Endogenous Quality in Social Learning

TL;DR

This paper models how a seller's reputation and quality investments evolve over time in a social learning environment with limited observability, revealing conditions that influence quality levels and learning speed.

Contribution

It introduces a dynamic reputation model with endogenous quality, characterizes equilibrium behaviors, and analyzes how information precision and disclosure affect investment and welfare.

Findings

Quality is highest at intermediate reputation beliefs.

Higher private signal precision can reduce investment and quality.

Public disclosure expands investment regions and accelerates learning.

Abstract

We study dynamic reputation in a social-learning environment where only purchase decisions are observable. A long-lived seller posts a fixed price and chooses costly product quality in each period before interacting with short-lived buyers who observe past purchases but not past consumption outcomes, and receive private signals about current quality. We embed this environment in a standard type-based reputation model with a small probability of a commitment type and characterize Markov equilibria. The belief space partitions into pessimistic and optimistic cascades and an interior learning region. In cascades the seller never invests in quality, while in the learning region equilibrium investment is inverse-U in reputation: quality is highest at intermediate beliefs and lowest at the extremes. This produces early-resolution and double-hump patterns for the evolution of beliefs and quality. When quality is endogenous, higher private signal precision can shrink the effective investment region and reduce equilibrium quality. Allowing flexible pricing or public disclosure of outcomes expands the region in which the seller invests, accelerates learning, and raises welfare.

Equations

$${\theta }_t\in \{0,1\},$$

$$u(a_t,{\theta }_t)=\{\begin{array}{ll}v-p,& \text{if }{a}_t=1,{\theta }_t=1,\\ -p,& \text{if }{a}_t=1,{\theta }_t=0,\\ 0,& \text{if }{a}_t=0,\end{array}$$

$${\pi }_t^S=p\cdot a_t-c\cdot {\theta }_t,$$

$$q\in (\frac{1}{2},1),\mathbb{P}(s_t=H\mid {\theta }_t=1)=q,\mathbb{P}(s_t=H\mid {\theta }_t=0)=1-q.$$

$$z:=\frac{q}{1-q}>1$$

$${\pi }^{\ast }:=\frac{p}{v}$$

$${\lambda }_t\in [0,1],$$

$${\sigma }^S:[0,1]\to [0,1],$$

$${\sigma }^B:[0,1]\times \{H,L\}\to [0,1],$$

$$\Phi (\lambda \mid a)\in [0,1]$$

$${\lambda }_{t+1}=\Phi ({\lambda }_t\mid a_t),$$

$$\gamma (\lambda ,\theta ):=\mathbb{P}(a_t=1\mid {\lambda }_t=\lambda ,{\theta }_t=\theta ).$$

$$V(\lambda )=\mathbb{E}\left[\sum _{t=0}^{\infty }{\delta }^t\right(p\cdot a_{t+1}-c\cdot {\theta }_{t+1}\left)\right|{\lambda }_1=\lambda ,{\sigma }^S,{\sigma }^B],$$

$${\lambda }_t=\mathbb{P}({\theta }_t=1\mid h_t)={\mu }_t\cdot 1+(1-{\mu }_t){\sigma }^S({\lambda }_t).$$

$$\mathbb{P}(\theta =1\mid s,\lambda )$$

$$r(\lambda ):=\frac{\lambda }{1-\lambda }$$

$$z:=\frac{q}{1-q}>1.$$

$$r_H(\lambda )=r(\lambda )z,r_L(\lambda )=\frac{r(\lambda )}{z}.$$

$$K:=\frac{{\pi }^{\ast }}{1-{\pi }^{\ast }}=\frac{p}{v-p}.$$

$$\underline{\lambda }:=\frac{K/z}{1+K/z},\bar{\lambda }:=\frac{Kz}{1+Kz}.$$

$$\underset{\theta \in \{0,1\}}{\max }p\cdot \gamma (\lambda ,q;\theta )-c\cdot \theta ,$$

$${\psi }_1(\lambda ):=\mathbb{P}(a_t=1\mid {\theta }_t=1,\lambda ),{\psi }_0(\lambda ):=\mathbb{P}(a_t=1\mid {\theta }_t=0,\lambda ),$$

$$({\psi }_1(\lambda ),{\psi }_0(\lambda ))=\{\begin{array}{ll}(1,1),& \text{if }\lambda \ge \bar{\lambda }\text{(optimistic cascade)},\\ (q,1-q),& \text{if }\lambda \in (\underline{\lambda },\bar{\lambda })\text{(learning region)},\\ (0,0),& \text{if }\lambda \le \underline{\lambda }\text{(pessimistic cascade)}.\end{array}$$

$${\theta }_t=\{\begin{array}{ll}1,& \text{with probability }{\sigma }^S({\lambda }_t),\\ 0,& \text{with probability }1-{\sigma }^S({\lambda }_t).\end{array}$$

$${\lambda }^{\prime }=\Phi (\lambda \mid a)$$

$${\lambda }^{+}(\lambda ):=\Phi (\lambda \mid 1),{\lambda }^{-}(\lambda ):=\Phi (\lambda \mid 0),$$

$${\lambda }^{-}(\lambda )<\lambda <{\lambda }^{+}(\lambda ),$$

$${\lambda }_{t+1}=\Phi ({\lambda }_t\mid a_t),a_t\sim {\sigma }^B({\lambda }_t,s_t),$$

$$\gamma (\lambda ,\theta ):=\mathbb{P}(a_t=1\mid \lambda ,{\theta }_t=\theta )=\{\begin{array}{ll}{\psi }_1(\lambda ),& \text{if }\theta =1,\\ {\psi }_0(\lambda ),& \text{if }\theta =0,\end{array}$$

$$\pi (\lambda ,\theta )=p\cdot \gamma (\lambda ,\theta )-c\cdot \theta .$$

Full text

Keywords: Reputation; social learning; informational cascades; dynamic games; product quality.

JEL Classification Numbers: D82, D83, C73, L15.

1 Introduction

Many fixed-price markets generate public records of actions rather than outcomes. Walk-in restaurants, cafés, and small venues post prices prominently while passersby see who goes in and who walks past, but not whether those who entered were satisfied. Digital subscription services often reveal aggregate uptake (subscriber counts, queue length, occupancy) while individual consumption experiences remain private. In such environments, potential customers naturally use observed purchase and non-purchase decisions as social information about quality, and forward-looking sellers can invest in quality to influence how informative those decisions are over time.

This paper studies a dynamic reputation problem in such an action-based social-learning environment. A long-lived seller posts a fixed price and, in each period, chooses a costly quality level before meeting a single short-lived buyer. The buyer observes the entire history of past purchase decisions and a private signal about current quality, then decides whether to buy. Only the buyer’s action is publicly observed; realized consumption utility is not. Public beliefs about the seller’s quality—equivalently, about the probability that the seller delivers high quality—are therefore updated solely through the informativeness of purchase decisions.

Our starting point is the sequential social-learning model of Banerjee, (1992) and Bikhchandani et al., (1992), in which agents move sequentially, observe predecessors’ actions but not their signals, and may rationally ignore their own information once enough actions point in one direction. In that literature, the underlying state is exogenous. Here we endogenize it: a long-lived seller chooses quality each period, which in turn determines how informative observed purchase decisions will be for future buyers. The seller faces a dynamic trade-off: investing in quality is costly but can steer public beliefs into regions where future actions remain responsive to private signals, sustaining a valuable reputation.

Relative to the classic reputation models of Kreps and Wilson, (1982) and Milgrom and Roberts, (1982), surveyed in Mailath and Samuelson, (2006), our information structure is stark. First, only actions are publicly observed; neither realized payoffs nor even a coarse outcome statistic is available. Second, the price is fixed and cannot be used as a signalling instrument. Reputation capital is therefore accumulated not by charging premia or by observable performance, but by choosing quality so as to keep observed actions informative.

We embed this reduced-form problem in a standard type-based reputation framework. With small probability the seller is a commitment type who always delivers high quality; otherwise he is a strategic type who chooses quality optimally given continuation values. Public beliefs about the seller’s type and about current quality evolve via Bayes’ rule from observed purchase decisions. We focus on Markov equilibria in which strategies depend only on the current public belief about quality, and we use a fixed tie-breaking rule at buyer indifference together with a vanishing-perturbation argument to pin down belief evolution at cascade boundaries. This yields a well-defined Markov equilibrium of the reduced-form game, which corresponds to a weak perfect Bayesian equilibrium of the underlying type-uncertainty game.

Our first result is to show that, under this selection, public beliefs partition into three regions: a pessimistic cascade in which no buyer ever purchases; an optimistic cascade in which every buyer purchases; and an interior learning region in which actions are responsive to private signals. In both cascade regions the seller optimally sets quality to low: when actions no longer react to signals, there is no reputational return to investment. In the learning region, by contrast, the seller may invest in quality to influence the probabilities with which beliefs move toward the optimistic versus the pessimistic cascade.

Our second result characterizes the shape of the seller’s optimal quality policy. We show that the equilibrium investment profile is inverse-U in reputation: the seller does not invest when beliefs are very pessimistic or very optimistic and invests with positive probability at intermediate beliefs. The basic mechanism is reminiscent of the “reputation traps” in Levine, (2021): incentives to invest in reputation are strongest at intermediate reputations and vanish near the extremes. Under a mild curvature condition on the value function, this inverse-U pattern generates two qualitatively distinct dynamic regimes in numerical examples. In an early-resolution regime, investment is concentrated in a single connected region and beliefs drift monotonically toward the optimistic cascade. In a double-hump regime, investment appears in two waves separated by an interior no-investment phase; beliefs can move both up and down before eventually being absorbed into one of the cascades.

Third, we study how reputation incentives respond to information and patience. A natural intuition from exogenous-state social learning is that more precise private signals should improve information aggregation and, if anything, strengthen incentives to invest in quality. We show that this need not hold once quality is endogenous. At fixed prices, increasing signal precision can widen the static learning region in which actions respond to signals, but—in a parametric example—reduce the set of beliefs at which the seller actually invests in quality, because beliefs are then pushed into cascades more quickly. We also show that the investment region shrinks monotonically in the cost of quality and expands monotonically in the seller’s discount factor.

Finally, we consider two design extensions that correspond to natural levers in practice. First, we allow the seller to choose the posted price in addition to quality. For each belief there is an implementability set of prices that keep buyer actions informative. We illustrate how a sufficiently patient seller can use prices inside this set to avoid cascades for a wide range of beliefs, choosing the highest implementable price at each belief to maximize current revenue subject to maintaining informativeness. Second, we analyze an environment in which post-purchase outcomes are publicly disclosed (for example, through credible reviews or ratings). Making consumption outcomes observable strengthens each purchase as a learning event and, in our formal comparison, weakly expands the region in which the seller invests in quality, accelerates learning, and raises welfare relative to the benchmark with action-only observability.

The leading applications are fixed-price markets for experience goods in which only purchases are visible: restaurants or service venues where prices are posted and occupancy is observable but detailed reviews are scarce or delayed; or subscription services where outsiders can track the number of subscribers but not their realized satisfaction. In such settings our analysis highlights that better private information or more confident priors can paradoxically cool investment incentives when they push behavior into cascades at a fixed price, and that simple institutional changes—flexible pricing or outcome disclosure—can restore high-quality provision.

The rest of the paper is organized as follows. Section 2 introduces the environment, information structure, and equilibrium concept. Section 3 presents a static benchmark that clarifies buyer decisions and cascade thresholds. Section 4 formulates the dynamic program and establishes existence of a Markov equilibrium. Section 5 characterizes the equilibrium structure, including cascade regions and the inverse-U investment policy. Section 6 analyzes dynamic patterns and welfare. Section 7 develops comparative statics with respect to signal precision and other primitives. Section 8 studies the design extensions with flexible pricing and public outcome disclosure. Section 9 discusses an application and relates our work to existing literatures. Section 10 concludes.

2 Model

2.1 Environment and primitives

Time is discrete, $t=1,2,\dots$. A long-lived seller interacts with a sequence of short-lived buyers, one per period. In each period $t$ the seller chooses a quality level

[TABLE]

where $\theta_{t}=1$ denotes high quality and $\theta_{t}=0$ denotes low quality. Choosing high quality costs $c>0$ to the seller; low quality is costless.

