manipulationin our model occurs through pricing choices and hence affects not only the learning process but
also the distribution of surplus directly. As we will discuss in more detail in Section 5.2, theseelements of our model generate different forms of dynamic interaction and different welfare
implications.
Several works analyze dynamic adverse selection models coupled with additional information sources. Daley and Green (2012) consider a dynamic lemons market where the value of along-run seller’s asset is gradually and publicly revealed to the market via some news process
and characterize trading dynamics in this environment. Kaya and Kim (2018) consider a similar setup but instead introduce a news process that provides a private signal to each potential
buyer and show that the private nature of the news process gives rise to diverse patterns of
trading dynamics. These models share an important feature with our model that buyers haveaccess to an additional, albeit imperfect, information source. A key distinction is that the news
6

This is in contrast to Cripps et al. (2004) who are the first to analyze the long-run consequences of
reputation building and point out that it is impossible to sustain a reputation in the long run under imperfectmonitoring.
7
In our analysis, signal jamming works in a slightly different way. As in Holmstrom (1999) and Fudenberg
and Tirole (1986), the buyers correctly anticipate this manipulation, but it can still affect the way they aggregate
information because strategic pricing effectively introduces noise into the rating system.

6

process in their models is exogenous whereas the rating system in our model is endogenous.8
Because of this feature, asymptotic learning fails in our model as even the high-quality seller
is driven out of the market with some probability.
Finally, while our analysis centers around the seller’s strategic incentive to influence theflow of information, there are some works that examine the role of rating systems with more
direct focus on the process of buyer learning and its asymptotic properties (Crapis et al., 2017;
Ifrach et al., 2019; Acemoglu et al., 2022). These works consider a rating system similar toours, where a buyer leaves a good rating if his utility is above some threshold with preference
heterogeneity among buyers, but abstract away from information asymmetry and the seller’s
strategic behaviors. Most notably, Acemoglu et al. (2022) construct a symmetric-information
model of Bayesian learning from online reviews and investigate the speed of learning under
different rating systems with an exogenously fixed price.9 Their analysis identifies a selection
effect which arises from the fact that the types of user who purchase the good depend on
the information available at the time of purchase. Although cast in a different framework,
this consideration is also important in our analysis because different pricing strategies attract
different types of buyer, which in turn affects the value of a good rating.

3

Model

3.1

Setup

Consider a dynamic lemons market with a long-lived seller and a sequence of short-lived
buyers who arrive randomly over time. Time is continuous and extends from 0 to infinity.
Seller. The seller, who has an unlimited supply of a good, lists it on a platform for sales byposted price. The seller is of fixed quality, which is either high (i = H) or low (i = L). The
seller’s type is her private information, and the common initial belief that the seller is of highquality is q0 . At each t, the seller posts a price pt , where the cost of production is normalized
8

Aside from the exogenous news process, another crucial difference is that the seller in their setting only
has a single indivisible good, so that the seller of an asset can signal her type by passing on transaction
opportunities. Signaling thus takes place in a different form. There are now several works which incorporate
signaling via exit decisions in dynamic adverse-selection models (Bar-Isaac, 2003; Gul and Pesendorfer, 2012;
Chen and Ishida, 2018).
9
Acemoglu et al. (2022) later extend their analysis and briefly consider the case with endogenous pricing.
Ifrach et al. (2019) also consider a similar symmetric-information environment, both with and without endogenous pricing. In either case, however, strategic interactions between the seller and buyers are not their main
focus. Crapis et al. (2017) analyze optimal monopoly pricing in an asymmetric-information environment with
non-Bayesian information updating, but restrict their attention to the case where the monopolist can adjust
the price at most once.

7

to 0 for both types. The seller maximizes the discounted sum of profits with discount rater > 0.
Buyers. The buyers have heterogeneous preferences, and hence what a good rating embodies
is potentially different across them. To capture this aspect in the simplest manner, we assume
that each arriving buyer is either a price shopper or a value shopper. Let uP (i, p) (uV (i, p)) be
the utility of a price (value) shopper when he purchases from the type i seller at price p. The
price shoppers do not care about quality, and we normalize their utility from consumption to0, i.e., for both i = L, H,
uP (i, p) = −p.
In contrast, the value shoppers appreciate quality, and their utility is given by1 − pif i = H,uV (i, p) =−k − p if i = L,
where k > 0 is a parameter that measures the degree to which they value quality.10 Each
type of buyer arrives according to a Poisson process: for an interval of time [t, t + dt), a priceshopper arrives with probability mdt, and a value shopper with probability dt. Each arriving
buyer is myopic and exits the market immediately, regardless of whether he makes a purchaseor not. In what follows, we refer to a buyer who arrives at time t as buyer t for clarity.
Rating system. Upon observing the price pt , buyer t, if any, decides whether to make apurchase given his belief about the seller. The true quality is discovered after consumption,and the buyer leaves a good rating with probability λ ∈ (0, 1] at time t if his utility is weakly
positive.
Remark: In our model, the buyers can only provide good ratings when they are satisfied,but have no choice but to remain silent when they are not. This specification is motivatedby the fact that ratings are predominantly positive in online platforms (Nosko and Tadelis,
2015; Tadelis, 2016; Zervas et al., 2021).11 For most part, we focus on this simplest setting
10

It is worth noting that works such as Crapis et al. (2017), Ifrach et al. (2019), and Acemoglu et al. (2022)
also incorporate buyer heterogeneity, but the nature of buyer heterogeneity in these works qualitatively differsfrom that in the current setting. In their works, the heterogeneity term is additive, so that buyers differ interms of the threshold to give a good rating. This is different from our setting where the heterogeneity term ismultiplicative, so that buyers differ in terms of the sensitivity to quality.
11
Fradkin et al. (2021) argue that this constitutes the largest source of inefficiency in the rating system.
There are many conceivable reasons for why buyers are reluctant to leave bad ratings, e.g., for fear of retaliation,
reciprocation or psychological discomfort of giving negative feedbacks. In this paper, we do not explore why

8for clarity, which allows us to illuminate how the seller’s pricing incentives are affected by therating system in a transparent manner. We will later discuss the case where there are more
rating categories in Section 6.2 and show that our main insight still holds even when the rating
system becomes “finer.”

3.2

Information and strategies

Buyer t observes all the past ratings left by previous buyers and the current price offer pt .
Observe that the ratings are publicly observable while the current price offer is only privately
observable to the current buyer. Let ht be the history of ratings (including the time at which
each rating is left) up to time t. The public belief at time t is defined as the probability thatthe seller is of high quality conditional on ht and is denoted by qt . In addition, buyer t observes
pt and further updates his belief. We denote the private belief of buyer t by q̂t , which is theprobability conditional on both ht and pt .
We look for a Markov perfect Bayesian equilibrium in pure strategies (hereafter, simply anequilibrium). Let π i : [0, 1] → R+ be the strategy of the type i seller, which is a mapping fromqt to a non-negative price pt .
Definition 1. The two types of seller adopt a pooling strategy at q if π L (q) = π H (q); otherwise,
they adopt a separating strategy. An equilibrium is a pooling equilibrium if π L (q) = π H (q) for
all q and a separating equilibrium if π L (q) 6= π H (q) for some q.
If the two types adopt a pooling strategy at some qt , the current price offer reveals no
information and q̂t = qt . It is important to note, however, that the implication of “pooling” inour model is qualitatively different from its conventional one and should thus be interpretedwith caution: even if the seller reveals her type by a separating price offer, this information
will not be transmitted to the future buyers, meaning that separation at this stage does notcontribute to building up her reputation (see Section 3.3 for more on this point).
Since the problem faced by myopic buyers is rather straightforward, instead of defining theirstrategies formally, we simply take them as if they are a non-strategic information-generatingprocess. First, a price shopper purchases the good and is satisfied if and only if the price is0, independently of his belief q̂t . Second, given q̂t , a value shopper purchases the good if pt is
no greater than p(q̂t ) ≡ q̂t − k(1 − q̂t ), and is satisfied if the seller is of high quality. As such,at any time t, a price shopper leaves a good rating with probability λ if pt = 0, and a value
this is the case, but rather take this nature of ratings as given and examine its consequences. In fact, since
positive feedbacks play a critical role for their success, platforms even try to impose costs for consumers to leave
negative ratings (Zervas et al., 2021). There are also reports on how Airbnb was accused of deleting negative
reviews (see, e.g., https://qz.com/1333242/airbnb-reviews/)

9

shopper does so if p(q̂t ) ≥ pt and the seller is of high quality, also with probability λ. Let
γ i (pt ; q̂t ), or γ i (pt ) for short, denote the rate at which the type i seller receives a good rating.
k
If q̂t ≥ q ≡ 1+k, we haveλm for p = 0,tLγ (pt ; q̂t ) =0for pt ∈ (0, p(q̂t )],λ(1 + m) for p = 0,tHγ (pt ; q̂t ) =λfor pt ∈ (0, p(q̂t )].

If q̂t < q, value shoppers would never purchase the good, so that γ H (0; q̂t ) = λm andγ H (pt ; q̂t ) = 0 for any pt > 0.

3.3

The static signaling problem and equilibrium selection

Although our analytical focus is on the seller’s strategic response to the rating system, ourmodel also entails a static signaling problem at each instant, given that an informed party—theseller—makes an observable pricing choice. The signaling problem in our model is somewhat
irregular, as it differs from a typical setting characterized by the single-crossing property. The
most distinctive feature of our signaling environment is that it is static in the sense that thecurrent price is observable only to the current buyer, and any information conveyed by the priceis not transmitted to the future buyers. As such, separation at this stage does not contributeto improving her future reputation, and hence the high-quality seller has no strong incentive
to separate from the low-quality one by pricing differently.
As typical in signaling models, our model admits a plethora of signaling equilibria, butstandard refinement arguments cannot be applied directly to obtain a sharp prediction due toits structure. Although it is possible to enumerate different signaling equilibria and analyze
trading dynamics for each possible case, such an exercise would be rather tedious and provide
little economic insight because all equilibria more or less exhibit similar dynamic properties(see Sections 5.1 and 5.2 for more detail). In the subsequent analysis, therefore, we focus ona class of equilibria which we deem as the most reasonable and admits a simple equilibrium
structure. Specifically, we restrict our attention to the following class of equilibria:
1. (Pooling equilibria) Both types of seller adopt a pooling strategy in equilibrium, i.e.,π L (q) = π H (q) for all q;
2. (Full surplus extraction) Whenever the seller offers a strictly positive price, she always
offers the price that equals the maximum willingness-to-pay of a value shopper, both on
and off the equilibrium path.

10Below, we comment on some technical issues specific to our signaling environment and provide
arguments for our equilibrium selection.
Pooling equilibria. Although one might be tempted to focus on separating equilibria, aclose inspection reveals that the structure of our signaling environment is not conducive toseparation. First, the static nature of the problem suggests that the seller has no incentive tobuild reputation by sacrificing current gains: separation is beneficial only to the extent that it
induces the current buyer to purchase the good, and any reputation gains can be capitalized
only by charging a higher price to the current buyer. This essentially rules out all separating
strategies in which the low-quality seller offers a higher price than the high-quality seller. Given
this, the high-quality seller must set a higher price to separate from the low-quality seller, but
separation in this direction is clearly difficult to achieve as the low-quality seller can easily
mimic. In Section 5.1, we formalize this intuition and show that pooling equilibrium is the
only possible equilibrium if we focus on a natural and intuitive class of equilibria, which we callmonotone equilibria, in which the value functions are monotonically increasing in the belief.
Full surplus extraction. Even if we restrict our attention to pooling equilibria, the multiplicity of equilibria still remains. To fix ideas, suppose that the seller chooses to offer a strictly
positive price at some belief qt , and the maximum willingness-to-pay of a value shopper is p(qt )in some pooling equilibrium (where q̂t = qt ). The value shopper is then willing to accept anoffer as long as p(qt ) ≥ pt . This means that we may construct an arbitrary equilibrium inwhich π L (qt ) = π H (qt ) = ηp(qt ) for some η < 1.12 We argue, however, that in our posted-priceenvironment where the seller retains all the bargaining power, it is unlikely that an equilibrium
in which the seller is forced to offer a suboptimal price survives in the long run.

4

Analysis

4.1

Preliminaries

The notion of full surplus extraction implies that the seller’s choice at each instant isessentially binary, either to choose a low price pt = 0 or a high price pt = p(qt ) > 0. For clarity,
we thus represent the seller’s strategy by a simpler and more intuitive form, defined over this12

For instance, this can be done by assigning q̂t = 0 to all off-path deviations. Standard refinement
arguments, such as the Intuitive Criterion and D1, have no bite in selecting among different pooling equilibria
because both types benefit equally from deviating to any pt ∈ (ηp(qt ), p(qt )]. To see this, consider an equilibrium
in which π(qt ) = π L (qt ) = π H (qt ) < p(qt ) for some qt . Suppose that we observe a deviation to some pt ∈(π(qt ), p(qt )]. For both types, this deviation is profitable if and only if the belief assigned to it is such that thecurrent buyer accepts pt , i.e., q̂t is high enough to satisfy p(q̂t ) ≥ pt .

11

binary choice with no direct reference to pt . Let σ i : [0, 1] → {0, 1} denote the (pure) strategy of
the type i seller, where she offers the low price if σ i (qt ) = 0 and the high price if σ i (qt ) = 1, with
the interpretation that the type i seller is “having a sale” at qt if σ i (qt ) = 0. In what follows,
we refer to σ i (qt ) = 0 and σ i (qt ) = 1 as the low-price and the high-price policy, respectively.
Define σ ≡ (σ L , σ H ) as the seller’s strategy profile and σ(q) ≡ (σ L (q), σ H (q)) ∈ {0, 1}2 as the
profile at belief q. We also denote by st ∈ {0, 1} the seller’s actual pricing choice at time t(as observed by buyer t) where st = σ L (qt ) = σ H (qt ) on the equilibrium path (in a pooling
equilibrium). Let q̂ts be the private belief after observing st = s ∈ {0, 1}.
Since the price cannot be negative, a value shopper purchases the good only if his belief
is at least as high as q. For qt ∈ [0, q), there is a continuation equilibrium in which bothtypes of seller adopt the low-price policy.13 In this region, only price shoppers would purchase,and hence any good ratings could only come from them. Since good ratings no longer reflect
quality, there is no way for the seller to improve her reputation once qt dips below q. The
seller’s continuation payoff is then 0 for both types, and the game effectively ends at thatpoint. We thus only define the seller’s strategy over Q ≡ [q, 1] for simplicity, and say that theseller “exits the market” once qt reaches q. Note also that if the initial prior q0 is below q,
no transactions take place on the equilibrium path. To rule out this trivial case, we make thefollowing assumption.
Assumption 1. q0 > q.
Given our binary-choice framework and focus on pooling equilibria, the seller’s equilibrium
strategy can be expressed by two mutually exclusive belief sets (Ql , Qh ), where σ(q) = (0, 0)
if q ∈ Ql and σ(q) = (1, 1) if q ∈ Qh , with the restriction that Ql ∪ Qh = Q and Ql ∩ Qh = ∅.
For clarity, we refer to Ql and Qh as the low-price and the high-price region, respectively. Notethat Ql and Qh may or may not be connected. When both Ql and Qh are connected, we
say that the strategy is a threshold strategy. Given this, we also say that an equilibrium is a
threshold equilibrium if both types of seller adopt a threshold strategy.

4.2

Belief updatingWith abuse of notation, we now redefine γ i (·; q̂t ), or γ i (·) for short, in terms of the binary
choice where for st = s ∈ {0, 1} and q̂ts ≥ q,γ L (s; q̂ts ) = λm(1 − s), γ H (s; q̂ts ) = λ + λm(1 − s).
13This constitutes an equilibrium if the belief after deviating to the high-price policy is low enough so thatno buyers would purchase. Under Assumption 2 which we will postulate below, this is the unique continuation
equilibrium within the class of equilibria under consideration.

12

When q̂ts < q, there is no difference between the low-price and the high-price policy, and
γ i (s; q̂ts ) = λm(1 − s) for i = L, H and s = 0, 1. Given the seller’s strategy σ and the current
belief qt , the belief at the next instant is updated to qt+dt = qt− (σ(qt )) if the seller receives norating and jumps up to qt+dt = qt+ (σ(qt )) if she receives a good rating, where
qt− (σ(qt )) = qt − γ H (σ H (qt )) − γ L (σ L (qt )) qt (1 − qt )dt,H

qt+ (σ(qt )) =(1)

H

qt γ (σ (qt )).
H
H
qt γ (σ (qt )) + (1 − qt )γ L (σ L (qt ))(2)

Observe that a good rating does not ensure high quality (because it might be left by a satisfied
price shopper), and its absence does not indicate low quality (either because no buyer arrives
or because a buyer arrives and is satisfied but fails to leave a rating). In what follows, we often
suppress σ(·) and simply write qt− and qt+ whenever it is not confusing.
It is easy to see that if γ H (σ H (qt )) > γ L (σ L (qt )), the belief decreases continuously as longas the seller receives no good rating. For qt ≥ q, this condition always holds in any pooling
equilibrium because γ H (s) − γ L (s) = λ for both s = 0, 1, which implies that the belief follows
q̇t = −λqt (1 − qt ),for all qt ∈ Q in the absence of good ratings. Once the seller receives a good rating, qt jumps
up to qt+ . In particular, if σ L (qt ) = 1, then γ L (1) = 0 and qt jumps up to 1 after receiving agood rating. In contrast, once qt dips below the threshold q, we have γ L (0) = γ H (0) = λm,
and the belief stays constant indefinitely as noted above. Both of these properties reflect anessential aspect of our model where the reputation value of a good rating is endogenous anddepends on the seller’s strategy profile at the time of purchase.

4.3

Seller’s optimal strategy

Let V i (·) denote the value function of the type i seller. For qt ∈ Q, the value function isdefined as
V i (qt ) = maxs∈{0,1}p(q̂ts )sdt + e−rdt E[V i (qt + dqt ) | σ, s] ,where
E[V i (qt + dqt ) | σ, s] = γ i (s)dtV i (qt+ ) + (1 − γ i (s)dt)V i (qt− ).

13

(3)

This formulation leads to an intuitive incentive compatibility constraint for each type. From(3), σ i (qt ) = 1 if and only if
e−rdt E[V i (qt + dqt ) | σ, s = 0] ≤ p(q̂t1 )dt + e−rdt E[V i (qt + dqt ) | σ, s = 1].
In the limit, the condition above is reduced to(γ i (0) − γ i (1))∆V i (qt ) ≤ p(q̂t1 ),(4)

where ∆V i (qt ) ≡ V i (qt+ ) − V i (qt ) is the reputation value of a good rating. The left-hand sideof (4) measures the future benefit brought by the low-price policy, and the right-hand side
measures the loss of current revenue. Condition (4) provides an important characterization of
the tradeoff faced by the seller in framing her pricing policy.

4.4

Equilibrium

There are two key observations which help us identify the equilibrium. First, since only thehigh-quality seller provides strictly positive benefits and is able to charge a strictly positive
price (to value shoppers), both types of seller have an incentive to present themselves as
the high-quality one. Second, since future buyers cannot distinguish good ratings from value
shoppers (informative signals) and those from price shoppers (uninformative signals), this canbe achieved by offering the low price and inducing good ratings from price shoppers. As is
made clear in (4), this gives rise to the tradeoff between the low-price and the high-price policy.
To balance this tradeoff, the seller may occasionally have a sale for qt > q, i.e., even when thewillingness-to-pay of a value shopper is strictly positive. Note that the incentive to have a saleis intensified as the belief gets closer to the lower bound, because the seller is forced to exit
the market once qt gets below q. These considerations suggest that there should exist someequilibria in which both types of seller adopt the high-price policy if and only if qt is greaterthan some threshold q > q.
Consider a threshold strategy characterized by a pair of thresholds (q, q) such that
1 if q ≥ q,t
σ H (qt ) = σ L (qt ) =0 if q > qt ≥ q.