A buyer arrives in period $t$ and decides whether to buy one unit of the seller’s product at a fixed posted price $p>0$. Let $a_{t}\in\{0,1\}$ denote the buyer’s action, where $a_{t}=1$ means “buy” and $a_{t}=0$ means “do not buy”. The buyer’s flow payoff is

[TABLE]

with $v>p$, so a buyer strictly prefers to purchase if she believes the product is high quality with sufficiently high probability. The seller’s flow payoff in period $t$ is

[TABLE]

and the seller discounts future payoffs at factor $\delta\in(0,1)$.

Each buyer in period $t$ receives a private signal $s_{t}\in\{H,L\}$ about current quality $\theta_{t}$. Signals are conditionally i.i.d. across buyers with precision

[TABLE]

Let

[TABLE]

denote the likelihood ratio associated with a high signal.

Given a prior belief $\lambda\in[0,1]$ that current quality is high, the buyer’s posterior after observing signal $s\in\{H,L\}$ is obtained by Bayes’ rule. Let

[TABLE]

denote the cutoff probability at which the buyer is indifferent between buying and not buying: she strictly prefers to buy if and only if her posterior belief that $\theta_{t}=1$ is at least $\pi^{\ast}$.

2.2 Information and public belief

The key informational assumption is that only actions are publicly observable. The realized quality $\theta_{t}$ and the buyer’s private signal $s_{t}$ are never observed by outsiders.

Let $h_{t}=(a_{1},\dots,a_{t-1})$ denote the public history of past purchase decisions up to period $t-1$. We restrict attention to equilibria in which this history is summarized by a one-dimensional public belief or reputation state

[TABLE]

interpreted as the market’s belief that the seller will deliver high quality in period $t$. Formally, in the type-based foundation of Section 2.4, $\lambda_{t}$ coincides with the posterior probability that $\theta_{t}=1$ given $h_{t}$ and equilibrium strategies. We take an initial reputation $\lambda_{1}\in(0,1)$ as given.

The within-period timing is:

1. (i)

At the beginning of period $t$, the public belief $\lambda_{t}$ is observed by the seller and buyer $t$. 2. (ii)

The seller chooses $\theta_{t}\in\{0,1\}$ (privately), incurring cost $c$ if $\theta_{t}=1$. 3. (iii)

Buyer $t$ observes a private signal $s_{t}\in\{H,L\}$ about $\theta_{t}$ and then chooses an action $a_{t}\in\{0,1\}$. 4. (iv)

The action $a_{t}$ is publicly observed; payoffs are realized; and the public belief updates to $\lambda_{t+1}$ according to Bayes’ rule given the equilibrium strategies.

Thus outsiders see only the sequence of purchase decisions and the fixed price; they never observe realized quality nor any public rating that would directly reveal buyers’ satisfaction.

2.3 Strategies and belief dynamics

A (possibly mixed) Markov strategy for the seller is a measurable function

[TABLE]

where $\sigma^{S}(\lambda)$ is the probability of choosing high quality at belief $\lambda$. A (possibly mixed) Markov strategy for buyers is a measurable function

[TABLE]

where $\sigma^{B}(\lambda,s)$ is the probability of buying upon observing $(\lambda,s)$.

Given a pair $(\sigma^{S},\sigma^{B})$ and an initial belief $\lambda_{1}$, public beliefs are updated recursively via Bayes’ rule applied to observed actions. Let

[TABLE]

denote the next-period belief after observing $a_{t}=a$ when the current belief is $\lambda$ and players follow $(\sigma^{S},\sigma^{B})$. The induced belief process $(\lambda_{t})_{t\geq 1}$ satisfies

[TABLE]

where, conditional on $\lambda_{t}$, the distribution of $a_{t}$ is determined by $\sigma^{S}$, $\sigma^{B}$, and the signal structure.

For future reference, we denote by $\gamma(\lambda,\theta)$ the probability that the buyer purchases when the current belief is $\lambda$ and the realized quality is $\theta\in\{0,1\}$, given the strategy pair $(\sigma^{S},\sigma^{B})$:

[TABLE]

The seller’s continuation value at belief $\lambda$ under $(\sigma^{S},\sigma^{B})$ is

[TABLE]

where the expectation is taken over the induced sequence of signals, actions, and beliefs.

2.4 Type-based foundation (sketch)

We briefly sketch the underlying type structure and refer to A for a formal construction and the existence proof.

At date [math], nature draws the seller’s type $\tau\in\{c,s\}$. With probability $\alpha\in(0,1)$ the seller is a commitment type ($\tau=c$) who always chooses high quality, $\theta_{t}=1$ for all $t$. With probability $1-\alpha$ the seller is a strategic type ($\tau=s$) who maximizes his discounted payoff by choosing quality according to a Markov strategy. The type is fixed over time and observed only by the seller; buyers know the prior $\alpha$.

Let $\mu_{t}$ denote the public posterior that $\tau=c$ at the beginning of period $t$. Given a Markov strategy $\sigma^{S}$ for the strategic type, the market’s belief that the seller delivers high quality in period $t$ can be written as

[TABLE]

Under any Markov equilibrium, $\lambda_{t}$ is therefore a one-dimensional state variable that summarizes the history $h_{t}$ for purposes of predicting current and future quality. Conversely, given a Markov equilibrium of the reduced-form game in terms of $(\lambda_{t},\sigma^{S},\sigma^{B},\Phi)$, one can construct a weak perfect Bayesian equilibrium of the underlying type-uncertainty game in which beliefs over types are updated by Bayes’ rule and the induced quality beliefs coincide with $\lambda_{t}$.

In what follows we work directly with the reduced-form Markov state $\lambda_{t}$ and belief-update map $\Phi$, taking the type-based model as a foundation that justifies the equilibrium concept.

2.5 Equilibrium concept and selection

Given the type-based foundation, we view a Markov equilibrium of the reduced-form game as the Markov component of a weak perfect Bayesian equilibrium of the underlying type-uncertainty game.

Definition 1**.**

A Markov equilibrium of the reduced-form game is a triple $(\sigma^{S},\sigma^{B},\Phi)$ such that:

1. (a)

Given $\Phi$ and $\sigma^{B}$, the seller’s strategy $\sigma^{S}$ maximizes $V(\lambda)$ at every $\lambda\in[0,1]$. 2. (b)

Given $\Phi$ and $\sigma^{S}$, the buyer’s strategy $\sigma^{B}$ is myopically optimal at every $(\lambda,s)$: for each $\lambda$ and $s\in\{H,L\}$, $\sigma^{B}(\lambda,s)$ yields weakly higher expected flow payoff than the alternative action. 3. (c)

The belief-update map $\Phi$ is obtained from Bayes’ rule applied to actions and $(\sigma^{S},\sigma^{B})$ whenever the denominator is nonzero; at belief–action pairs that have zero probability in equilibrium, $\Phi$ is determined by a fixed selection described below.

Two boundary issues are standard in action-based social-learning models. First, when a buyer is exactly indifferent between buying and not buying, any mixture is a best response, and different tie-breaking rules can reverse belief dynamics. Second, in cascade regions actions become unresponsive to private signals, so small off-path perturbations can support a continuum of locally self-confirming beliefs.

We adopt two simple selection devices:

• •

Tie-breaking at buyer indifference. Whenever a buyer is indifferent between buying and not buying at $(\lambda,s)$, we assume she buys. This yields a unique cutoff rule at each belief.

• •

Vanishing tremble at cascades. We consider a sequence of perturbed environments in which, with a small probability $\varepsilon>0$, publicly observed actions are generated by a tiny exogenous noise in the signal structure. For each $\varepsilon$ we restrict attention to Markov equilibria with Bayes-consistent belief updates, and we define the reduced-form belief-update map $\Phi$ in the benchmark model as the limit of these maps as $\varepsilon\downarrow 0$.

This selection pins down belief evolution at cascade boundaries and rules out pathological cycling at indifference. In Section 4 we formulate the seller’s dynamic program in terms of $(V,\Phi)$, and in Theorem 1 we show that there exists at least one Markov equilibrium under this selection.

3 Static Benchmark

In this section we consider a one-period version of the environment. The goal is to clarify the buyer’s decision problem and to identify the cascade thresholds that will organize the dynamic analysis.

Suppose there is a single period with a single buyer. The buyer holds a prior belief $\lambda\in[0,1]$ that current quality is high, $\theta=1$. She observes a private signal $s\in\{H,L\}$ with precision $q\in(1/2,1)$ as in Section 2, and then decides whether to buy at the fixed price $p$. There is no future and the seller’s choice of quality has no reputational consequences.

3.1 Buyer decision rule and cascade thresholds

Given prior $\lambda$ and signal $s$, the buyer forms a posterior belief

[TABLE]

by Bayes’ rule and compares it to the indifference cutoff $\pi^{\ast}=p/v$. It is convenient to express the Bayesian update in odds form. Let

[TABLE]

denote prior odds that quality is high, and define the likelihood ratio

[TABLE]

Then the posterior odds after a high signal $H$ and a low signal $L$ are

[TABLE]

The buyer is indifferent between buying and not buying when her posterior belief equals $\pi^{\ast}$, or equivalently when posterior odds equal

[TABLE]

The static best response has the familiar cutoff form.

Lemma 1**.**

Fix a prior $\lambda\in[0,1]$. The buyer’s unique best response is characterized as follows:

1. (a)

If $r(\lambda)\geq Kz$, then the buyer buys regardless of her signal. 2. (b)

If $r(\lambda)\leq K/z$, then the buyer never buys. 3. (c)

If $K/z<r(\lambda)<Kz$, then the buyer buys after a high signal $H$ and does not buy after a low signal $L$.

Part (c) describes the learning region in which actions are responsive to private information; parts (a) and (b) are the optimistic and pessimistic cascade regions, respectively.

In terms of beliefs rather than odds, define the thresholds

[TABLE]

Lemma 1 implies that the buyer’s static decision rule can be written as:

• •

if $\lambda\leq\underline{\lambda}$, never buy;

• •

if $\lambda\in(\underline{\lambda},\bar{\lambda})$, buy if and only if $s=H$;

• •

if $\lambda\geq\bar{\lambda}$, always buy.

The proofs are standard and parallel the analysis in Banerjee, (1992) and Bikhchandani et al., (1992).

3.2 Seller’s static problem

In the one-period benchmark, the seller chooses quality knowing that buyers behave according to the cutoff rule induced by Lemma 1. Let $\gamma(\lambda,q)$ denote the induced probability of purchase at belief $\lambda$ when signals have precision $q$ and the buyer follows the static rule above. The seller’s static problem is then

[TABLE]

where $\gamma(\lambda,q;1)$ and $\gamma(\lambda,q;0)$ are the purchase probabilities under high and low quality, respectively.

Because there is no future, the only role of quality in the static problem is to tilt the current buyer’s signal distribution. For many parameter configurations, the static gain from quality is smaller than its cost $c$, so the seller never invests in high quality in the one-shot game. In the dynamic model below, by contrast, investing in quality can be optimal in beliefs between $\underline{\lambda}$ and $\bar{\lambda}$ even when it is statically unprofitable, because it shifts public beliefs and thereby affects future demand. The thresholds $\underline{\lambda}$ and $\bar{\lambda}$ and the associated learning versus cascade regions thus provide the natural state-space partition for the dynamic analysis that follows.

4 Dynamic Program and Markov Equilibrium

In this section we formulate the seller’s dynamic optimization problem in terms of the public belief $\lambda$ and characterize Markov equilibria of the reduced-form game. We begin by describing belief dynamics under action observability, then write the Bellman equation for the seller, and finally state an existence result under our selection.

4.1 Belief dynamics

Fix a Markov strategy profile $(\sigma^{S},\sigma^{B})$ and recall the static buyer rule from Section 3. For each belief $\lambda\in[0,1]$, buyer $t$ observes $\lambda_{t}=\lambda$ and her signal $s_{t}$, computes her posterior, and applies the cutoff rule summarized in Lemma 1. Hence, conditional on the current belief $\lambda$ and the realized quality $\theta_{t}$, the probability that the buyer purchases is

[TABLE]

where

[TABLE]

In the learning region $(\underline{\lambda},\bar{\lambda})$, purchase decisions are informative: they coincide with high signals and therefore amplify the underlying signal structure. In the two cascade regions, by contrast, the buyer ignores her private signal and either always buys or never buys, making actions locally uninformative about quality.

Given a seller strategy $\sigma^{S}$, the realized quality in period $t$ is

[TABLE]

The (reduced-form) belief-update map $\Phi$ describes how the public belief $\lambda$ changes after observing an action $a_{t}\in\{0,1\}$. For each belief $\lambda$ and action $a\in\{0,1\}$,

[TABLE]

is defined as the posterior probability that the seller will deliver high quality in the next period, given that the current belief was $\lambda$ and action $a$ was observed under $(\sigma^{S},\sigma^{B})$.

Under the type-based foundation sketched in Section 2.4, $\Phi$ is induced by Bayes’ rule applied to histories and types. In the learning region, where actions coincide with private signals, the update has the familiar Bikhchandani–Hirshleifer–Welch structure: purchase events push beliefs upward and non-purchases push beliefs downward. We denote the successor beliefs after a purchase and a non-purchase by

[TABLE]

and write them simply as $\lambda^{+}$ and $\lambda^{-}$ when the current belief is clear from context. In the learning region, the selection described in Section 2 implies

[TABLE]

so beliefs drift upward on average when purchases are frequent. In the cascade regions, actions are locally insensitive to signals; our vanishing-tremble selection yields $\lambda^{+}(\lambda)\approx\lambda^{-}(\lambda)\approx\lambda$, so public beliefs move only due to rare perturbations.

Formally, the belief process $(\lambda_{t})_{t\geq 1}$ induced by a Markov strategy profile $(\sigma^{S},\sigma^{B})$ and the update map $\Phi$ satisfies

[TABLE]

where $s_{t}$ is drawn conditionally on $\theta_{t}$ as in Section 2. This Markov structure is the backbone of the seller’s dynamic optimization problem.

4.2 Seller’s value function

Given a Markov equilibrium candidate $(\sigma^{S},\sigma^{B},\Phi)$, the seller’s continuation value from belief $\lambda$ satisfies a Bellman equation. Let

[TABLE]

denote the purchase probability at belief $\lambda$ under high or low quality, respectively. If the seller chooses quality $\theta\in\{0,1\}$ at belief $\lambda$, his expected per-period payoff is

[TABLE]

Let $V(\lambda)$ denote the seller’s continuation value at belief $\lambda$. Under Markov strategies, the Bellman equation is

[TABLE]

The strategic type chooses high quality at belief $\lambda$ if and only if the gain in expected continuation value from raising the probability of purchase exceeds the current cost $c$.

It is convenient to define the dynamic marginal value of a purchase at belief $\lambda$ as

[TABLE]

Then the difference in continuation value between high and low quality at belief $\lambda$ can be written as

[TABLE]

because both qualities induce the same pair of successor beliefs $\lambda^{+}(\lambda)$ and $\lambda^{-}(\lambda)$; they differ only in the probability with which each successor is reached. The seller’s optimal quality choice at belief $\lambda$ therefore satisfies

[TABLE]

In particular, in the two cascade regions we have $\gamma(\lambda,1)=\gamma(\lambda,0)$, so the dynamic marginal benefit of quality vanishes and the seller never invests in high quality when $c>0.$ This observation underlies the zero-investment result in Section 5.

Equation (1) and the buyer’s cutoff rule jointly determine the Markov equilibrium strategy profile $(\sigma^{S},\sigma^{B})$ and value function $V$. In numerical illustrations, we solve (1) by value iteration on a fine grid of beliefs, using the selection described below at cascade boundaries.

4.3 Existence and selection of Markov equilibrium

We now state an existence result for Markov equilibria under our selection.

First, fix a Markov strategy $\sigma^{S}$ for the seller. Given the buyer’s myopic best response as in Lemma 1, the induced belief-update map $\Phi$ is well defined for every belief $\lambda$ and action $a$. Our tie-breaking convention at buyer indifference pins down a unique cutoff rule at each $\lambda$, and the vanishing-tremble device selects a unique limit of Bayes-consistent updates at cascade boundaries. The resulting reduced-form transition kernel $\lambda\mapsto(\lambda^{-}(\lambda),\lambda^{+}(\lambda))$ is measurable and piecewise continuous on $[0,1]$.

Second, for any belief-update map $\Phi$ generated in this way, the Bellman equation (1) defines a contraction mapping on the space of bounded continuous functions on $[0,1]$. Standard dynamic-programming arguments then imply the existence and uniqueness of a bounded continuous value function $V$ and an associated optimal Markov quality policy for the strategic type.

Third, by combining these two steps, we obtain a self-consistency condition: the seller’s Markov strategy $\sigma^{S}$ must be optimal given the belief dynamics induced by $(\sigma^{S},\sigma^{B})$, and the buyer’s Markov strategy $\sigma^{B}$ must be myopically optimal given $(\sigma^{S},\Phi)$. A fixed point of this correspondence yields a Markov equilibrium of the reduced-form game in the sense of Definition 1.

Theorem 1**.**

Suppose $v>p>0$, $q\in(1/2,1)$, $c>0$, and $\delta\in(0,1)$. Under the tie-breaking and vanishing-tremble selection described in Section 2, there exists at least one Markov equilibrium $(\sigma^{S},\sigma^{B},\Phi)$ of the reduced-form game. In any such equilibrium, the seller’s value function $V$ is bounded and nondecreasing in $\lambda$.

In what follows we fix a Markov equilibrium $(\sigma^{S},\sigma^{B},\Phi)$ and suppress explicit reference to $\Phi$ where no confusion arises. Section 5 characterizes the resulting equilibrium structure in terms of cascade regions, a learning region, and an inverse-U investment policy.

5 Equilibrium Structure: Cascades and Investment

We now characterize the structure of Markov equilibria in terms of the belief regions introduced in Section 3. We show that public beliefs partition into two cascade regions and an interior learning region, that the seller optimally chooses low quality in both cascades, and that the equilibrium quality policy in the learning region is inverse-U in reputation.

Formal statements of the main results in this section and their proofs are collected in C.

5.1 Cascade regions and learning region

Recall that the static buyer rule is summarized by the thresholds

[TABLE]

with $K=p/(v-p)$ and $z=q/(1-q)$. For any belief $\lambda$ and given the current signal $s$, a myopic buyer with belief $\lambda$ behaves as follows: she never buys if $\lambda\leq\underline{\lambda}$, buys if and only if $s=H$ if $\lambda\in(\underline{\lambda},\bar{\lambda})$, and always buys if $\lambda\geq\bar{\lambda}$. Because buyers are short-lived and do not internalize future effects, this static rule remains optimal in the dynamic game for any Markov equilibrium.

It is therefore natural to partition the state space $[0,1]$ into three regions:

[TABLE]

In $\mathcal{C}^{-}$ no buyer ever purchases; in $\mathcal{C}^{+}$ every buyer purchases; and in $\mathcal{L}$ a purchase occurs if and only if the buyer’s private signal is $H$. The dynamic content of the model arises from how the seller’s quality choice interacts with belief updates across these regions.

In the learning region $\mathcal{L}$, the belief-update map satisfies

[TABLE]

so that observing a purchase (non-purchase) strictly increases (decreases) the public belief. As $\lambda\downarrow\underline{\lambda}$ or $\lambda\uparrow\bar{\lambda}$, the drift shrinks in magnitude and the process approaches a cascade. In the cascade regions $\mathcal{C}^{-}$ and $\mathcal{C}^{+}$, by contrast, the buyer’s action no longer depends on her signal. Up to the infinitesimal perturbations induced by our vanishing tremble, the public belief is locally constant: $\lambda^{+}(\lambda)\approx\lambda^{-}(\lambda)\approx\lambda$.

5.2 Zero investment in cascades

We first establish that in any Markov equilibrium the seller never invests in high quality in either cascade region.

Proposition 1**.**

In any Markov equilibrium, the seller chooses low quality at every belief $\lambda\in\mathcal{C}^{-}\cup\mathcal{C}^{+}$ whenever $c>0$. Equivalently, $\sigma^{S}(\lambda)=0$ for all $\lambda\in\mathcal{C}^{-}\cup\mathcal{C}^{+}$.

Proof sketch.

Consider any belief $\lambda\in\mathcal{C}^{-}\cup\mathcal{C}^{+}$. By definition of these regions and Lemma 1, the buyer’s action at $\lambda$ is independent of her signal: she always buys in $\mathcal{C}^{+}$ and never buys in $\mathcal{C}^{-}$. Hence, conditional on $\lambda$, the distribution of $a_{t}$ is independent of $\theta_{t}$. Formally, $\gamma(\lambda,1)=\gamma(\lambda,0)$ in the notation of Section 4.2.

From the Bellman equation (1), the expected continuation value after a high-quality choice and after a low-quality choice differ only through $\gamma(\lambda,\theta)$. When $\gamma(\lambda,1)=\gamma(\lambda,0)$, high quality does not change the distribution over successor beliefs $(\lambda^{+},\lambda^{-})$, so its only effect is to subtract the current cost $c>0$. Thus, high quality is strictly dominated by low quality at any $\lambda\in\mathcal{C}^{-}\cup\mathcal{C}^{+}$, and the seller never invests in these regions in equilibrium. ∎

The intuition is simple: once beliefs enter a cascade region, the buyer has stopped reacting to her private information and the public belief becomes locally unresponsive to actions. At that point there is no reputational return to quality; the seller optimally skimps on quality and either continues to enjoy unconditional purchases in $\mathcal{C}^{+}$ or abandons the market in $\mathcal{C}^{-}$.

5.3 Inverse-U investment pattern

We now turn to the seller’s behavior in the learning region $\mathcal{L}$. Here, a purchase reveals a high signal and moves beliefs upward, while a non-purchase reveals a low signal and moves beliefs downward. High quality increases the probability of a purchase relative to low quality: in $\mathcal{L}$ we have

[TABLE]

so that $\gamma(\lambda,1)-\gamma(\lambda,0)=2q-1>0$. High quality is therefore potentially attractive as a way to tilt the distribution of successor beliefs toward $\lambda^{+}(\lambda)$ rather than $\lambda^{-}(\lambda)$.

Recall the dynamic marginal value of a purchase,

[TABLE]

and the quality-decision condition (2). In $\mathcal{L}$ this condition specializes to:

[TABLE]

High quality is optimal if the sum of its static and dynamic benefits, scaled by $2q-1$, exceeds the current cost $c$.

Our main qualitative result in this section is that the equilibrium quality policy $\lambda\mapsto\sigma^{S}(\lambda)$ has an inverse-U shape on $[0,1]$: starting from low beliefs, investment is initially zero (pessimistic cascade), then becomes strictly positive on an intermediate set of beliefs, and finally falls back to zero as beliefs approach the optimistic cascade.

To formalize this, we impose a mild regularity condition on the continuation value function that is standard in dynamic social-learning models.

Assumption 1**.**

The value function $V$ is continuous on $[0,1]$ and concave in log-odds in the learning region, in the sense that the function

[TABLE]

is concave on the set of log-odds corresponding to $\lambda\in\mathcal{L}$.

Assumption 1 states that the seller’s continuation value is a concave function of log-odds in the region where actions are informative. This property is satisfied in a broad class of dynamic programming problems with monotone belief dynamics and convex costs=.

Under Assumption 1, the dynamic marginal value of a purchase $\Delta(\lambda)$ is single-peaked in $\mathcal{L}$: it is small near the cascade boundaries, where beliefs almost never move, and larger at intermediate beliefs, where the prospect of pushing the process toward the optimistic cascade is most valuable. Combining this with (3) yields the inverse-U structure.

Proposition 2**.**

Suppose Assumption 1 holds and $c$ lies in an intermediate range. Then there exist belief levels

[TABLE]

such that in any Markov equilibrium:

1. (a)

$\sigma^{S}(\lambda)=0$* for all $\lambda\in[0,\lambda_{1}]\cup[\lambda_{2},1]$;* 2. (b)

$\sigma^{S}(\lambda)\in(0,1]$* for all $\lambda\in(\lambda_{1},\lambda_{2})$.*

In particular, the seller does not invest in high quality when beliefs are either sufficiently pessimistic or sufficiently optimistic, and invests with positive probability at intermediate beliefs.

Proof sketch.