(5)

We will henceforth call q the lower threshold and q the upper threshold. Since the lowerkthreshold is exogenously fixed at q = 1+k, each threshold equilibrium is uniquely characterized
by the upper threshold q. Under this class of strategies, we denote by V i (·; q) the value function
14

of the type i seller when the upper threshold is set at q. In the case of threshold strategies,
the belief sets (Ql , Qh ) reduce to Ql = [q, q) and Qh = [q, 1].
The following proposition states that we can always construct an equilibrium by a thresholdstrategy described in (5). The class of threshold equilibria identified here is a natural one tofucus on because it contains all the key insights of our model in a simple and intuitive form.
Proposition 1. For any given (λ, m, r), let q ∗ be the solution toλm

1− V L (q ∗ ; q ∗ )r= p(q ∗ ),(6)

if it exists for q ∗ < 1; let q ∗ = 1 if not. Then, for every q ∈ [q ∗ , 1], there is a threshold
equilibrium with upper threshold q.
Proof. See Section 4.5 and Appendix A.
Proposition 1 states that we can construct a continuum of threshold equilibria with anyq ∈ [q ∗ , 1]. The threshold equilibria identified in Proposition 1 have a simple and intuitive
structure: both types of seller adopt a bang-bang-type strategy by charging a positive pricep(qt ) if qt is higher than q, and having a sale if it is lower. What is especially important is thefact that both types of seller must adopt the low-price policy at the lower end of the belief
space: when qt ∈ (q, q), although the willingness-to-pay of a value shopper is still strictly
positive, the seller nonetheless charges the low price to boost her reputation. Since the belief
enters this region with positive probability for any prior q0 > q, occasional flash sales mustoccur with positive probability. The occurrence of flash sales is indeed a robust prediction of
our model, as we will discuss later in Section 5.2.
The case when q ∗ = 1 merits some remark. It occurs when either λ or m is large, or whenr is small. When this happens, no tradeoff characterized by (4) exists, and both types of seller
simply charge the low price for all qt ∈ Q in the unique equilibrium. As this case is rather
trivial, we make the following assumption to focus our attention to more relevant cases.
Assumption 2. There is an interior solution to (6) so that q ∗ < 1.

4.5

Sketch of the proof

Here, we provide a brief sketch of our equilibrium construction to develop some intuition
and illustrate how the equilibrium looks; technical details are relegated to Appendix A. For

15

the subsequent analysis, it is convenient to define T (q, q 0 ) for any q > q 0 ≥ q such that0

qe−λT (q,q ).
q = −λT (q,q0 )
qe
+1−q
0

(7)

In words, T (q, q 0 ) is the length of time it takes for the belief to drop from q to q 0 in a pooling
equilibrium when the seller receives no good rating.14 Solving (7) for T (q, q 0 ) gives
T (q, q 0 ) =1 q(1 − q 0 )ln.
λ q 0 (1 − q)Also, for any q ∈ Ql , define t as the length of time it takes for the belief to drop back to q
when the seller receives a good rating now and no rating afterwards. Since the belief jumps
q(1+m)
up to q(1+m)+(1−q)mafter receiving a good rating, t is the solution toq=

q(1 + m)e−λt,q(1 + m)e−λt + (1 − q)mfrom which we obtain
t=

1 1+m
ln
.
λ
m

Note that t is independent of the initial belief, implying that if the seller receives a good rating
at time t and no rating afterwards, the belief goes back to qt at time t + t regardless of thevalue of qt (as long as qt ∈ Ql ). This fact reflects a convenient property of Poisson process.
To illustrate how the equilibrium looks like, it is instructive to categorize the lower-priceregion Ql into different sets, based on how many ratings the seller needs to enter the high-priceregion Qh . From (1) and (2), we know that qt− < qt under a pooling strategy, and that there
is a jump to qt+ when the seller receives a good rating. Given any threshold strategy describedin (5) and any qt ∈ Ql , we define Qn ≡ [q n , q n−1 ), n = 1, 2, ..., recursively as follows. Letqm
qm}, where qm+(1−q)(1+m)
is the value of
Q1 = [q 1 , q 0 ), with q 0 = q and q 1 = max{q, qm+(1−q)(1+m)+
h
qt such that qt = q. This means that for any qt ∈ Q1 , the belief enters Q if the seller receives
a good rating within a period of length T (qt , q 1 ). For n > 1, if q n−1 > q, we recursively defineq n (qt ) = max{q, q

q n−1 mm+(1−q n−1 )(1+m)
n−1}. Similarly, for any qt ∈ Qn , it takes n good ratings withina period of length T (qt , q n ) for the belief to enter Qh . We continue this process until we findq n−1 msome n such that max{q, q m+(1−q} = q. For any n 6= 1, Qn ⊆ Ql and both types of)(1+m)n−1

n−1

Note that T (q, q 0 ) can be determined independently of the strategy σ in any pooling equilibrium because
γ (s) − γ L (s) = λ for both s = 0, 1.
14

H

16

Figure 1: Construction of Qn and sample belief path. The figure depicts a sample belief pathof the high-quality seller. Each jump represents the arrival of a good rating. Once a goodrating arrives in Qh , the belief jumps up to 1 and stays there indefinitely.
seller adopt the low-price policy. Figure 2 plots the case with n = 2, which provides a succinct
view of how the belief evolves in equilibrium.
We now show by construction that any threshold strategy with q ≥ q ∗ constitutes anequilibrium. In the proof (in Appendix A), we prove that the following properties hold under
this class of strategies.
Lemma 1. V i (q; q) is increasing in q for all q ∈ [q, 1) and i = H, L.
Lemma 2. Given any q ∈ [q ∗ , 1), for all q ∈ [q, 1), (i) V H (q; q) − V L (q; q) > 0, and (ii)V i (q; q) is increasing in q for i = H, L.
Lemma 2 is easy to understand. It simply says that, for any given belief, the continuation
payoff of the high-quality seller is always greater than that of the low-quality seller, and alsothat the continuation payoff is strictly increasing in the belief for both types. Lemma 1 is lessobvious, but it says that the benefit of high reputation always dominates the potential loss ofan increase in q.
Given these results, we can verify that as long as q ≥ q ∗ , any pair (q, q) can constitute thethresholds of an equilibrium. First, pick any q ≥ q ∗ . For qt ∈ Ql , we need to show that it is
the best response for both types of seller to adopt the low-price strategy. Suppose that the17

off-path belief after deviating to the high-price policy is low enough so that q̂t1 < q.15 Then,from (4), the condition for adopting the low-price policy is given by
λm∆V L (qt ) > max{p(q̂t1 ), 0} = 0 and λ∆V H (qt ) > max{p(q̂t1 ), 0} = 0.
These conditions hold ince ∆V i (q) > 0 for all q ∈ Q by Lemma 2. For qt ∈ Qh , to prevent
downward deviation, it must be thatii

(γ (0) − γ (1))




1
i
− V (qt ; q) ≤ p(q).
r

To show that this holds for the low-quality seller, note that

λm






1
1
1
L
L
L ∗ ∗
− V (qt ; q) < λm
− V (q; q) ≤ λm
− V (q ; q ) = p(q ∗ ) ≤ p(q) ≤ p(qt ),rrr

where the first inequality is by Lemma 2, and the second inequality is by Lemma 1. For thehigh-quality seller, since γ H (0; q̂t0 ) − γ H (1; q̂t1 ) ≤ λm,16 we have

λm




1
1
H
L
− V (qt ; q) < λm
− V (qt ; q) ≤ p(qt ),rrwhich holds by Lemma 2. Again, it is optimal for both types to adopt the high-price policy
when qt ∈ Qh .

5

Discussion

5.1

Monotone equilibria

Our model admits a plethora of equilibria, including some unrealistic ones, which requires
us to adopt some restrictions to reduce the set of equilibria. One such restriction is our exclusive
focus on pooling equilibria, which at a glance does not appear to have any solid foundation.
15

As noted above, it is difficult to restrict off-path beliefs by the standard refinement arguments in ourmodel. The current choice of off-path beliefs reflects an intuitive restriction that there are “no reputation
gains from (upward) deviation.” If the off-path belief q̂t1 is very high for some reason, we cannot construct
any monotone equilibrium in pure strategies. Since any equilibrium involving mixed strategies is substantiallymore complicated, we leave it for future research. We must emphasize, however, that the prospect of such an
exercise is unclear since any monotone equilibrium must possess similar dynamic properties even if the seller
randomizes in some intermediate range of the belief.
16
The value of γ H (0; q̂t0 ) − γ H (1; q̂t1 ) depends on the off-path belief assigned to a downward deviation: if0q̂t ≥ q, γ H (0; q̂t0 ) − γ H (1; q̂t1 ) = λm; if not, γ H (0; q̂t0 ) − γ H (1; q̂t1 ) = λ(m − 1).

18

Before we discuss equilibrium properties of our model, we first deal with this issue and argue
that our focus on pooling equilibria is hardly restrictive, as it can be derived from a morestructural restriction. To this end, we consider an intuitive class of equilibria, which we callmonotone equilibria, defined as follows.
Definition 2. An equilibrium is a monotone equilibrium if the value function V i (·) is monotonically increasing for both i = L, H.
The notion of monotone equilibrium is a natural restriction to impose, as it captures anessential premise of all reputation models, i.e., higher reputation is always better for the seller.
The following statement provides a structural foundation of pooling equilibria in our setting.
Proposition 2. Any pure-strategy monotone equilibrium must be a pooling equilibrium.
Proof. See Appendix A.
It is out of the scope of the current analysis to explore whether there exists any pure-strategy
separating equilibrium. Proposition 2 suggests, however, that even if such an equilibrium
exists, it must possess an unappealing property of reputation non-monotonicity, i.e., the seller’s
continuation payoff decreases when her reputation improves at some point. It should be noted,
though, that the set of monotone equilibria is still potentially very large, as Proposition 2
does not imply that all pure-strategy monotone equilibria have been identified in Proposition
1: for instance, we can construct an equilibrium in which Ql and Qh are disconnected. In
what follows, we say that an equilibrium property holds robustly if it holds in all pure-strategy
monotone equilibria of our model.