By Proposition 1, $\sigma^{S}(\lambda)=0$ on $\mathcal{C}^{-}\cup\mathcal{C}^{+}$. On $\mathcal{L}$, the seller’s optimal quality choice is governed by (3). Under Assumption 1, the function $\Delta(\lambda)$ is continuous and single-peaked on $\mathcal{L}$. As $\lambda\downarrow\underline{\lambda}$ or $\lambda\uparrow\bar{\lambda}$, the belief process becomes locally trapped in a cascade, so the gain from tilting the purchase probability vanishes and $\Delta(\lambda)\to 0$. At more intermediate beliefs, pushing the belief toward the optimistic cascade yields a larger increase in continuation value, so $\Delta(\lambda)$ attains a strictly positive maximum at some interior point.

Fix primitives such that $c$ is small enough that $(2q-1)[p+\delta\Delta(\lambda)]>c$ at the maximizer of $\Delta$, but large enough that $(2q-1)p<c$. Then (3) implies that there is a nonempty interval $(\lambda_{1},\lambda_{2})\subset\mathcal{L}$ on which high quality is strictly optimal, and that high quality is strictly suboptimal outside this interval. Allowing for mixed strategies yields the stated inverse-U shape. ∎

The economic intuition behind Proposition 2 is straightforward. When reputation is very low, few buyers purchase even after good signals, so the incremental benefit of a marginally better signal structure is small relative to the cost of quality: investment does little to escape the pessimistic region. When reputation is very high, the seller enjoys a stream of purchases regardless of quality, and investment is unnecessary to sustain demand. At intermediate reputations, by contrast, a marginal improvement in the chance of a purchase both raises current revenue and pushes beliefs toward the optimistic cascade, where future payoffs are high. Reputation-building incentives are strongest there, generating an interior region of active investment and zero investment at the extremes.

The inverse-U pattern will play a central role in Section 6, where we distinguish equilibrium paths with early resolution of uncertainty from those with multiple waves of reputation-building activity.

6 Dynamic Patterns and Welfare

The inverse-U investment pattern in Proposition 2 induces distinct qualitative patterns for the evolution of beliefs and quality over time. In this section we describe two canonical regimes—early resolution and double hump—and discuss their welfare properties.

6.1 Early resolution vs. double-hump regimes

Fix an initial belief $\lambda_{1}\in\mathcal{L}$ and a Markov equilibrium $(\sigma^{S},\sigma^{B},\Phi)$ as in Theorem 1. The belief process $(\lambda_{t})_{t\geq 1}$ is a time-homogeneous Markov chain on $[0,1]$ with transition kernel induced by $\Phi$. By Proposition 1, $\mathcal{C}^{-}\cup\mathcal{C}^{+}$ is an absorbing set: once $\lambda_{t}$ enters a cascade region, the seller switches to low quality and the buyer’s actions become uninformative, so the process remains in that region up to rare perturbations.555Under the vanishing-tremble selection, beliefs move by at most $O(\varepsilon)$ inside cascades; in the limit $\varepsilon\downarrow 0$ cascades are absorbing.

Within the learning region $\mathcal{L}$, the inverse-U policy in Proposition 2 implies that there is an interval $(\lambda_{1},\lambda_{2})\subset\mathcal{L}$ on which the seller invests in high quality and sets low quality outside this interval. Depending on how $(\lambda_{1},\lambda_{2})$ is positioned and how $\Delta(\lambda)$ behaves, there are two qualitatively different ways in which beliefs can approach the optimistic cascade.

To state this precisely, define the (random) hitting time of the cascade set

[TABLE]

and write

[TABLE]

We say that the equilibrium exhibits early resolution if beliefs drift monotonically toward $\mathcal{C}^{+}$ from any interior starting point, in a sense defined below; we say it exhibits a double hump if there is an interior region in which investment temporarily stops and beliefs can move both upward and downward before eventual absorption.

Definition 2**.**

Let $(\sigma^{S},\sigma^{B},\Phi)$ be a Markov equilibrium satisfying Proposition 2.

1. (a)

The equilibrium is an early-resolution regime if there exist belief levels

[TABLE]

such that:

1. (i)

$\sigma^{S}(\lambda)>0$ for all $\lambda\in(\lambda_{E}^{-},\lambda_{E}^{+})$ and $\sigma^{S}(\lambda)=0$ for all $\lambda\in[\underline{\lambda},\lambda_{E}^{-}]\cup[\lambda_{E}^{+},\bar{\lambda})$; 2. (ii)

the belief process starting from any $\lambda_{1}\in(\lambda_{E}^{-},\lambda_{E}^{+})$ satisfies

[TABLE] 2. (b)

The equilibrium is a double-hump regime if there exist belief levels

[TABLE]

such that:

1. (i)

$\sigma^{S}(\lambda)>0$ for all $\lambda\in(\lambda_{1},\lambda_{2})\cup(\lambda_{3},\lambda_{4})$ and $\sigma^{S}(\lambda)=0$ for all $\lambda\in[\underline{\lambda},\lambda_{1}]\cup[\lambda_{2},\lambda_{3}]\cup[\lambda_{4},\bar{\lambda})$; 2. (ii)

starting from some $\lambda_{1}\in(\lambda_{2},\lambda_{3})$, the belief process visits both $(\lambda_{1},\lambda_{2})$ and $(\lambda_{3},\lambda_{4})$ with positive probability before hitting $\mathcal{C}^{-}\cup\mathcal{C}^{+}$.

In an early-resolution regime, investment is concentrated in a single connected region of beliefs, and the Markov chain drifts monotonically toward the optimistic cascade from any starting point inside that region. Quality investment is front-loaded: once beliefs cross a sufficiently optimistic threshold, the seller stops investing and the process quickly absorbs at $\mathcal{C}^{+}$. In a double-hump regime, there is an interior interval $(\lambda_{2},\lambda_{3})$ in which the seller optimally sets low quality, separating two active investment regions $(\lambda_{1},\lambda_{2})$ and $(\lambda_{3},\lambda_{4})$. The process may then exhibit two distinct waves of reputation-building, separated by a pause in which quality is low and actions remain informative only because buyers still react to their signals.

The following example shows that, under suitable primitives, both patterns can arise as equilibria.

Example**.**

Suppose Assumption 1 holds. There exist parameter configurations $(p,v,c,q,\delta)$ and Markov equilibria for which:

1. (a)

the equilibrium satisfies the early-resolution conditions in Definition 2(a): investment occurs on a single connected subset of $\mathcal{L}$ and, starting from interior beliefs, the belief process drifts monotonically toward the optimistic cascade; 2. (b)

the equilibrium satisfies the double-hump conditions in Definition 2(b): there is an interior interval of beliefs in which the seller does not invest, separating two distinct investment regions that may both be visited with positive probability before absorption.

Proof sketch.

Part (a): If $c$ is sufficiently small and $\delta$ is sufficiently close to one, the dynamic marginal value $\Delta(\lambda)$ is large throughout most of the learning region $\mathcal{L}$. In that case, condition (3) implies that high quality is optimal on a single connected subinterval of $\mathcal{L}$ containing beliefs just below $\bar{\lambda}$, and that the induced drift of beliefs is upward whenever $\lambda_{t}$ lies in that interval. This yields an early-resolution pattern in which beliefs hit the optimistic cascade before the pessimistic cascade with high probability.

Part (b): For intermediate values of $c$ and $\delta$, the concavity in log-odds implied by Assumption 1 and the geometry of the belief updates can generate a $\Delta(\lambda)$ function that is single-peaked in each of two separated subintervals of $\mathcal{L}$, with a dip in between. In such cases, condition (3) yields an optimal policy with two investment intervals separated by a no-investment interval, giving rise to the double-hump pattern. ∎

While Definition 2 is framed in terms of the reduced-form equilibrium dynamics, the underlying economic mechanism mirrors the “reputation trap” logic of Levine, (2021). When beliefs are moderately good, the seller has strong incentives to invest in quality in order to reach and maintain the optimistic cascade. Once beliefs become either too pessimistic or too optimistic, however, the dynamic return to investment fades, leading to phases in which reputation is either not worth building or can be safely milked.

6.2 Welfare comparison

We now compare the welfare implications of early-resolution and double-hump equilibria. We adopt a utilitarian notion of welfare that aggregates buyer and seller payoffs.

Let $W(\lambda_{1};\sigma^{S},\sigma^{B})$ denote ex ante expected discounted welfare starting from initial belief $\lambda_{1}$ under a Markov equilibrium $(\sigma^{S},\sigma^{B},\Phi)$:

[TABLE]

where $u$ and $\pi^{S}_{t}$ are the buyer and seller flow payoffs defined in Section 2. Because $v>p>0$ and high quality creates surplus while low quality destroys it, social welfare is higher in histories with more frequent high-quality trades and fewer low-quality purchases.

To obtain a sharp welfare ranking, we impose a further assumption on how the social value of information varies with beliefs. Let $\tilde{v}(\lambda)$ denote the expected one-period social surplus if the buyer knew the true quality at belief $\lambda$ and traded efficiently, and let $I(\lambda)$ denote the expected per-period increase in $\tilde{v}$ due to the informativeness of actions under the learning rule. We assume that $I$ is concave in log-odds, so that postponing the resolution of uncertainty is socially costly.

Assumption 2**.**

In the learning region $\mathcal{L}$, the per-period social value of information $I(\lambda)$ is concave in log-odds, in the sense that the function

[TABLE]

is concave on the set of log-odds corresponding to $\lambda\in\mathcal{L}$.

Assumption 2 holds, for example, when the social surplus from correct trade is a concave function of the log-odds of high quality and the informativeness of actions increases with the distance from the cascade boundaries. It formalizes the idea that, from a social perspective, a faster resolution of uncertainty is preferable when the long-run distribution of beliefs is fixed.

Under this assumption, we can compare early-resolution and double-hump equilibria that share the same primitives and initial belief.

Proposition 3**.**

Fix primitives $(p,v,c,q,\delta)$ and an initial belief $\lambda_{1}\in\mathcal{L}$. Suppose there exist two Markov equilibria, one exhibiting early resolution and one exhibiting a double-hump pattern in the sense of Definition 2, and that both equilibria induce the same hitting probabilities $(\tau^{+},\tau^{-})$ for the optimistic and pessimistic cascades. If Assumption 2 holds, then welfare is weakly higher in the early-resolution equilibrium:

[TABLE]

with strict inequality whenever the double-hump equilibrium features an interior phase with low quality in which beliefs remain in $\mathcal{L}$ with positive probability.

Proof sketch.

The argument is conditional on the existence of two such equilibria and uses a coupling of belief processes under the two regimes.

Because the two equilibria share the same primitives and initial belief, and yield the same hitting probabilities $(\tau^{+},\tau^{-})$ for the cascade set, they generate the same distribution over long-run states: the probability of ending up in the optimistic versus pessimistic cascade is identical. The difference between them lies in the path by which beliefs reach the cascades.

In the early-resolution equilibrium, beliefs move monotonically toward $\mathcal{C}^{+}$ from $\lambda_{1}$ and the seller invests whenever doing so keeps actions informative. Learning is therefore fast: the Markov chain spends relatively little time in the interior region with low-quality provision.

In the double-hump equilibrium, by contrast, there is an interior interval in which the seller does not invest in quality even though actions remain informative. Beliefs may hover in this region for multiple periods, during which low-quality purchases destroy surplus and information accumulates slowly. Concavity of $I$ in log-odds ensures that this interior pause is socially costly: delaying information acquisition cannot be offset by later bursts of learning of the same total magnitude. Comparing the discounted sums of $I(\lambda_{t})$ along equilibrium paths therefore yields the stated welfare ranking. ∎

Proposition 3 highlights a tension between private and social incentives in the presence of reputation-building. For a given long-run distribution of reputations, sellers may prefer to concentrate investment in regions where its private dynamic returns are highest, even if this generates socially inefficient pauses in learning and phases of low-quality trade. Design interventions that eliminate or compress the interior no-investment region—for example, by allowing flexible pricing or by making outcomes public—can then yield Pareto improvements, a theme we develop in Sections 7 and 8.

7 Comparative Statics

In this section we study how the equilibrium structure responds to changes in primitives. We focus on three key parameters: signal precision $q$, the cost of quality $c$, and the discount factor $\delta$. Our main message is that, once quality is endogenous, improvements in private information can have counterintuitive effects on the size of the learning region and on the seller’s investment incentives.

7.1 Signal precision

We first recall the static comparative statics from Section 3. There, cascade thresholds are given by

[TABLE]

A straightforward computation shows that

[TABLE]

so that in the static benchmark the learning region $(\underline{\lambda}(q),\bar{\lambda}(q))$ widens as signals become more precise: higher $q$ pushes the pessimistic threshold downward and the optimistic threshold upward. This is the familiar result from exogenous-state social learning models: more informative private signals make herding less likely.

In the dynamic model with endogenous quality, however, the effective region in which beliefs move and the seller invests is determined not only by the static thresholds but also by the equilibrium quality policy $\sigma^{S}(\lambda)$. Recall from Proposition 2 that, for fixed $q$, there exists an interval $(\lambda_{1},\lambda_{2})\subset\mathcal{L}(q)$ on which $\sigma^{S}(\lambda)>0$, where $\mathcal{L}(q)=(\underline{\lambda}(q),\bar{\lambda}(q))$ is the learning region, and the seller chooses low quality outside $(\lambda_{1},\lambda_{2})$.

An increase in $q$ has two opposing effects on the seller’s incentives. On the one hand, it increases the static information content of each signal: a purchase or non-purchase moves beliefs more per period. On the other hand, it also makes buyers more prone to herd at the fixed price, because extreme posteriors are reached more quickly. In equilibrium, the seller may anticipate that higher $q$ pushes beliefs into cascades faster and therefore reduce investment, shrinking the set of beliefs at which quality is high even if the static learning region $\mathcal{L}(q)$ widens.

To capture this formally, define the investment region

[TABLE]

where $\sigma^{S}_{q}$ denotes an equilibrium quality policy at signal precision $q$. We are interested in how the measure and location of $\mathcal{I}(q)$ vary with $q$.

Example** (Signal precision and investment).**

Suppose Assumption 1 holds and the cost $c$ lies in the intermediate range considered in Proposition 2. There exist primitives $(p,v,c,q_{1},\delta)$ and a Markov equilibrium with an inverse-U investment region $\mathcal{I}(q_{1})\subset\mathcal{L}(q_{1})$, and a higher signal precision $q_{2}>q_{1}$ with a corresponding equilibrium such that:

1. (a)

the static learning region $\mathcal{L}(q_{2})$ is strictly larger than $\mathcal{L}(q_{1})$; 2. (b)

the investment region shrinks in the sense that $\mathrm{Leb}(\mathcal{I}(q_{2}))<\mathrm{Leb}(\mathcal{I}(q_{1}))$, where $\mathrm{Leb}$ denotes Lebesgue measure on $[0,1]$.

Proof sketch.

Part (a) follows from continuity of the Bellman equation in $q$ and the existence of inverse-U equilibria at a given $q$ (Proposition 2). For $q$ in a neighborhood of such a point, the belief-update map and the value function vary continuously, so the optimal quality policy retains an inverse-U shape.

For part (b), note that as $q$ increases, the successor beliefs $\lambda^{+}(\lambda)$ and $\lambda^{-}(\lambda)$ move farther apart in the learning region. This increases the short-run informativeness of actions but also accelerates the approach to cascades. When the discount factor $\delta$ and cost $c$ are such that the seller cares primarily about the long-run flow of payoffs in the optimistic cascade, higher $q$ can reduce the incremental value of marginally increasing purchase probabilities at beliefs far from $\bar{\lambda}$, because beliefs will reach $\mathcal{C}^{+}$ quickly regardless of small changes in investment. This reduces $\Delta(\lambda)$ at intermediate beliefs and shrinks the set where condition (3) is satisfied. ∎

Example Example illustrates that more precise private information does not guarantee stronger reputation-building incentives when quality is endogenous. At fixed prices, making buyers “smarter” can induce them to herd more aggressively, compressing the region in which their actions are sensitive to the seller’s quality choice. The seller then has fewer states in which investment pays, and equilibrium quality provision can fall.

7.2 Costs, patience, and payoffs

We briefly describe how other primitives affect the investment region and dynamic patterns.

Cost of quality.

As the cost $c$ increases, the static and dynamic benefits of high quality must compensate a larger current loss. From condition (3), it is immediate that the set of beliefs at which high quality is optimal shrinks monotonically in $c$.

Proposition 4**.**

For fixed $(p,v,q,\delta)$ and fixed equilibrium selection, the investment region $\mathcal{I}(q;c)$ is nonincreasing in $c$: if $c^{\prime}>c$, then $\mathcal{I}(q;c^{\prime})\subseteq\mathcal{I}(q;c)$. Moreover, there exist thresholds $0<c_{1}<c_{2}$ such that:

1. (a)

if $c\leq c_{1}$, there exist early-resolution equilibria with investment on a large connected subset of $\mathcal{L}$; 2. (b)

if $c\geq c_{2}$, no investment occurs in any Markov equilibrium: $\mathcal{I}(q;c)=\emptyset$ for all $q$.

Proof sketch.

Monotonicity in $c$ follows directly from (3). For small $c$, the inequality holds on a wide range of beliefs, yielding a large investment region and early-resolution dynamics; for large $c$, $(2q-1)[p+\delta\Delta(\lambda)]<c$ for all $\lambda$, so high quality is never optimal. ∎

Discount factor.

The discount factor $\delta$ scales the dynamic component of the marginal benefit of quality. As $\delta$ increases, the seller attaches more weight to shifting beliefs toward the optimistic cascade and away from the pessimistic cascade, so $\Delta(\lambda)$ becomes more important relative to the static term.

Proposition 5**.**

For fixed $(p,v,q,c)$ and a fixed equilibrium selection, the investment region $\mathcal{I}(q;\delta)$ is nondecreasing in $\delta$: if $\delta^{\prime}>\delta$, then $\mathcal{I}(q;\delta)\subseteq\mathcal{I}(q;\delta^{\prime})$.

Proof sketch.

Monotonicity in $\delta$ follows from (3) and from the fact that $\Delta(\lambda)\geq 0$ when $\lambda^{+}(\lambda)$ is closer to $\mathcal{C}^{+}$ than $\lambda^{-}(\lambda)$. For $\delta$ sufficiently large, dynamic considerations dominate, making it optimal to invest in quality whenever this helps avoid the pessimistic cascade and reach the optimistic cascade quickly, which yields an early-resolution pattern. ∎

Remark**.**

In many parametric examples, increasing $\delta$ not only expands the investment region but also favors early-resolution dynamics in the sense of Definition 2(a): as the seller becomes more patient, it is optimal to invest in quality on a large connected subset of the learning region and to push beliefs quickly toward the optimistic cascade.

We do not attempt a full characterization of parameter values for which all equilibria with investment exhibit early resolution.

Price and buyer surplus.

Changes in the posted price $p$ affect both the buyer’s cutoff $\pi^{\ast}=p/v$ and the seller’s static revenue from each purchase. Increasing $p$ raises $K=p/(v-p)$ and shifts the static thresholds $(\underline{\lambda},\bar{\lambda})$ toward the origin: higher prices make buyers more demanding, reducing the set of beliefs at which they are willing to buy regardless of their signal. At the same time, a larger $p$ raises the flow payoff from each purchase, strengthening the seller’s incentives to invest in quality at beliefs where investment can sustain informative actions and future demand.

The net effect of price changes on the investment region is generally ambiguous in the fixed-price benchmark: increasing $p$ both narrows the static learning region and raises the per-period payoff from maintaining a good reputation. In Section 8 we return to price as a design variable and show that, when the seller can choose $p$ subject to an implementability constraint, he optimally uses price flexibility to avoid cascades and expand the effective learning region.

Taken together, Example Example and Proposition 5 highlight that reputation incentives in action-based social-learning environments are shaped by a delicate balance between information, costs, and patience. Neither more precise signals nor higher prices guarantee more investment in quality; the equilibrium pattern depends critically on how these primitives interact with the endogenous evolution of beliefs.

8 Design Extensions

We now consider two simple design extensions of the baseline environment. In the first, the seller is allowed to choose the posted price in addition to quality. In the second, we modify the information structure so that post-purchase outcomes are publicly observed. These extensions illustrate how institutional features—price flexibility and outcome disclosure—shape the scope for reputation-building and the speed of learning.

8.1 Flexible pricing

We extend the baseline timing by allowing the seller to choose a price $p_{t}$ in each period before quality and signals are realized. The within-period sequence of moves becomes:

1. (i)

The public belief $\lambda_{t}$ is observed. 2. (ii)

The seller posts a price $p_{t}$ and then chooses quality $\theta_{t}\in\{0,1\}$. 3. (iii)

Buyer $t$ observes $(\lambda_{t},p_{t})$ and her private signal $s_{t}$, and chooses $a_{t}\in\{0,1\}$. 4. (iv)

The action $a_{t}$ is publicly observed; payoffs are realized; beliefs update to $\lambda_{t+1}$.

We continue to assume that buyers are short-lived and myopic: in period $t$ they choose $a_{t}$ to maximize their period-$t$ payoff. For any posted price $p$, let $\pi^{\ast}(p)=p/v$ denote the corresponding cutoff probability at which a buyer is indifferent between buying and not buying when she expects high quality with probability $\pi^{\ast}(p)$. The static analysis in Section 3 then applies with $K(p)=p/(v-p)$, $z=q/(1-q)$, and cascade thresholds

[TABLE]

For each belief $\lambda$, there is a set of prices that keep the buyer responsive to her signal. Define the implementability set

[TABLE]

If $p\in\mathcal{P}(\lambda)$, then a buyer with prior $\lambda$ reacts to her signal according to the learning rule (buy if and only if $s=H$), so that actions remain informative. If $p$ is chosen too high or too low relative to $\lambda$, the buyer falls into a cascade region and actions cease to convey information.

A simple computation shows that $\mathcal{P}(\lambda)$, when nonempty, is an interval. Moreover, for beliefs bounded away from [math] and $1$, this interval is nonempty for a range of signal precisions $q$. We state this formally.

Lemma 2**.**

Fix $\lambda\in(0,1)$ and $q\in(1/2,1)$. The set $\mathcal{P}(\lambda)$ is either empty or a nondegenerate interval of prices. For beliefs $\lambda$ in any compact subinterval of $(0,1)$, there exists a nonempty open set of $(p,v,q)$ with $0<p<v$ and $q\in(1/2,1)$ such that $\mathcal{P}(\lambda)$ is nonempty.

Proof sketch.

For fixed $q$ and $v$, the inequalities $\underline{\lambda}(p)<\lambda<\bar{\lambda}(p)$ define an open interval of permissible odds ratios $K(p)$, and hence an open interval of prices $p\in(0,v)$. Continuity of $K(p)$ in $p$ and of the thresholds in $K$ yields the desired properties. ∎

Lemma 2 implies that, for beliefs that are not too extreme, there is a continuum of prices that keep buyer actions responsive to signals. When the seller can choose $p_{t}$ in addition to $\theta_{t}$, he faces a dynamic trade-off: higher prices yield higher current margins conditional on purchase, but may push buyers into a cascade and thereby eliminate future informational gains from quality.

In line with our earlier Markov structure, we restrict attention to seller strategies that map beliefs into price-quality pairs, $\lambda\mapsto(p(\lambda),\sigma^{S}(\lambda))$. The buyer’s myopic best response is then determined by $\lambda$, $p(\lambda)$, and her signal, and the induced belief dynamics can be summarized by a belief-update map $\Phi$ as before.

The following proposition summarizes the qualitative effect of flexible pricing on equilibrium behavior.

Remark**.**

When the seller can adjust the posted price, the implementability set $\mathcal{P}(\lambda)$ describes prices that keep buyers in the learning region at belief $\lambda$. In parametric examples with moderate costs and high patience, one can construct Markov equilibria in which, at each belief $\lambda$ in a suitable interval, the seller chooses the highest price in $\mathcal{P}(\lambda)$ and invests in quality on a nontrivial subset of that interval. Along such paths, beliefs avoid the cascade regions for a long time and actions remain informative.

Remark Remark highlights a design insight: when the seller can adjust the posted price, he can use price reductions at low beliefs and moderate price increases at high beliefs to keep buyers in the learning region and avoid cascades. In contrast to the fixed-price benchmark, sufficiently patient sellers then have the option of sustaining long-run investment in quality and informative actions, rather than being pulled into pessimistic or optimistic cascades in which investment incentives disappear.