5.2

Rating systems and occasional flash sales

One of the main contentions of our analysis is that any rating system entails an unintended
consequence of inducing strategic behaviors from sellers, which in turn introduces noise into
the buyers’ inference process and impedes social learning. In our model, the seller fluctuates
stochastically between the two policies, and hence occasionally engages in a sale to boost her
reputation. From the viewpoint of social learning, the existence of the low-price region createsan efficiency loss. To see this, suppose q = q, so that σ(q) = (1, 1) for all q ∈ Q. Under thisstrategy profile, a good rating is always a sure sign of high quality, and the low-quality seller
exits the market at the earliest timing T (q0 , q). As we have seen above, however, this cannot
happen in equilibrium, since the seller always has an incentive to switch to the low-price policy
when the belief gets close enough to the lower threshold. Although we make this point undera set of restrictions, slow social learning due to the occurrence of occasional flash sales is aprediction that holds robustly in our setting.
19

Proposition 3. In all pure-strategy monotone equilibria, σ(q) = (0, 0) for all q < q ∗ . Moreover, in all pure-strategy monotone equilibria in which both types of seller offer the high pricewith positive probability, σ(q) = (1, 1) for q sufficiently close to 1.
Proof. Let V i (·) denote the value function of a monotone equilibrium. As we will show in
Proposition 5, the threshold equilibrium with q = q ∗ maximizes the low-quality seller’s expected
payoff, i.e., V L (q; q ∗ ) ≥ V L (q) for all q ∈ Q. This means that for all q ∈ Q, we have

λm




1
1
L
L∗
− V (q) ≥ λm
− V (q) > p(q),rrwhich implies that the incentive compatibility constraint cannot be satisfied in any monotone
equilibrium.
To show that the seller must adopt the high-price policy when the belief is sufficiently high,fix some ε > 0 such that σ(q) = (0, 0) for all q ∈ (1 − ε, 1). Then, in the limit, we havelim V i (q) ≤ lim e−rT (q,1−ε) V i (1 − ε) = 0,

q→1

q→1

where the second term is the best-case payoff where the seller receives no good rating until
qt reaches 1 − ε. This means that if there is such an ε, V i (·) must decrease at some point,
contradicting monotonicity. Therefore, the seller must adopt the high-price policy when the
belief is sufficiently high in any monotone equilibrium in which she offers the high price withany positive probability.
It is important to note that Proposition 3 is not a trivial restatement of Proposition 1: itsays that the seller adopts the low-price policy when qt is below q ∗ and the high-price policy
when qt is close to 1 in any pure-strategy monotone equilibrium in which the seller offers
the high price with positive probability,17 including those that do not belong to the class
identified in Proposition 1. What is especially important is the fact that there is a threshold q ∗below which the seller cannot adopt the high-price policy. To see this, suppose that there is a
threshold equilibrium with q < q ∗ . In this equilibrium, the value of a good rating is extremely
high when qt ∈ [q, q ∗ ) because qt jumps up to 1 after one good rating, which allows the seller
to charge pt = 1 indefinitely. More precisely, the value of a good rating in this range is given
17

We can trivially construct an equilibrium in which the seller never charges a positive price. This equilibrium is perhaps of little interest because the seller’s expected payoff is invariably 0, making it highly unlikely
for such a market to survive.

20

by∆V i (qt ) = V i (1) − V i (qt ),which is strictly decreasing in qt in any monotone equilibrium by Lemma 2. By definition,
therefore, we have
∆V L (qt ) > ∆V L (q ∗ ) = p(q ∗ ) > p(qt ),for any qt ∈ [q, q ∗ ). In a nutshell, the incentive to deviate and (secretly) set a low price togarner a good rating is too strong at this point.
Observe that our model exhibits two absorbing states: qt = 1 and qt = q. Once qt reaches
either one of them, no additional information will be revealed thereafter, and the belief stays
constant over time. On the equilibrium path, the low-quality seller can never receive a goodrating in Qh : once qt jumps above q, it will gradually fall back. This means that the rating
system never identifies a low-quality seller as a high-quality one, i.e, it never displays a false
negative. On the other hand, there is a strictly positive probability that qt reaches q for bothtypes. Since the low-quality seller can never reach the upper absorbing state, it is easy to
see that the low-quality seller will eventually be revealed as such as t tends to infinity. The
problem is that there is also a strictly positive probability that the high-quality seller reaches
the lower absorbing state and is forced to exit the market.
The fact that the rating system may display a false positive is the reason why even thehigh-quality seller needs to engage in a sale when the belief is closer to the lower threshold.
The occurrence of flash sales by the high-quality seller generates an important welfare implication that stands in sharp contrast to standard adverse selection models. In standard adverse
selection models, either static or dynamic, buyers usually cannot benefit from the presence of
information asymmetry. This is not the case in our setting because information asymmetry
induces the high-quality seller to hold a sale occasionally, which generates potential surplus forvalue shoppers. Since the expected payoff of a price shopper is always 0 by assumption, thisresult implies that all buyers are weakly better off with information asymmetry than without.
A formal statement goes as follows.
Proposition 4. The expected payoff of a value shopper is always 0 if q0 = 1 (no asymmetric
information), whereas it is strictly positive in all pure-strategy monotone equilibria if q0 ∈ (q, 1).
Proof. The proof is straightforward. Suppose that the seller is known to be of high quality, sothat qt = 1 for any history of the game. Then, there is clearly no incentive for the high-quality
seller to offer the low price, and the expected payoff of a value shopper is always 0 in any
21equilibrium. If q0 < 1, on the other hand, Proposition 3 states that the high-quality seller
must offer σ H (q) = 0 for q ≤ q ∗ . Since any arriving value shopper earns a payoff of p(qt ) inthis region and also there is a strictly positive probability that the belief enters this region,
the expected payoff must be strictly positive in any monotone equilibrium with information
asymmetry.
Two elements of our model are essential for Propositions 3 and 4. First, it is important thatboth types of seller we consider are strategic. This is a departure from canonical reputation
models which assume at least one commitment type and analyze how strategic types mimic
the commitment type. If we applied this standard approach to our setting, we would take thehigh-quality seller as the commitment type, assuming that she always offers the high price,
and analyze how the low-quality seller mimics the high-quality type. While such an approachmakes the analysis more tractable and clarifies the distortion generated by the low-quality
seller, it cannot describe how the high-quality seller would respond. Our assumption that bothtypes are strategic allows us to capture dynamic interactions between the two types, which we
believe are crucial for the problem we consider.
Second, it is also important that the rating system is an endogenous information-generatingprocess in that the value of a good rating depends on the seller’s strategy profile at the timeof purchase. The contrast is clear with Daley and Green (2012) who consider an exogenous
news process in a dynamic lemons market. In their model, there is a belief lower bound which
serves as a reflecting barrier rather than an absorbing state. At the lower bound, low-quality
sellers exit gradually (by accepting a low offer) while high-quality sellers never exit. There isno false positive in their setting because news arrives exogenously, and high-quality sellers canafford to wait for the arrival of good news even if the belief is approaching the lower bound. In
our setting, on the other hand, once the belief reaches the lower threshold, the seller can only
sell to price shoppers, and therefore has no chance to improve her reputation thereafter.18
In our model, the value of a good rating is endogenous because the rating system aggregates
information anonymously from buyers with diverse preferences. The importance of anonymity
is evident, as our entire argument rests on the fact that buyers cannot differentiate informative
signals and uninformative ones. To see the role of preference diversity, note that if all buyers
are price shoppers, both types of seller obviously cannot charge any positive price. If all
buyers are value shoppers, on the other hand, there is no incentive to have a sale because thelikelihood of receiving a good rating is unaffected in this case; as such, the seller is willing tohave a sale only when there are buyers who are satisfied by low price. This suggests that the
combination of anonymity and preference diversity is the crucial driving force of the trading
18

Indeed, sellers with low ratings choose to leave the market (Fan et al., 2016).

22

dynamics (Proposition 3) as well as of the welfare implication (Proposition 4).

5.3

Seller optimality

As noted above, Proposition 1 does not exhaust all possible forms of pooling equilibrium.
Moreover, even if we restrict attention to the class of equilibria identified in Proposition 1,
there is still a continuum of equilibria. Those different equilibria obviously entail different
efficiency properties, with some more favorable for the seller and others more favorable forthe buyers. In this subsection, we propose a selection criterion in order to identify the mostdesirable equilibrium from the viewpoint of the high-quality seller.
Definition 3. An equilibrium is the seller-optimal equilibrium if it maximizes the high-quality
seller’s expected payoff among all pure-strategy monotone equilibria.
The notion of seller optimality is an important criterion for platforms when the seller’s
participation decision is endogenous: if the high-quality seller’s expected payoff from some
platform is not high enough, she may choose not to join the platform in the first place; inthe worst case, the initial prior q0 might dip below q, which forces the platform to shut down
entirely. The following result provides a characterization based on this notion.
Proposition 5. The threshold equilibrium with q = q ∗ is the seller-optimal equilibrium. Moreover, it also maximizes the low-quality seller’s expected payoff.
Proof. See Appendix A.
Proposition 5 demonstrates the importance of Proposition 1 as it suggests that, if there isan equilibrium that does not belong to the class in Proposition 1, then we can identify one inthat class that is more favorable to the seller. This result stems from the fact that the seller
has an excessively strong incentive to deviate unilaterally to the low-price policy: as noted in
Section 5.2, even if the high-price policy benefits the seller, she would be tempted to secretly
undercut the price and garner a good rating in the high-price region. As a consequence, the setof beliefs in which the high-price policy benefits the seller is always larger and strictly contains
the set of beliefs in which the incentive compatibility constraints are satisfied for both types.
This observation implies that the seller-optimal equilibrium is the one in which the seller offers
the high price as much as possible, i.e., the threshold equilibrium with q = q ∗ .

5.4

Comparative statics

We now conduct some comparative statics to derive more specific implications of our model.
Since there is a continuum of equilibria, our comparative statics will focus on the seller-optimal
23

equilibrium characterized in Proposition 5, i.e., we focus on how the value of q ∗ changes in
response to changes in underlying parameters. Note that although we look at the seller-optimalequilibrium, the value of q ∗ itself has relevance beyond this particular equilibrium because thisindicates the lower bound of all possible upper thresholds.
Proposition 6. (i) There exists some r > 0 such that q ∗ = 1 for all r < r and q ∗ → q as
r → ∞, where q ∗ is strictly decreasing in r for r ∈ [r, ∞). (ii) There exists some λ > 0 suchthat q ∗ = 1 for all λ > λ and q ∗ → q as λ → 0. (iii) There exists some m > 0 such that q ∗ = 1
for all m > m and q ∗ → q as m → 0.
Proof. We prove each part of the proposition in turn.
(i) Given some qt < 1, the expected payoff at each τ > t is bounded away from 1. Therefore,
as r increases, 1r −VL (q ∗ ) must decrease. To satisfy (6), q ∗ must decrease, which simultaneously
increases the left-hand side of (6) and decreases the right-hand side. Note also that the righthand side of (6) is bounded by 1, while the left-hand side goes to infinity as r goes to 0.
Therefore, when r is close enough to 0, (6) does not have a solution, so that q ∗ = 1 bydefinition. Finally, as r → ∞, 1r − VL (q ∗ ) converges to 0, which implies that q ∗ must also
converge to q.
(ii) As λ approaches 0, since 1r − V L (q ∗ ) is bounded from above, the right-hand side of (6)must also go to 0. The solution q ∗ thus converges to the lower bound q as λ goes to 0. Also,
when λ becomes sufficiently large, (6) does not have a solution, so that q ∗ = 1 by definition.
(iii) The proof is essentially the same as in case (ii) and is omitted.
The first result concerns the effect of the discount rate r. It is intuitively clear that as rincreases (more discounting), the seller’s preferences move towards the immediate revenue andaway from the future gains. Correspondingly, there is less incentive to offer the low price, andthe low-price region reduces. At the other end, when r is small enough, (6) does not admit asolution, and q ∗ = 1. This presents an intriguing situation given that the expected payoff is 0
for both types in this equilibrium. Nevertheless, when q ∗ = 1, it is impossible to construct a
pooling equilibrium with q < 1 because the incentive to (secretly) set the low price and garnera good rating is too strong, not just for the low-quality seller but also for the high-quality
seller.
The discount rate r can be interpreted in different ways, ranging from the seller’s patienceto the (exogenous) probability of exiting the market; in principle, it can include all factorsthat make the seller more or less forward-looking. When the seller is more forward-looking,the frequency of sales increases, which generally hurts the seller but benefits the value shoppers
as we have seen above. This suggests a counterintuitive possibility that the seller is made better

24

off, while the buyers are worse off, in a more competitive market environment where the seller
is driven out of the market with higher frequency.
A comparative statistics result that might be of particular interest to platform designers isthe effect of λ, which measures the frequency of providing a rating after a transaction. Intuition
suggests that an increase in λ would provide more information to buyers and expedite social
learning; indeed, platforms usually adopt various measures to encourage buyers to leave ratings.
As it turns out, though, this type of intervention can backfire in our setting: with a higher
λ, the seller expects to receive more ratings in a given interval of time, which generates asimilar effect to a decrease in r and raises q ∗ . Table 1 shows this general tendency where an
increase in λ induces an increase in q ∗ .19 This finding leads to a conclusion, also emphasizedin other works such as Daley and Green (2012), Kaya and Kim (2018), Ekmekci et al. (2022),
and Acemoglu et al. (2022), that more information does not necessarily improve learning instrategic environments.
λ
q∗

0.2
0.53

0.4
0.70

0.6
0.84

0.8
0.90

Table 1: Effect of λ on q ∗Finally, we also discuss the effect of m, which can be interpreted as measuring the extent
of buyer heterogeneity. Proposition 6 suggests that there is an equilibrium in which the upper
threshold q is arbitrarily close to q as the number of price shoppers goes to 0. To see this,consider the case of homogeneous buyers where m = 0 and all arriving buyers are value
shoppers. Then, γ L (s) = 0 and γ H (s) = λ for both s = 0, 1, and there is no benefit fromhaving a sale. In equilibrium, therefore, σ(q) = (1, 1) for all q ∈ Q, which obviously benefits
the seller but hurts the buyers. This is a formal representation of the argument made in Section
5.2 that an important qualifier of Proposition 4 is m > 0, i.e., there is some preference diversity
among buyers.

5.5

Buyer preferences and asymptotic learning

As noted above, one distinctive feature of our model is that the value of a good rating is
endogenously determined: one good signal is enough to identify the high-quality seller when
the belief is above q while it has no reputation value once it gets below q. Because of thisfeature, even the high-quality seller may be forced to exit the market, i.e., social learning not
19

The values in the table are computed by using the value functions derived in Appendix B. The parametervalues used in the exercise are r = 0.2, m = 0.3, and q = 31 .

25

only takes time but may also fail in the long run. Since only the high-quality seller can generate
surplus, the likelihood of this event obviously yields a crucial welfare implication.
To measure the extent of this efficiency loss, let Pti ≡ P[qt > q | i] be the probability thatthe type i seller stays in the market at time t in a given equilibrium, and let P i ≡ limt→∞ Pti .
Since the low-quality seller never reaches the upper absorbing state, we clearly have P L = 0,i.e., the low-quality seller is eventually driven out of the market. The asymptotic efficiency oflearning is thus determined by P H which we call the asymptotic probability of learning. Notethat there are infinitely many belief paths that can be realized under each equilibrium strategy
profile. The following result shows, however, that the asymptotic probability is independent of
the seller’s strategy and depends only on two parameters, q0 and k, in any pooling equilibrium.
Proposition 7. In all pooling equilibria, the asymptotic probability of leaning is
PH = 1 −q(1 − q0 )1 − q0=1−k.
q0 (1 − q)q0

Proof. Although the low-quality seller exits the market at various points in time, the distribution of exiting times actually does not matter for the asymptotic probability. Fix someequilibrium σ ∗ , and let f i (t) be the probability of the type i seller exiting at time t underσ ∗ . Define a discrete set T of t such that f i (t) > 0. Under this formulation, the asymptotic
P
probability can be written as P i = 1 − t∈T f i (t). Since the low-quality seller eventually exits
Pat some point, we know t∈T f L (t) = 1.
Observe that in any pooling equilibrium, both types of seller are driven out of the market
if and only if qt reaches q. Therefore, for each t ∈ T , if the seller is forced to exit, we have
qt = q ⇔q0 f H (t)= q,
q0 f H (t) + (1 − q0 )f L (t)which can be written as
q0 (1 − q)f H (t) = q(1 − q0 )f L (t).
Summing them up for all t ∈ T then gives
q0 (1 − q)(1 − P H ) = q(1 − q0 )(1 − P L ) = q(1 − q0 ).
Solving this for P H yields the result.
What is remarkable about this result is that the asymptotic probability of learning is
26

constant across all pooling equilibria of this model. Observe that the asymptotic probability
depends linearly on and is strictly decreasing in k. In other words, aside from the seller’s initial
reputation q0 , the asymptotic probability is determined by how selective the value shoppers
are. If the value shoppers care little about quality so that k goes to 0, the rating system canalmost surely identify the high-quality seller, despite its obvious limitations on informationtransmission. If they are very selective, on the other hand, learning opportunities are limited,and social learning fails asymptotically.

6

Extensions

In our baseline model, the buyers are only given coarse information, i.e., ratings which
only indicate that some buyers are satisfied. What then happens if we allow the buyers tohave access to more information? In this section, we discuss two possibilities—one in which
the buyers can observe past prices (instead of only the current one) and the other in which
the buyers are given the option to leave bad ratings—to address this issue and illuminate theimpact of providing more information to buyers.

6.1

Observable price history

Suppose that in addition to the history of ratings, the buyers can also observe the historyof prices, i.e., each buyer t can observe the whole history of prices {pτ }tτ =0 . Alternatively, since
the price when there is no rating is irrelevant, we may simply assume that the price at thetime of purchase is appended to each rating; as we show below, these two cases are actuallyequivalent. In our baseline model, given the current belief qt , there are two possible belief
realizations for the next instant, qt− and qt+ , depending on whether a good rating is left or not.
When the price information is appended to a rating, there are three possible belief realizations,depending not only on whether a good rating is left, but also on whether the price is high or
low. We denote these new possibilities by qts+ which indicates the updated belief when theseller receives a good rating with st = s.
With this augmented rating system, we can construct an equilibrium in which the lowprice policy is never adopted. The reason for this is that the current price is now observable tothe future buyers as well, which suppresses the temptation to (secretly) undercut the price toobtain a good rating. In this equilibrium, price shoppers never purchase, and hence the learning
process is as efficient as when there are no price shoppers (although this is detrimental to thevalue shoppers as we have discussed above).
Proposition 8. Consider a rating system in which the price at the time of purchase is appended27

to each rating. Under this rating system, there exists an equilibrium in which σ(q) = (1, 1) for
all q ∈ Q.
Proof. The proof is straightforward. Given that both types of seller always adopt the highprice policy, st = 0 is off the equilibrium path. For clarity, suppose we assign q̂t0 = 0 to all
deviations. We then have qt0+ = 0 even if the seller receives a good rating.20 This entirely
eliminates any deviation incentive, because this information is now transmitted to the futurebuyers (in contrast to the case where the price can only be observed by the current buyer),and the seller is forced to exit the market immediately after receiving a good rating.
Although not many platforms provide this information, there are actually several pricetracking tools available online such as CamelCamelCamel, Honey, and Keepa. Proposition 8
indicates that these price trackers not only help buyers find the best timing to make a purchase
but also have the potential to improve social learning. More generally, the efficiency of social
learning can be improved substantially by providing the “right information” to buyers.
We must note, however, that it could be a tall order to implement such an outcome inpractice, because the augmented rating system requires far more cognitive effort from buyers.
The argument circles back to our original premise that the rating system is subject to various
communication frictions, which makes it difficult to convey detailed information to all buyers.
In fact, the augmented rating system requires buyers to figure out how ratings and prices arecorrelated over time, whereas many buyers in reality do not bother to go into such details. It
is also not entirely clear whether there exist any summary statistics that are intuitively and
immediately understandable for all buyers. The simple rating system we consider has a definite
advantage in this sense since at any t such that qt < 1, a buyer simply needs to count how
many good ratings the seller has received.21

6.2

Bad ratings

The previous subsection shows that a rating system which provides two-dimensional information can work quite well. Here, we enrich the rating system in a different way, by making
it “finer” with more rating categories. Suppose that, in addition to leaving a good rating with20

Of course, this is a rather extreme, though simple, way to construct an equilibrium. A more conservative
approach would be to suppose that the buyers believe that the two types deviate randomly with equal probability such that q̂t0 = qt . In this case, the two types receive a good rating with the same probability and henceq̂t0 = qt . Since the value of a good rating is 0, no type has an incentive to offer the low price.
21
The fact that qt < 1 implies that the seller has not received a good rating in the high-price region.
Given this, the current belief depends only on the number of good ratings due to the memoryless property of
exponential distribution.