8.2 Public outcome disclosure

We next modify the information structure by making consumption outcomes publicly observable. We return to the fixed-price environment of Sections 2–5 and assume that, in addition to the purchase decision $a_{t}$, the realization of the buyer’s payoff is observed by future buyers and the seller. Equivalently, one can think of each buyer as posting a verified review after consuming the product.

Formally, suppose that in period $t$ the tuple $(a_{t},y_{t})$ is publicly observed, where

[TABLE]

A review $y_{t}\in\{v-p,-p\}$ fully reveals whether the buyer who purchased in period $t$ experienced high or low quality; a review of [math] corresponds to a non-purchase. Public beliefs are then updated on the basis of the history of $(a_{s},y_{s})_{s\leq t-1}$.

In this environment, a purchase now generates a more informative public signal about quality than in the benchmark: in the benchmark, observing $a_{t}=1$ reveals only that the buyer’s signal was $H$ and that the seller’s quality choice may have influenced the probability of that signal; under outcome disclosure, observing $a_{t}=1$ and $y_{t}=v-p$ directly reveals that quality was high. This strengthens the learning externality associated with quality investment.

We can again summarize public histories by a one-dimensional belief state $\lambda_{t}$, now interpreted as the posterior probability that the seller will deliver high quality given all past $(a_{s},y_{s})$ and equilibrium strategies. The buyer’s myopic decision problem remains unchanged: for a given $\lambda_{t}$ and signal $s_{t}$, she trades off the expected gain from buying against the certainty of not buying. The difference lies in the belief-update map $\Phi^{\mathrm{out}}$ induced by outcome disclosure.

Let $\lambda^{\prime}=\Phi^{\mathrm{out}}(\lambda\mid a,y)$ denote the next-period belief after observing $(a_{t},y_{t})=(a,y)$ at belief $\lambda$. When $a=1$ and $y\in\{v-p,-p\}$, the signal about quality is perfectly informative; when $a=0$ and $y=0$, the only information comes from the fact that the buyer preferred to stay out given her signal. As a result, the successor beliefs $\lambda^{+}_{\mathrm{out}}$ and $\lambda^{-}_{\mathrm{out}}$ satisfy

[TABLE]

whenever $\lambda\in\mathcal{L}$, where $(\lambda^{+},\lambda^{-})$ are the benchmark successor beliefs. Outcome disclosure thus expands the range of beliefs over which actions significantly move posterior beliefs.

Define the learning region and investment region under outcome disclosure as

[TABLE]

where $\sigma^{S,\mathrm{out}}$ denotes an equilibrium quality policy under the outcome-disclosure information structure. Because the buyer’s static cutoff rule is unchanged, the thresholds $(\underline{\lambda},\bar{\lambda})$ are the same as in the benchmark; what changes is the dynamic payoff from quality.

Proposition 6**.**

Fix primitives $(p,v,c,q,\delta)$ and an initial belief $\lambda_{1}\in(0,1)$. Let $(\sigma^{S},\sigma^{B},\Phi)$ be a Markov equilibrium in the benchmark environment and $(\sigma^{S,\mathrm{out}},\sigma^{B,\mathrm{out}},\Phi^{\mathrm{out}})$ a Markov equilibrium under outcome disclosure. Then, under the selection described in Section 2, we have:

1. (a)

$V^{\mathrm{out}}(\lambda)\geq V(\lambda)$* for all $\lambda\in[0,1]$, where $V^{\mathrm{out}}$ and $V$ are the seller’s value functions under disclosure and in the benchmark, respectively;* 2. (b)

$\mathcal{I}^{\mathrm{out}}$* weakly contains $\mathcal{I}$, in the sense that whenever $\lambda$ lies in the interior of $\mathcal{I}$ under the benchmark, it also lies in $\mathcal{I}^{\mathrm{out}}$ under outcome disclosure;* 3. (c)

welfare is weakly higher under outcome disclosure for any initial belief, with strict inequality whenever the benchmark equilibrium induces positive-probability histories with low-quality purchases in the learning region.

Proof sketch.

Part (a): Outcome disclosure increases the informativeness of each purchase, so that for any given quality policy the induced belief process under $\Phi^{\mathrm{out}}$ is a mean-preserving spread of the benchmark process under $\Phi$. Because the seller is risk-neutral and cares about reaching the optimistic cascade while avoiding the pessimistic cascade, more informative signals increase his continuation value at each belief. Dynamic-programming arguments yield $V^{\mathrm{out}}(\lambda)\geq V(\lambda)$.

Part (b): The quality decision condition analogous to (3) holds under outcome disclosure with $\Delta^{\mathrm{out}}(\lambda)=V^{\mathrm{out}}(\lambda^{+}_{\mathrm{out}})-V^{\mathrm{out}}(\lambda^{-}_{\mathrm{out}})$ in place of $\Delta(\lambda)$. Because successor beliefs are farther apart and value functions are higher, $\Delta^{\mathrm{out}}(\lambda)\geq\Delta(\lambda)$ for all $\lambda\in\mathcal{L}$. It then follows that any belief at which high quality is optimal in the benchmark remains a point of positive investment under disclosure.

Part (c): Welfare aggregates buyer and seller payoffs. Outcome disclosure increases the frequency with which high-quality trades are followed by future high-quality trades, while reducing the time spent in regions where low-quality purchases are common. Combining parts (a) and (b) with Assumption 2 yields the welfare ranking. ∎

Proposition 6 formalizes a natural intuition: making consumption outcomes public enhances the reputational return to quality and therefore expands the set of beliefs at which the seller finds it optimal to invest. Compared to the action-only benchmark, public reviews or ratings accelerate learning and reduce the scope for reputational milking, raising both the seller’s expected payoff and social welfare.

Proposition 6 shows that modest institutional changes can substantially alter the equilibrium dynamics of quality and reputation. Flexible prices allow patient sellers to steer beliefs away from cascades, and credible outcome disclosure restores strong investment incentives in a wide range of states. In the next section we connect these design insights to a concrete class of fixed-price experience-good markets in which only purchase decisions are observable.

9 Application and Discussion

9.1 A fixed-price experience-good market

We briefly illustrate how the model can be interpreted in a concrete fixed-price market for an experience good in which only purchase decisions are publicly visible.

Consider a restaurant located on a busy street. Each evening a sequence of potential customers passes by; in period $t$ a single representative customer arrives, observes whether the restaurant has attracted patrons on previous evenings, and decides whether to dine there. The restaurant posts a fixed price $p$ for its standard menu and cannot tailor this price to individual customers or to current demand. Each evening the restaurant chooses whether to incur a cost $c$ to deliver high quality—for example, by using fresh ingredients, hiring an extra cook, or spending more time on preparation—or to cut corners and deliver low quality.

The period-$t$ customer observes the restaurant’s posted price and a public history of past occupancy (how many customers appeared to go in on previous evenings), which we summarize by the public belief $\lambda_{t}$. She also receives a private signal $s_{t}$ about the restaurant’s current quality: this may reflect, for example, the appearance and smell of the dishes served that evening, any informal comments overheard from earlier diners, or indirect information about the chef’s reputation. The signal is noisy but informative, as in Section 2. The customer then decides whether to dine at the restaurant. If she does, she privately experiences either high or low quality and pays the price $p$; if she does not, she receives her outside option. Crucially, her realized satisfaction is not publicly recorded: there is no reliable review platform, or reviews reach future customers only with substantial delay. Future customers thus observe only her action (whether she dined there), not her outcome.

This environment maps directly into our baseline model. The restaurant is the long-lived seller; customers are short-lived buyers. The restaurant’s choice of high or low quality corresponds to $\theta_{t}\in\{1,0\}$, and the fixed price $p$ is exogenous. Each customer’s private signal $s_{t}$ is conditionally i.i.d., with precision $q$, and the public belief $\lambda_{t}$ summarizes the history of past purchase decisions and the equilibrium quality strategy. The restaurant’s flow payoff is $p$ whenever a customer chooses to dine and [math] otherwise, net of any quality cost; the customer earns $v-p$ when she dines and finds high quality, $-p$ when she dines and finds low quality, and [math] if she stays out.

Our analysis then characterizes when and how the restaurant finds it optimal to invest in high quality in order to influence future demand. At very low reputations, few customers will dine even after favorable signals, so incurring the cost $c$ today has only a limited effect on future beliefs and on the long-run probability of sustaining high occupancy. At very high reputations, by contrast, customers will dine almost regardless of their signals; the restaurant can “skim” on reputation and save on quality costs without jeopardizing demand in the short run. In both cases, the dynamic marginal value of quality is small, and the model predicts that the restaurant will not invest in quality.

At intermediate reputations, however, the dynamic return to quality is substantial. By choosing high quality in these states, the restaurant increases the probability that the current customer chooses to dine and that her action is interpreted by future customers as a favorable signal. Beliefs are then pushed toward the optimistic cascade region, where the restaurant enjoys long runs of high demand. Proposition 2 predicts that these incentives generate an inverse-U investment pattern: quality is lowest in restaurants that are either widely regarded as poor or widely regarded as excellent and highest in restaurants with intermediate reputations.

The dynamic patterns described in Section 6.1 have a natural interpretation in this setting. In an early-resolution regime, a restaurant with a moderately good initial reputation quickly moves into the optimistic cascade and then ceases to invest in quality; customers continue to arrive, but quality is low. In a double-hump regime, quality investment occurs in two waves, separated by a period in which the restaurant cuts corners even though customers still react to their private impressions. From a policy perspective, the design extensions in Section 8 suggest how institutional changes could modify this dynamic. Allowing price flexibility would let the restaurant adjust prices to keep customers responsive to their signals and avoid cascades, while credible dissemination of reviews (public outcome disclosure) would increase the reputational return to quality, expanding the set of beliefs at which high quality is sustained.

9.2 Relation to existing literature

Our analysis connects three strands of work: (i) social learning and informational cascades, (ii) dynamic reputation for quality in long-run relationships, and (iii) recent models in which a long-lived agent shapes the information environment faced by short-lived agents.

On the social-learning side, we build on the canonical cascade models of Banerjee, (1992) and Bikhchandani et al., (1992), in which agents move sequentially, observe predecessors’ actions but not their signals, and may come to ignore their own information once enough past actions point in the same direction. In these models, the state is exogenous and fixed, and the main question is whether behavior eventually aggregates dispersed information or becomes trapped in an informational cascade. Bikhchandani et al., (1998) provide a broad overview of this literature, while Smith and Sørensen, (2000) show how richer signal structures generate both efficient learning and “pathological” herds. Subsequent work has extended the basic cascade logic to richer signals, networks, and observation structures; see, for example, the monograph Chamley, (2004) and the literature on Bayesian learning in networks.666For example, Acemoglu et al., (2011) study learning in social networks with exogenous states and rich communication patterns. Relative to this work, we retain the action-based information structure of Bikhchandani et al., (1992) but endogenize the state: the “quality” that buyers care about is chosen by a long-lived seller who internalizes how current quality affects the informativeness of future actions. This feedback from reputation investment to the scope for social learning is responsible for the inverse-U shape of investment and for the possibility that more precise private information shrinks the region in which actions are responsive.

On the reputation side, our starting point is the classic IO literature in which firms build reputations for unobservable quality. Early contributions such as Klein and Leffler, (1981) and Shapiro, (1983) show how premium prices can sustain high quality when quality is not directly observed but repeat business is important. The game-theoretic reputation models of Kreps and Wilson, (1982) and Milgrom and Roberts, (1982), and the survey in Mailath and Samuelson, (2006), embed a long-lived player with a possible commitment type interacting with a sequence of short-lived opponents. In this tradition, quality (or effort) is typically learned from publicly observed outcomes or payoffs, and prices can often be used as additional signals. In contrast, we assume a fixed price and action-only observability: future buyers see who bought but not what those buyers experienced.777A large branch of the reputation literature allows prices to serve as an additional signal of quality or type; see, for example, Shapiro, (1983) and the survey in Bar-Isaac and Tadelis, (2008). We deliberately abstract from price choice in the benchmark to isolate the pure effect of action-based social learning on investment incentives. Section 8 then reintroduces price flexibility as a design extension. Quality therefore affects beliefs only indirectly, via the informativeness of subsequent actions rather than via directly observed outcomes or price choices.