28

probability λg when a buyer is satisfied, he can also leave a bad rating with probability λb
if he is unsatisfied. We say that a buyer is unsatisfied when his utility from consumption isnegative. Given this classification, a value shopper is unsatisfied when he purchases from thelow-quality seller, regardless of the price, implying that only the low-quality seller can receive
a bad rating. Therefore, after a bad rating is left, qt drops down to 0, and the seller is forced
to exit the market. On the other hand, if no rating is left, the belief is updated toqt− = qt − (γ H (σ H (qt )) − γ L (σ L (qt )) − λb )qt (1 − qt )dt,
where for q̂ts ≥ q,γ L (s) = λg m(1 − s), γ H (s) = λg + λg m(1 − s).
Notice a slight difference from (1), where we now have λb in the second term. If λb < λg , thebelief would go down when there is no rating, but at a slower rate than in our baseline model
where bad ratings are not allowed, i.e., λb = 0. In this case, the analysis remains essentiallythe same. The only notable difference is that the low-quality seller may receive a bad rating,which drives her out of the market immediately. Since this event occurs whenever the good ispurchased by a value shopper, it is as if the low-quality seller is driven out of the market at aconstant rate for some exogenous reasons. The possibility of bad ratings then yields an effectthat is equivalent to an increase in the discount rate r, as long as she offers a price that is
acceptable for value shoppers. Other than this, the tradeoff involved in setting a low price stillworks in the same way as before, and an equilibrium characterized by some upper threshold q
still exists. Furthermore, Proposition 6 implies that the presence of bad ratings benefits thehigh-quality seller while it hurts the value shoppers.
If λb ≥ λg , on the other hand, the belief increases as long as the seller receives no rating,i.e., “no news is good news.” Unless the seller receives a bad rating, in which case qt drops to0, qt will always stay above q for any q0 ≥ q. This completely changes the incentive structureof the seller, and may produce equilibria that are very different from what we characterize in
Proposition 1. However, though theoretically possible, it is highly unlikely that buyers leaves
bad ratings with higher frequency than good ones in online platforms, given that reviews are
predominantly positive (Nosko and Tadelis, 2015; Tadelis, 2016; Zervas et al., 2021); even atsome intuitive level, it is hard to conceive of a situation where a seller establishes her reputationby having received no ratings. In all likelihood, λb may be positive but should lie somewhere
between 0 and λg , so that “no news is bad news” is the appropriate specification in online
platforms. It is nonetheless of some interest, from the purely theoretical point of view, to29

investigate this “no news is good news” case because this difference in the specification oflearning often leads to markedly different dynamics.22
Finally, one may conjecture that the efficiency of social learning can be improved by extending the rating scale in the other direction, e.g., creating an additional category of “verygood.” Suppose that a buyer can leave a very good rating when he is “very satisfied,” i.e., when
his utility is above some positive threshold. For instance, in our setting, we may allow a value
shopper to leave a very good rating when he purchases from the high-quality seller at the low
price. While this rating system allows the high-quality seller to separate from the low-quality
seller, and discourages the low-quality seller from having a sale, it does not fundamentally solve
the problem at hand. In our model, we assume that the utility of a price shopper is always 0
simply for clarity, but the same problem persists if we introduce a third type, say a bargain
shopper, who derives a very high utility regardless of the quality when he can purchase at thelow price. Ultimately, this problem boils down to the fact that we cannot compare utilities
across individuals. Different buyers not only have different preferences, with some emphasizing
quality and others emphasizing price, but also have different evaluation standards—different
rules to map a realized utility to a rating—and hence it is fundamentally infeasible for platforms to device a clear rule of what should constitute a good rating, no matter how elaborate
the rating system becomes.

7

Conclusion

Under a rating system, buyers share their evaluations anonymously without knowing theidentities of the raters. When the buyers have diverse preferences, this creates a room forstrategic sellers to exploit. This paper explores the consequences of this inherently coarse
nature of rating systems and its efficiency implications on social learning. In our model,
both types of seller switch back and forth between the two pricing policies and occasionallyhold a flash sale to garner good ratings. This form of strategic pricing introduces noise into
the buyers’ inference process and impedes social learning but at the same time generates a
non-trivial welfare effect that stands in sharp contrast to standard adverse selection models:
information asymmetry, when combined with preference heterogeneity among buyers, strictly
benefits buyers who value quality. Since information is generated endogenously, social learning
fails in the long run, as the high-quality seller is forced to exit the market after a string of bad
luck. Incentivizing buyers to leave ratings exacerbates the seller’s strategic pricing incentive
and can be detrimental to social learning.
22

See, for instance, Board and Meyer-ter-Vehn (2013) who show that reputation dynamics can be markedly
different between these two cases.

30

References
Acemoglu, Daron, Ali Makhdoumi, Azarakhsh Malekian, and Asu Ozdaglar,“Learning from Reviews: The Selection Effect and the Speed of Learning,” Econometrica,
2022, 90 (6), 2857–2899.
Bar-Isaac, Heski, “Reputation and Survival: Learning in a Dynamic Signalling Model,”Review of Economic Studies, 2003, 70 (2), 231–251.
Board, Simon and Moritz Meyer-ter-Vehn, “Reputation for Quality,” Econometrica,
2013, 81 (6), 2381–2462.
Chen, Chia-Hui and Junichiro Ishida, “Hierarchical Experimentation,” Journal of Economic Theory, 2018, 177, 365–404.
Crapis, Davide, Bar Ifrach, Costis Maglaras, and Marco Scarsini, “Monopoly Pricing
in the Presence of Social Learning,” Management Science, 2017, 63 (11), 3586–3608.
Cripps, Martin W., George J. Mailath, and Larry Samuelson, “Imperfect Monitoring
and Imperfect Reputations,” Econometrica, 2004, 72 (2), 407–432.
Daley, Brendan and Brett Green, “Waiting for News in the Market for Lemmons,” Econometrica, 2012, 80 (4), 1433–1504.
Einav, Liran, Chiara Farronato, Jonathan Levin, and Neel Sundaresan, “Auctions
versus Posted Prices in Online Markets,” Journal of Political Economy, 2018, 126 (1), 178–
215.
Ekmekci, Mehmet, “Sustainable Reputations with Rating Systems,” Journal of Economic
Theory, 2011, 146 (2), 479–503.
, Leandro Gorno, Lucas Maestri, Jian Sun, and Dong Wei, “Learning from Manipulable Signals,” American Economic Review, 2022, (forthcoming).
Faingold, Eduardo and Yuliy Sannikov, “Reputation in Continuous-Time Games,” Econometrica, 2011, 79 (3), 773–876.
Fan, Ying, Jiandong Ju, and Mo Xiao, “Reputation Premium and Reputation Management: Evidence from the Largest e-Commerce Platform in China,” International Journal ofIndustrial Organization, 2016, 46, 63–76.

31

Fradkin, Andrey, Elena Grewal, and David Holtz, “Reciprocity and Unveiling in TwoSided Reputation Systems: Evidence from an Experiment on Airbnb,” Marketing Science,
2021, 40 (6), 1013–1029.
Fudenberg, Drew and David K. Levine, “Reputation and Equilibrium Selection in Games
with a Patient Player,” Econometrica, 1989, 57 (4), 759–778.
and , “Maintaining a Reputation when Strategies are Imperfectly Observed,” Review of
Economic Studies, 1992, 59 (3), 561–579.
and Jean Tirole, “A “Signal-Jamming” Theory of Predation,” RAND Journal of Economics, 1986, 17 (3), 366–376.
Gul, Faruk and Wolfgang Pesendorfer, “The War of Information,” Review of Economic
Studies, 2012, 79 (2), 707–734.
Holmstrom, Bengt, “Managerial Incentive Problems: A Dynamic Perspective,” Review of
Economic Studies, 1999, 66 (1), 169–182.
Ifrach, Bar, Costis Maglaras, Marco Scarsini, and Anna Zseleva, “Bayesian Social
Learning from Consumer Reviews,” Operations Research, 2019, 67 (5), 1209–1221.
Kaya, Ayca and Kyungmin Kim, “Trading Dynamics with Private Buyer Signals in theMarket for Lemons,” Review of Economic Studies, 2018, 85 (4), 2138–2352.
Kreps, David M. and Robert Wilson, “Reputation and Imperfect Information,” Journalof Economic Theory, 1982, 27 (2), 253–279.
Liu, Qingmin, “Information Acquisition and Reputation Dynamics,” Review of Economic
Studies, 2011, 78 (4), 1400–1425.
and Andrzej Skrzypacz, “Limited Records and Reputation Bubbles,” Journal of Economic Theory, 2014, 151, 2–29.
Milgrom, Paul and John Roberts, “Predation, Reputation, and Entry Deterrence,” Journal of Economic Theory, 1982, 27 (2), 280–312.
Nosko, Chris and Steven Tadelis, “The Limits of Reputation in Platform Markets: An
Empirical Analysis and Field Experiment,” 2015. NBER Working Paper No. 20830.
Pei, Harry, “Reputation Effects under Interdependent Values,” Econometrica, 2020, 88 (5),2175–2202.
32

, “Reputation Building under Observational Learning,” Review of Economic Studies, forthcoming, 2022.
Royden, Halsey and Patrick Fitzpatrick, Real Analysis, Pearson Education Asia Limited,2010.
Tadelis, Steven, “Reputation and Feedback System in Online Platform Markets,” Annual
Review of Economics, 2016, 8 (2), 321–340.
Zervas, Georgios, Davide Proserpio, and John W. Byers, “A First Look at Online
Reputation on Airbnb, Where Every Stay is Above Average,” Marketing Letters, 2021, 32(1), 1–16.