More recent work develops fully dynamic models of quality and reputation in which investment, stochastic quality shocks, and public signals interact. Board and Meyer-ter Vehn, (2013) study a continuous-time model in which firm effort affects a hidden quality process that becomes publicly observable at random times; they characterize how reputation and investment respond to public news. Marinovic et al., (2018) analyze dynamic certification and ratings for quality when a principal chooses when to certify outcomes. Dilmé, (2019) considers reputation building through costly adjustments of a persistent state variable, generating nontrivial dynamics for effort and beliefs. These papers share with us the view that quality is endogenous and that reputation is the market’s belief about that quality, but they work with outcome-based public signals and typically allow the firm to influence either the stochastic process of quality or the arrival of public information directly. Our contribution is to develop a tractable dynamic model in which the only public signal is the sequence of purchase decisions, and quality investment operates by keeping those actions informative rather than by shifting a publicly observed outcome process.

A closely related and rapidly growing literature studies how long-lived firms manage their reputations by manipulating the information that customers observe. Hauser, (2023) analyzes a firm that both invests in quality and censors negative news; when censorship is cheap, the threat of bad news disappears and investment incentives can collapse. In Hauser, (2024), a firm chooses both quality and costly promotion, which affects the arrival rate of public news about quality; promotion and investment are complements at low reputation but can crowd out quality investment at high reputation. These models highlight that the ability to design or filter information can create nonmonotonic incentives to invest in quality.888Hauser, (2023, 2024) and Dilmé, (2019) obtain inverse-U or nonmonotonic effort incentives because the firm can censor bad news, promote good news, or adjust a persistent state. In our setting the seller cannot directly manipulate the information technology: all past purchase decisions are observed and no outcomes are ever disclosed. Nonmonotonic investment arises instead because quality affects the informativeness of actions and hence the speed at which beliefs drift into cascade regions. Our analysis is complementary: the seller has no direct control over the information technology—buyers always observe all past purchase decisions and never see outcomes—so the only way to keep information flowing is to invest in quality so that actions remain signal-responsive. We show that this yields an inverse-U investment pattern and reputation “plateaus” even in the absence of censorship or promotion technologies.

Our paper is also connected to models in which the object of social learning is a strategic player’s endogenous actions rather than an exogenous state. Pei, (2023) studies reputation building under observational learning when a patient seller chooses quality and each buyer observes a limited window of past actions together with private signals. He characterizes how the observation network affects the speed of learning and the seller’s payoff in the patient limit. We likewise combine a long-lived seller with action-based social learning, but we impose full observability of the action history and a fixed price, and we focus on how cascade regions, the learning region, and the shape of the optimal quality policy interact. Our selection of a Markov equilibrium via a type-based foundation and a vanishing public tremble is tailored to the BHW-style cascade environment and ensures that the equilibrium value function and policy are well defined even when actions cease to respond to signals.

The inverse-U pattern of investment that we obtain is closely related to the “reputation trap” phenomenon studied by Levine, (2021), and to the nonmonotonic investment incentives that arise in recent dynamic reputation models with endogenous information. In Levine, (2021), a long-run player can be stuck in a bad reputation state where short-run players decline to interact, so that the marginal value of investment is very low; at very good reputations, incentives to invest also vanish, while intermediate reputations generate the strongest incentives. Hauser, (2023, 2024) and Dilmé, (2019) likewise obtain nonmonotonic effort incentives in dynamic settings where the state or the information flow is endogenous. We show that a similar inverse-U shape arises in an action-based social-learning environment with fixed prices and cascade regions. Moreover, because quality determines the informativeness of purchase decisions, we can characterize how changes in signal precision or in the cost of quality shift the size of the interior learning region and the location of investment peaks, and we show that more precise private information can reduce equilibrium investment by expanding cascade regions.999The idea that richer information can weaken effort incentives also appears in other dynamic settings, for instance when more precise performance evaluation makes it easier to spot and replace underperforming agents, thereby reducing their ex ante incentives to invest. Our mechanism is different: higher signal precision accelerates movement into cascades at a fixed price, shrinking the range of beliefs at which actions are both informative and sensitive to the seller’s quality choice.

Finally, our reduced-form Markov formulation can be interpreted through the lens of misspecified learning. If buyers treat the seller’s quality as an exogenous state and neglect the strategic link between current quality and the informativeness of others’ actions, their reasoning resembles the cursed-equilibrium logic of Eyster and Rabin, (2005).101010We do not attempt a full analysis of misspecified or cursed beliefs here. Our formal results are derived under fully Bayesian reasoning with a type-based foundation. The misspecification interpretation is best viewed as a reduced-form lens: it suggests that similar inverse-U investment patterns and cascade dynamics could emerge when buyers use simplified mental models of quality, even in environments that do not literally fit our type structure. The type-based foundation we adopt ensures that our Markov equilibrium can be rationalized as a weak perfect Bayesian equilibrium with small probability of a commitment type, but the misspecification interpretation highlights that similar inverse-U investment patterns and cascade dynamics may arise when buyers employ simplified mental models of quality. More broadly, our results complement the broader literature on seller reputation and feedback mechanisms surveyed in Bar-Isaac and Tadelis, (2008), by focusing on an extreme benchmark with fixed prices and action-only observability and asking how a forward-looking seller uses quality investment to keep social learning alive.

10 Conclusion

We have studied a dynamic reputation problem in an action-based social-learning environment with fixed prices. A long-lived seller repeatedly chooses costly quality before interacting with short-lived buyers who observe past purchase decisions but not past consumption outcomes. Buyers receive private noisy signals about current quality and decide whether to buy. Public beliefs about the seller’s quality therefore evolve solely through the informativeness of actions.

From an applied perspective, the model is meant to capture fixed-price markets for experience goods in which only purchase decisions are observed—such as restaurants or subscription services without reliable public reviews. In such environments, our analysis suggests that reputation-building may be most intense for intermediate-reputation providers, that high reputation can be associated with both high demand and low quality, and that more precise private information need not improve equilibrium quality. Policies that make outcomes more transparent or that allow sellers to adjust prices in order to keep actions informative can therefore have sizable effects on both investment incentives and long-run welfare.

Several directions for future work remain. One is to relax the binary-quality assumption and study richer quality dynamics, including continuous effort or persistent quality shocks. Another is to combine action-based social learning with richer forms of misspecification or bounded rationality on the buyers’ side, building on the cursed equilibrium concept of Eyster and Rabin, (2005). A third is to embed the present framework in more structured industrial-organization environments with competition between multiple sellers, heterogeneous buyers, or explicit regulatory interventions. We hope that the tractable Markov structure developed here can serve as a building block for these extensions and for further work at the intersection of social learning and dynamic reputation.

Acknowledgments

This paper previously circulated under the title “Reputation and Social Learning.” We thank Kirill Bukin, Harry Di Pei, Miaomiao Dong, Olivier Gossner, Christian Hellwig, Natalia Kovaleva, Yukio Koriyama, and participants at the 30th Stony Brook International Conference on Game Theory, the 2019 Econometric Society European Meeting (ESEM) in Manchester, and internal seminars at ICEF and NES for helpful comments. All remaining errors are our own.

Appendix A Type-based foundation and existence of Markov equilibrium

This appendix provides a type-based foundation for the reduced-form model in Section 2 and outlines the existence proof for a Markov equilibrium under our selection. The aim is to show that the Markov equilibrium defined in the main text can be viewed as the reduced-form representation of a weak perfect Bayesian equilibrium (WPBE) of a standard type-uncertainty game.

A.1 Underlying type-uncertainty game

At date [math], Nature draws the seller’s type $\tau\in\{c,s\}$. With probability $\alpha\in(0,1)$, the seller is a commitment type ($\tau=c$) who always chooses high quality, $\theta_{t}=1$ for all $t$. With probability $1-\alpha$, the seller is a strategic type ($\tau=s$) who maximizes a discounted sum of period payoffs by choosing quality in each period.

Time is discrete, $t=1,2,\dots$. At each date $t$:

1. (i)

The public history $h_{t}=(a_{1},\dots,a_{t-1})$ is observed by the seller and by buyer $t$. 2. (ii)

The seller (of type $\tau$) chooses a quality level $\theta_{t}\in\{0,1\}$.

• •

If $\tau=c$, then $\theta_{t}=1$ for all $t$.

• •

If $\tau=s$, then $\theta_{t}$ is chosen according to a strategy $\sigma^{S}$. 3. (iii)

Buyer $t$ observes $h_{t}$ and receives a private signal $s_{t}\in\{H,L\}$ drawn with precision $q$ conditional on $\theta_{t}$ as in Section 2. 4. (iv)

Buyer $t$ chooses an action $a_{t}\in\{0,1\}$. 5. (v)

The action $a_{t}$ becomes part of the public history; payoffs are realized.

A (pure) strategy for the strategic seller is a sequence of functions

[TABLE]

where $\mathcal{H}_{t}$ is the set of feasible public histories in period $t$. A (pure) strategy for buyer $t$ is a function

[TABLE]

Mixed strategies are defined in the usual way. Let $\mu_{t}$ denote the public posterior probability that the seller is the commitment type at the beginning of period $t$, given $h_{t}$ and the strategy profile. A weak perfect Bayesian equilibrium (WPBE) of this game consists of a strategy profile

[TABLE]

and a belief system $\{\mu_{t}\}_{t\geq 1}$ such that:

1. (a)

strategies are sequentially rational given beliefs; 2. (b)

beliefs are updated by Bayes’ rule whenever possible.

We restrict attention to WPBE in which both the strategic seller and buyers use Markov strategies, in the sense that there exists a one-dimensional state variable summarizing the public history, and strategies depend on histories only through that state. We now formalize this reduction.

A.2 Reduction to the Markov state

Fix a WPBE with Markov strategies. Let $\mu_{t}$ be the public posterior that $\tau=c$ at the beginning of period $t$. Given the commitment type’s behavior (always high quality) and the strategic type’s Markov strategy $\sigma^{S}:[0,1]\to[0,1]$, the market’s belief that the seller delivers high quality in period $t$ can be written as

[TABLE]

This equation implicitly defines $\lambda_{t}$ as a function of the underlying posterior $\mu_{t}$ and the strategic type’s policy $\sigma^{S}$. For a given WPBE, we can therefore treat $\lambda_{t}$ as the canonical one-dimensional Markov state summarizing the public history $h_{t}$.

Conversely, given a Markov strategy profile $(\sigma^{S},\sigma^{B})$ on the state space $[0,1]$, we can construct a WPBE of the underlying type game with the following properties:

• •

The public belief $\mu_{t}$ about the seller’s type is updated by Bayes’ rule from $(a_{s})_{s\leq t-1}$ and the induced likelihood of actions under each type.

• •

The induced belief $\lambda_{t}$ about current quality solves the fixed-point equation

[TABLE]

and is used as the state variable in the reduced-form model.

• •

Buyers in each period condition their actions on $(\lambda_{t},s_{t})$ and use the Markov strategy $\sigma^{B}(\cdot,\cdot)$.

Under our tie-breaking and vanishing-tremble selection (Section 2), there is a unique way to extend any Markov strategy profile $(\sigma^{S},\sigma^{B})$ and induced belief-update map $\Phi$ in the reduced-form game to a WPBE of the underlying type game. We will therefore identify Markov equilibria of the reduced-form model with their type-based counterparts when convenient.

A.3 Existence of Markov equilibrium

We now outline the steps leading to Theorem 1 in the main text.

Step 1: Given $\sigma^{S}$, define buyer behavior and belief updates.

Fix a Markov quality policy $\sigma^{S}:[0,1]\to[0,1]$ for the strategic type. Given the type-based foundation, the buyer in period $t$ observes the current belief $\lambda$ and her signal $s\in\{H,L\}$ and faces a static decision problem identical to that in Section 3. Her myopic best response is therefore the cutoff rule in Lemma 1, with tie-breaking at indifference in favor of buying.

The Markov strategy $\sigma^{S}$ and the buyer’s static best response induce, via Bayes’ rule and the vanishing-tremble selection, a belief-update map

[TABLE]

and hence successor beliefs $\lambda^{+}(\lambda)$ and $\lambda^{-}(\lambda)$ as in Section 4.1. Because the tremble smooths out knife-edge cases at cascade boundaries, the map $\Phi$ is measurable and piecewise continuous in $\lambda$.

Step 2: Seller’s dynamic program.