Appendix A: Proofs
Proof of Proposition 1 : In the main text, we prove Proposition 1 based on Lemmas 1 and 2.
Here, we provide the proof of each statement.
Proof of Lemma 1. Consider two hypothetical situations: in situation 1, the thresholds are
given by (q 0 , q 0 ) with initial prior q 0 ; in situation 2, they are given by (q 00 , q 00 ) with initial prior(1−q 0 )q 0

(1−q 00 )q 00

q 00 . Assume q0 (1−q0 ) = q00 (1−q00 ) , so that T (q 0 , q 0 ) = T (q 00 , q 00 ) ≡ T . Since we here consider
different lower thresholds, with abuse of notation, we write V i (·; q, q) to explicitly denote its
depends on the lower threshold as well.
We first show V i (q 0 ; q 0 , q 0 ) < V i (q 00 ; q 00 , q 00 ). To this end, note that in the absence of any goodrating, the seller exits the market at time T in both scenarios by assumption. Moreover, sinceboth types of seller adopt the low-price policy when the belief is below the upper threshold,the probability of receiving a good rating, and hence entering the high-price region, is the same
between the two situations. Since q 00 > q 0 , the seller can charge a higher price in situation 2
than in situation 1. Also, in the high-price region, a single good rating allows the seller toearn a continuation payoff of 1r . Therefore, V i (q 00 ; q 00 , q 00 ) > V i (q 0 ; q 0 , q 0 ). Given this, we lower
q 00 to q 0 so as to compare the two situations with the same lower threshold. As this change
only makes the seller’s exit less likely, we in general have V i (q 00 ; q 0 , q 00 ) > V i (q 00 ; q 00 , q 00 ), which
in turn implies V i (q 00 ; q 0 , q 00 ) > V i (q 0 ; q 0 , q 0 ). As this holds for any q 00 > q 0 and q 0 = q 00 = q,
V i (q; q, q) is increasing in q.
Proof of Lemma 2. We first prove V H (q) > V L (q) for all q ∈ Q. For any belief higher thanq, the high-quality seller always has a higher probability of receiving a good rating under the33

pooling strategy considered here. This implies that, starting with any qt and at any timet0 > t, the belief distribution of the high-quality seller first-order stochastically dominates thatof the low-quality seller. Moreover, since the instantaneous payoff weakly increases with qt(and strictly increases for qt ∈ Qh ), the expected payoff of the high-quality seller is always
higher than that of the low-quality seller at any time t0 > t, which proves the claim.
We next show that V i (·) is increasing. For this purpose, consider two initial beliefs q0 = q 0
and q0 = q 00 , with q 00 > q 0 , and compare V i (q 0 ) and V i (q 00 ). Let {q̃τ (q)}∞τ =0 denote the belief
path that is realized when it starts from q and the seller receives no good rating afterwards.
First consider the case q 00 > q 0 > q, so that the seller charges the high price. With no goodrating, both q̃t (q 0 ) and q̃t (q 00 ) gradually decrease. However, as long as q̃t (q 00 ) > q̃t (q 0 ) ≥ q,
the seller charges a positive price under both cases, and receives a good rating with equal
probability. Moreover, the belief will jump to 1 with a single good rating, in which case theseller receives a continuation payoff of 1r . This means that before time T (q 0 , q), the expected
payoff under p00 is higher than that under q 0 . If the seller has received no good rating, we haveq̃t (q 00 ) > q > q̃t (q 0 ) for t > T (q 0 , q). The comparison in this case is harder because although theseller can charge a higher price under q 00 , the probability of receiving a good rating is lower.
To show that the expected payoff under q 00 is higher in this contingency, we need to show thatV i (q 00 ) > V i (q 0 ) when q 00 > p > q 0 , i.e., V i (q 00 ) > V i (q 0 ) when q 00 > q 0 > p if V i (q 00 ) > V i (q 0 )when q 00 > p > q 0 . We will deal with this case in the final step of the proof.
Next consider the case where q > q 00 > q 0 . The argument in this case is similar to theprevious one. Since the seller charges the low price under both cases, the probability of receivinga good rating is the same. However, given that q 00 > q 0 , it requires fewer good ratings to jumpback to the high-price region under q 00 than under q 0 . Moreover, the probability of hitting thelower threshold is higher under q 0 than under q 00 . Again, this implies that V i (q 00 ) > V i (q 0 )when p > q 00 > q 0 if V i (q 00 ) > V i (q 0 ) when q 00 > p > q 0 .
The argument above implies that the proof boils down to establishing V i (q 00 ) > V i (q 0 ) whenq 00 > p > q 0 . Note that when the seller receives a good rating before time min{T (q 00 , q), T (q 0 , q)},the belief under q 00 reaches 1 while the belief under q 0 jumps to a value strictly smaller than 1.
Therefore, the expected payoff under q 00 is higher than that under q 0 . If the seller has received
no rating until T (q 00 , q), the seller charges a higher price under q 00 but has a lower probability ofreceiving a good rating. Also, since q̃t (q 00 ) > q ≥ q ∗ , we know that λm( 1r −V L (q, q)) < p(q̃t (q 00 )).
Given that V H (q; q) > V L (q; q), this implies λm( 1r − V H (q, q)) < p(q̃t (q 00 )) as well. Therefore,by (4), the high-price policy yields a higher continuation payoff for both types of seller than the
low-price policy before time T (q 00 , q). Note that, if the seller were to adopt the low-price policy
before reaching q, her continuation payoff would be exactly the same as the continuation payoff
under q 0 . Since the seller can do even better by adopting the high-price policy, V i (q 00 ) > V i (q 0 )34when q 00 > p > q 0 , which completes the proof.
Proof of Proposition 2. The following proof exploits the value functions derived in Appendix
B. It is clear that there cannot be a separating equilibrium in which the low-quality seller offers
the high price and the high-quality seller offers the low price, i.e., σ(qt ) = (1, 0) for some qt .
Under this profile of strategies, the belief jumps up to 1 after one good rating. The low-quality
seller surely finds it profitable to deviate since λm( 1r − V L (qt ; q)) > p(q̂t1 ) = 0. The remainderof the proof thus focuses on ruling out a separating strategy in which the low-quality seller
offers the low price and the high-quality seller offers the high price.
Consider σ(qt ) = (0, 1) for some qt . We then have q̂t1 = 1 and qt+ < 1. The proof is still
straightforward in this case if m ≥ 1 because qt+ ≤ qt by (2), which implies ∆V i (qt ) < 0.
Since λm∆V i (qt ) ≤ 0 < p(q̂t ) = 1, a deviation to the high-price policy is profitable for the
low-quality seller. In what follows, we thus assume m < 1.
We now claim that if σ(qt ) = (0, 1) for some qt , then there must be some interval of qt suchthat σ(qt ) = (1, 1). Suppose otherwise. Then, it is either σ(qt ) = (0, 1) or σ(qt ) = (0, 0) for all
qt , in which case the low-quality seller’s expected payoff is always 0. She would then deviate
to the high-price policy to receive a positive payoff every time the high-quality seller offers thehigh price. This means that there must be some qt such that

λm


1
L
− V (qt ) ≤ p(qt ).
r

Given that both V L (·) and p(·) are increasing, we can define




1
L
z ≡ inf qt | λm
− V (qt ) ≤ p(qt ) .
rThen, for all qt ∈ (z, 1),Lλm∆V (qt ) ≤ λm

1
− V L (qt )r≤ p(qt ) < 1.

(8)This implies that, by (4), we cannot have σ(qt ) = (0, 1) for any qt ∈ (z, 1).
For notational simplicity, defineV̇ti ≡

dV i ˙q̃t ,dq

as the time derivative of the value function given that there is no rating. Following (13) and35

(14) derived in Appendix B, for qt such that σ(qt ) = (1, 1), it must be
V̇tL = rV L (qt ) − p(qt ) > −1.
Also, for qt such that σ(qt ) = (0, 0), it must be thatV̇tL = rV L (qt ) − λm∆V L (qt ) > −1by (8). This argument establishes that V̇tL > −1 for all qt ∈ [z, 1).
For qt < z, since λm( 1r − V L (qt )) > p(qt ), we cannot have σ(qt ) = (1, 1). Definez ≡ sup {qt | σ(qt ) = (0, 1)} .
Note that z < z by their definitions and by (8). Moreover, σ(qt ) = (0, 1) requires that
λm∆V L (qt ) > 1. For qt < z, only σ(qt ) = (0, 0) or σ(qt ) = (0, 1) is possible. On the other
hand, for qt > z, we cannot have σ(qt ) = (0, 1) by definition. Therefore, for qt ∈ (z, z), it must
be that σ(qt ) = (0, 0). If σ(qt ) = (0, 0) for all qt ≤ z, the proof is complete. If not, by thedefinition of z, we can find a small number ε > 0 such that σ(qt ) = (0, 1) andV̇tL = rV L (qt ) − λm∆V L (qt ) > −

λm
,
r

(9)

for all qt ∈ (z − ε, z), given that V L (qt+ ) < 1r . The equality in (9) comes from the fact that for
qt ∈ (z − ε, z), since σ L (qt ) = 0, the continuation payoff can be written as
V L (qt ) = λmV L (qt+ )dt + (1 − rdt)(1 − λmdt)V L (qt− ),which can be reduced toλm∆V L (qt ) = rV L (qt− ) − V̇tL .
1
Also, for qt ∈ (z − ε, z), it will take at most λ(1−m)
ln m1 for the belief to drop from qt+ to qt .23

Note that qt+ can be either smaller or larger than z. If it is smaller, then σ(qt+ ) = (0, 1). If it is larger,
then σ(qt+ ) = (0, 0). The high-quality seller has a higher probability of receiving a good rating in the latter
case, since price shoppers can also leave good ratings. This means that when no rating is left after time t, thebelief will decrease faster when qt+ > z than when qt+ < z. In this case, the length of time it takes for the belief
23

−λt̃

λqt e
1
1
to drop back to qt given by t̃ such that λq e−λt̃ +λm(1−q= t̃, which yields t̃ = λ(1−m)
ln m.
)e−λmt̃tt36

If λm ≤ r, then V̇tL > −1 for all qt ∈ (z − ε, z) by (9), and therefore
Z t

Lλm∆V (qt ) = −λm

V̇τL dτ ≤ λt−t̃(qt )

λm
1
m
1
ln
=
ln
< 1,
λ(1 − m) m
1−m m

given m < 1, which in turn implies that λm∆V L (qt ) < 1. Therefore, a deviation to the highprice policy is profitable for the low-quality seller. This implies that for qt ∈ [z − ε, z], it mustbe σ(qt ) = (0, 0) as well. We can then repeat this process to show that σ(qt ) = (0, 0) for all
qt ≤ z − ε, which completes the proof.
If λm > r, we can only ensure V̇tL > −1 for qt > z. In this case, we first show that there
exists some l > 0 such that V̇tL > −1 for qt ∈ (z − l, z). To this end, consider qt = z − l forsome l ∈ (0, z). Then,L

Z t

V̇τL dτλm∆V (qt ) = −λm
t−T (z,z−l)Z t−tlV̇τL dτ +Z tV̇τL dτ= −λmt−T (z,z−l)
t−t


 l
1
1
λm
ln − tl + tl
≤ λm
λ(1 − m) m
r


λm
m
1
=
ln + λmtl
−1 ,
1−m m
r
where the inequality comes from the facts that: (i) V̇tL > −1 for qt > z; (ii) V̇tL > − λm
for
r
1
1
qt ∈ (z − l, z] by (9); (iii) T (qt , q) ≤ λ(1−m) ln m . If
T (z, z − l) <m
1 − 1−mln m1λm( λm− 1)r,then λm∆V L (qt ) < 1. Therefore, if l < l(z), where l(q) is such thatm1 − 1−mln m1
1 q[1 − (q − l(q))]
ln=,
λ (q − l(q))(1 − q)λm( λm− 1)rthen λm∆V L (qt ) < 1, which implies V̇tL > −1. Since l(q) is continuous and strictly positiveon q ∈ [z, z], by the extreme value theorem (Royden and Fitzpatrick, 2010, p.26), there existsl ≡ minq∈[z,z] l(q) > 0, which proves the claim.

37

This argument suggests thatV̇tL = rV L (qt ) − λm∆V L (qt ) > −1,
for qt > z − l. By applying the same argument, we can next show that given that V̇tL > −1
for qt > z − l, there again exists l > 0 such that for qt ∈ (z − 2l, z − l], λm∆V L (qt ) < 1. Bygoing through this procedure repeatedly, we can have λm∆V L (qt ) < 1 for some qt ∈ (z − ε, z).
However, this contradicts the previous fact that σ(qt ) = (0, 1) for qt slightly below z. Therefore,
we cannot have σ(qt ) = (0, 1) for any qt ∈ Q in equilibrium, and the only possibility isσ(qt ) = (0, 0) for all qt < z. This proves that no separating strategies can constitute anequilibrium, and hence that only pooling equilibrium is possible.
Proof of Proposition 5. For the following proof, we denote by V i (·; σ) the value function of thetype i seller under strategy σ. Let σ ∗ be the strategy of the seller-optimal equilibrium andV i∗ (·) ≡ V i (·; σ ∗ ) for brevity.
To obtain the seller-optimal equilibrium, consider a social planner who determines (Ql , Qh )to maximize the type i seller’s expected payoff among all pure-strategy pooling equilibria.
Alternatively, the social planner determines whether to implement σ(qt ) = (0, 0) or σ(qt ) =(1, 1) at each qt . Note that the social planner can implement σ(qt ) = (0, 0) at any qt while she
can implement σ(qt ) = (1, 1) only if qt is sufficiently large. Specifically, the social planner can
implement σ(qt ) = (1, 1) only if

λm


1
i
− V (qt ; σ) ≤ p(qt ),r(10)

for both i = L, H. We impose (10) as a constraint because if this fails to hold, there is no
way to induce both types to offer the high price in a pooling equilibrium. Note that in any
monotone equilibrium with strategy σ, there is a unique q i (σ) such that

λm


1
i i
− V (q (σ); σ) = p(q i (σ)).
r

Under this formulation, the incentive compatibility constraint for σ(qt ) = (1, 1) can be written
as qt ≥ max{q L (σ), q H (σ)}.
We first claim that the value function of the seller-optimal equilibrium must satisfy
V H∗ (qt ) = max {p(qt )sdt + e−rdt E[V H∗ (qt + dqt ) | s]}s∈{0,1}38

for qt ≥ max{q L (σ ∗ ), q H (σ ∗ )},24 whereE[V i∗ (qt + dqt ) | s] = γ i (s)dtV i (qt+ (s, s)) + (1 − γ i (s)dt)V i (qt− (s, s)).
Alternatively, for qt ≥ max{q L (σ ∗ ), q H (σ ∗ )}, the social planner chooses the high-price policy if
and only if
e−rdt E[V H∗ (qt + dqt ) | s = 0] ≤ p(qt )dt + e−rdt E[V H∗ (qt + dqt ) | s = 1],
which is in the limit reduced toλ(1 + m) VH∗

(qt+ (0, 0)) − V H∗ (qt )≤ p(qt ) + λ
1H∗− V (qt ) .
r(11)

Now suppose on the contrary that there is a subset Q0 ⊂ [max{q L (σ ∗ ), q H (σ ∗ )}, 1] such thatfor qt ∈ Q0 , (11) holds strictly but σ ∗ (qt ) = (0, 0).25 We can then pick a q 0 ∈ Q0 and construct
an alternative strategy σ 0 such that σ 0 (q 0 ) = (1, 1) and σ 0 (qt ) = σ ∗ (qt ) for all qt 6= q 0 . Then,
when (11) is satisfied, we have V i (qt ; σ 0 ) > V ∗ (qt ). This means that the incentive compatibility
constraint is relaxed in that q i (σ 0 ) < q i (σ ∗ ). This is, however, a contradiction because σ 0 raises
the expected payoff while satisfying the incentive compatibility constraint. This shows that
(11) is a necessary condition for the seller optimality.
Given this, we now claim that the seller-optimal equilibrium must be a threshold equilibrium. Suppose on the contrary that there are q and q 0 < q such that σ ∗ (q) = (0, 0) but
σ ∗ (q 0 ) = (1, 1) in the seller-optimal equilibrium. Observe that σ ∗ (q 0 ) = (1, 1) implies q 0 ≥ q H .
Observe also that qt ≥ max{q L (σ ∗ ), q H (σ ∗ )} implies (11) because V H∗ (qt ) ≤ 1r . These two facts
in turn imply σ ∗ (q) = (1, 1) because q > q 0 ≥ q H and (11) must hold at q, a contradiction.
Therefore, the seller-optimal equilibrium must be a threshold equilibrium where σ ∗ (qt ) = (1, 1)
if and only if qt ≥ max{q L , q H }. Then, by statement (i) of Lemma 2 and the definition of q ∗ ,we have q L = q ∗ > q H , suggesting that the threshold equilibrium with q = q ∗ achieves theseller optimality.
By the same argument, to maximize the low-quality seller’s expected payoff, we must havee−rdt E[V L∗ (qt + dqt ) | s = 0] ≤ p(qt )dt + e−rdt E[V L∗ (qt + dqt ) | s = 1],
24

Notice a difference from the seller’s problem where each type chooses her own policy taking the othertype’s strategy as given. Here, a unilateral deviation by one type is not considered, and the maximum pricethe seller can charge depends on the public belief rather than the private belief as a consequence.
25
When (11) holds with equality, the social planner is indifferent between the low-price and high-pricepolicy. In this case, we assume that the social planner implements the high-price policy.

39

which becomes
λm V L∗ (qt+ (0, 0)) − V L∗ (qt ) ≤ p(qt ).
Again, since V L∗ (qt+ (0, 0)) < 1r , this condition holds if qt ≥ q L . Given that the seller-optimalequilibrium must be a threshold equilibrium, this condition holds for all qt ≥ q L = q ∗ . Thisshows that the threshold equilibrium with q = q ∗ also maximizes the low-quality seller’s expected payoff.

Appendix B: The value functions
In this appendix, we show how to derive the value function for each type, which we use
for the numerical example in Section 5.4 as well as for Proposition 2. Given the equilibrium
strategy of our focus, this can be done piecewise for Qh and each Qn . In what follows, wedenote by {q̃τ (q)}∞τ =0 the realized belief path, starting from q, when the seller receives no goodrating as defined in the proof of Lemma 2.
First, if the high-quality seller receives a good rating for any qt ∈ Qh , she can reveal hertrue type and earns a continuation payoff of 1r . For an interval [t, t + dt), this event occurs
with probability λdt. The value function of the high-quality seller must hence satisfyλV (qt ) = p(qt ) +dt + (1 − rdt)(1 − λdt)V H (qt− ).
rHTaking the limit, we obtain the following first-order ordinary differential equation:p(qt ) +λdV H ˙= (r + λ)V H (qt ) −
q̃t ,
r
dq

where q̃˙t = −λqt (1 − qt ) is the rate of change of q̃t . Solving this differential equation gives−(r+λ)T (q,q ∗ )

e

H

H

∗

Z T (q,q∗ )V (q) = V (q ) −−(r+λ)τ

e0λp(q̃τ (q )) +
r
∗


dτ,

(12)

q(1+m).26 Similarly, the value function of the low-quality
for any q ∈ Qh , where q ≤ q ∗ ≡ q(1+m)+(1−q)m

For the purpose of this proof, it suffice to construct a value function for q ∈ [q, q ∗ ], because qt never goes
above q ∗ on the equilibrium path (unless the initial prior q0 is greater than q ∗ ). Obviously, we can derive V i (q)for q ∈ (q ∗ , 1) by following the same procedure.
26

40

seller must satisfy
V L (qt ) = p(qt )dt + (1 − rdt)V L (qt− ),which leads top(qt ) = rV L (qt ) −dV L ˙q̃t .
dq

Again, solving the differential equation gives−rT (q,q ∗ )

e

L

L

Z T (q,q∗ )∗V (q) = V (q ) −e−rτ p(q̃τ (q ∗ ))dτ,

(13)

0

for any q ∈ Qh .
We now go down the belief space and consider each Qn in turn. For qt ∈ Qn , the valuefunction of the type i seller must satisfy
V i (qt ) = γ i (0)V i (qt+ )dt + (1 − rdt)(1 − γ i (0)dt)V i (qt− ),which leads toγ i (0)∆V i (qt ) = rV i (qt− ) −dV i ˙q̃t ,dq

where q̃˙t = −λmqt (1 − qt ). Solving this differential equation, we obtaine−(r+γ i (0))T (q,q 0 )

i

i

0

Z T (q,q0 )V (q) = V (q ) −ie−(r+γ (0))τ γ i (0)V i (q̃t (q 0 ))dτ,

(14)

0

for q 0 , q ∈ Qn .
Given these, we can obtain the value function for the whole belief space. We first obtainiV (q) for q ∈ Qh by assigning a value V i (q ∗ ) to (12) and (13). Then, by continuity of thevalue function, we can further derive V i (q) for q ∈ Q1 , q ∈ Q2 , and so forth, until we pin down
V i (q). Note that the value of V i (q) continuously increases with the initial value of V i (q ∗ ). The
correct value of V i (q ∗ ) should be the one that leads to V i (q) = 0.

41