Given $(\sigma^{S},\Phi)$ and the buyer’s best reply, the strategic seller’s problem is a standard infinite-horizon Markov decision problem with state space $[0,1]$, action space $\{0,1\}$, per-period payoff $\pi(\lambda,\theta)$, and transition kernel determined by $(\lambda^{+}(\lambda),\lambda^{-}(\lambda))$ and $\gamma(\lambda,\theta)$. The Bellman operator $T$ acting on bounded continuous functions $f:[0,1]\to\mathbb{R}$ is

[TABLE]

Standard arguments imply:

Lemma 3**.**

For any belief-update map $\Phi$ induced by a Markov policy $\sigma^{S}$ and buyer best responses, the Bellman operator $T$ is a contraction on the space of bounded continuous functions on $[0,1]$ with modulus $\delta$. There exists a unique bounded continuous value function $V$ satisfying $V=TV$.

The proof is standard and omitted.

Given $V$, the seller’s optimal Markov quality policy $\hat{\sigma}^{S}$ can be recovered pointwise from the argmax in the Bellman equation. Under our tie-breaking convention for the seller (e.g. choosing $\theta=0$ at indifference), the optimal policy is uniquely defined.

Step 3: Fixed point in seller policies.

Steps 1 and 2 define a self-map on the space of Markov quality policies. Given a candidate $\sigma^{S}$, we obtain:

• •

a buyer best response $\sigma^{B}(\cdot,\cdot)$ and belief-update map $\Phi$;

• •

a value function $V$ and an optimal Markov policy $\hat{\sigma}^{S}$ from the Bellman equation.

We are interested in fixed points $\sigma^{S}=\hat{\sigma}^{S}$. Under mild regularity conditions on $(p,v,q,c,\delta)$, the correspondence from candidate policies to optimal policies has nonempty, convex values and a closed graph in an appropriate topology (e.g. the sup norm). By Kakutani’s fixed-point theorem, there exists at least one fixed point $\sigma^{S,\ast}$, with corresponding buyer strategy $\sigma^{B,\ast}$ and belief-update map $\Phi^{\ast}$.

Lemma 4**.**

Under the tie-breaking and vanishing-tremble selection, there exists a Markov strategy profile $(\sigma^{S,\ast},\sigma^{B,\ast})$ and belief-update map $\Phi^{\ast}$ such that $\sigma^{S,\ast}$ is optimal given $(\sigma^{B,\ast},\Phi^{\ast})$ and $\sigma^{B,\ast}$ is a best response given $(\sigma^{S,\ast},\Phi^{\ast})$.

Step 4: From reduced-form fixed points to WPBE.

Given a fixed point $(\sigma^{S,\ast},\sigma^{B,\ast},\Phi^{\ast})$, we can embed it in the underlying type-uncertainty game by letting the commitment type always choose $\theta=1$, the strategic type follow $\sigma^{S,\ast}$, and each buyer follow $\sigma^{B,\ast}$. Beliefs over types are updated by Bayes’ rule using the induced likelihoods of actions under each type; beliefs over quality are summarized by the Markov state $\lambda_{t}$, which evolves according to $\Phi^{\ast}$. Sequential rationality of the strategic type and of buyers follows from Lemma 3 and from the static optimality of the buyer rule. By construction, the resulting profile and beliefs form a WPBE of the type game.

This completes the proof outline of Theorem 1 in the main text.

A.4 On Assumption 1

Assumption 1 requires that the seller’s value function be concave in log-odds on the learning region. Here we provide a brief rationale.

Define the log-odds variable $\ell=\log(\lambda/(1-\lambda))$ and write

[TABLE]

In the learning region $\mathcal{L}$, belief updates correspond to adding or subtracting a fixed increment to $\ell$: when actions coincide with private signals, observing a purchase (high signal) moves log-odds from $\ell$ to $\ell+\log z$, whereas observing a non-purchase (low signal) moves log-odds to $\ell-\log z$. The Bellman equation for $W$ therefore takes the form

[TABLE]

with suitable payoffs $\tilde{\pi}$ and transition probabilities $\gamma$. When the seller’s per-period payoff is affine in $\lambda$ and the transition probabilities are independent of $\ell$ within $\mathcal{L}$, the Bellman operator maps concave functions $W$ into concave functions. Under parameter restrictions ensuring that the optimal policy does not introduce nonconcavities (e.g. uniqueness of the argmax in the interior), iterating the Bellman operator from any concave initial guess preserves concavity. The unique fixed point $W$ is then concave on the subset of log-odds corresponding to $\mathcal{L}$.

This argument is standard for dynamic programs with linear payoffs and symmetric belief shocks. We therefore treat Assumption 1 as a mild regularity condition that holds in a large subset of the parameter space relevant for our comparative-statics analysis.

Appendix B Parametric example: dynamic patterns and signal precision

This appendix provides a simple parametric example that underpins Example Example (dynamic patterns) and Example Example (signal precision and investment). We fix primitives and solve the seller’s dynamic program numerically on a grid of beliefs. The resulting Markov equilibria display both early-resolution and double-hump patterns for different cost levels, and for a given cost we exhibit two signal precisions $q_{1}<q_{2}$ such that the static learning region expands while the equilibrium investment region shrinks.

B.1 Parametric environment and numerical scheme

We consider the following parametrization:

[TABLE]

The buyer’s indifference cutoff is

[TABLE]

We will work with two signal precisions $q_{1}$ and $q_{2}$,

[TABLE]

corresponding to likelihood ratios

[TABLE]

For each $q\in\{q_{1},q_{2}\}$, the static cascade thresholds are

[TABLE]

Numerically,

[TABLE]

Thus the static learning region $\mathcal{L}(q)=(\underline{\lambda}(q),\bar{\lambda}(q))$ is strictly larger at $q_{2}$ than at $q_{1}$.

We discretize the belief space $[0,1]$ on a grid

[TABLE]

with $N$ large (e.g. $N=1001$). For each grid point $\lambda\in\mathcal{G}$ we treat $\lambda$ as the current public belief and use the static buyer rule in Lemma 1 to compute the purchase probabilities $\gamma(\lambda,1)$ and $\gamma(\lambda,0)$. In the learning region, these are $\gamma(\lambda,1)=q$ and $\gamma(\lambda,0)=1-q$; in the pessimistic cascade region, both are zero; in the optimistic cascade region, both are one.

To construct successor beliefs we work in log-odds. For any interior belief $\lambda\in(0,1)$, let

[TABLE]

Within the learning region, if actions coincide with private signals, then observing a purchase (high signal) moves log-odds from $\ell$ to $\ell+\log z$, while observing a non-purchase (low signal) moves log-odds to $\ell-\log z$. We discretize this by setting

[TABLE]

and mapping back to probabilities via

[TABLE]

We then project $(\tilde{\lambda}^{+},\tilde{\lambda}^{-})$ onto the grid $\mathcal{G}$ by interpolation. In the cascade regions we set $\lambda^{+}\approx\lambda^{-}\approx\lambda$, reflecting that actions do not respond to signals; in practice, we use small perturbations (or the nearest grid point) consistent with the vanishing-tremble interpretation.

Given this transition structure, the Bellman equation on the grid takes the form

[TABLE]

where $\lambda^{+},\lambda^{-}$ are the interpolated successor beliefs for the current $q$. We solve this dynamic program by value iteration, starting from $V_{0}(\lambda)=0$ for all $\lambda$ and iterating until convergence. The equilibrium quality policy $\sigma^{S}(\lambda)$ is then the argmax in the Bellman equation at each grid point (with a tie-breaking rule at indifference).

B.2 Early-resolution pattern

We first illustrate an early-resolution regime. Fix

[TABLE]

For these parameters the static gain from quality is small relative to the cost $c_{\mathrm{low}}$, but the dynamic return from pushing beliefs toward the optimistic cascade is substantial when $\delta=0.95$. The numerical solution to the Bellman equation yields a quality policy $\sigma^{S}(\lambda)$ with the following features:

• •

$\sigma^{S}(\lambda)=0$ for all $\lambda\leq\underline{\lambda}(q_{1})\approx 0.22$ and for all $\lambda\geq\lambda^{\ast}$ for some $\lambda^{\ast}\in(\bar{\lambda}(q_{1})-\varepsilon,\bar{\lambda}(q_{1}))$;

• •

$\sigma^{S}(\lambda)\in(0,1]$ for all $\lambda$ in a connected interval $(\lambda_{E}^{-},\lambda_{E}^{+})$ with

[TABLE]

and $[\lambda_{E}^{-},\lambda_{E}^{+}]$ relatively close to the optimistic cascade threshold;

• •

starting from any $\lambda_{1}\in(\lambda_{E}^{-},\lambda_{E}^{+})$, simulated belief paths $\{\lambda_{t}\}$ drift monotonically upward with high probability and hit the optimistic cascade region $\mathcal{C}^{+}$ in relatively few periods.

This configuration supports part (a) of Example Example.

B.3 Double-hump pattern

We now increase the cost of quality to an intermediate level,

[TABLE]

keeping all other parameters fixed. Re-solving the Bellman equation with $c=c_{\mathrm{mid}}$ yields a value function $V(\lambda)$ that is still concave in log-odds on the learning region but flatter overall, reflecting reduced dynamic gains from reputation-building.

For these parameters, the equilibrium quality policy $\sigma^{S}(\lambda)$ exhibits two separated investment regions:

• •

there exist belief levels

[TABLE]

such that $\sigma^{S}(\lambda)\in(0,1]$ on $(\lambda_{1},\lambda_{2})$ and $(\lambda_{3},\lambda_{4})$, and $\sigma^{S}(\lambda)=0$ on $[\underline{\lambda}(q_{1}),\lambda_{1}]\cup[\lambda_{2},\lambda_{3}]\cup[\lambda_{4},\bar{\lambda}(q_{1}))$;

• •

the interval $[\lambda_{2},\lambda_{3}]$ is nonempty and lies strictly inside the learning region: buyers are still responsive to their signals in this interval, but the seller optimally chooses low quality there;

• •

for starting beliefs $\lambda_{1}\in(\lambda_{2},\lambda_{3})$, simulated belief paths visit both investment regions $(\lambda_{1},\lambda_{2})$ and $(\lambda_{3},\lambda_{4})$ with positive probability before eventually being absorbed into $\mathcal{C}^{-}\cup\mathcal{C}^{+}$.

B.4 Signal precision and the size of the investment region

We now use the same primitives to illustrate the effect of signal precision on the investment region, as in Example Example. Fix the intermediate cost $c_{\mathrm{mid}}=0.035$ and consider precisions

[TABLE]

As noted in B.1, the static learning region $\mathcal{L}(q)$ strictly expands when we move from $q_{1}$ to $q_{2}$:

[TABLE]

We solve the dynamic program for both $q_{1}$ and $q_{2}$ and compute the corresponding investment regions

[TABLE]

Numerically, the following pattern emerges:

• •

at $q_{1}=0.70$, the investment region $\mathcal{I}(q_{1})$ is the union of two intervals as described in B.3, with a total length (Lebesgue measure) of about $0.28$;

• •

at $q_{2}=0.88$, the static learning region $\mathcal{L}(q_{2})$ is larger, but the numerically computed investment region $\mathcal{I}(q_{2})$ consists of a narrower set of beliefs near the optimistic cascade, with total length about $0.17$;

• •

in particular,

[TABLE]

even though $\mathcal{L}(q_{2})$ is strictly larger than $\mathcal{L}(q_{1})$.

The underlying intuition is that higher signal precision makes each action more informative but also speeds up belief movements. At $q_{2}$, beliefs reach the optimistic cascade quickly even under relatively low investment, reducing the marginal value of investing in quality at intermediate beliefs. In contrast, at $q_{1}$ beliefs move more slowly, so the seller finds it worthwhile to invest in quality over a wider range of beliefs in order to steer the process toward the optimistic cascade.

Appendix C Proofs of main results

This appendix collects proofs and proof sketches for the main results in the text.

C.1 Zero investment in cascades

Proof of Proposition 1.

Fix a Markov equilibrium $(\sigma^{S},\sigma^{B},\Phi)$ and a belief $\lambda\in\mathcal{C}^{-}\cup\mathcal{C}^{+}$. By definition of the pessimistic and optimistic cascades and Lemma 1, the buyer’s equilibrium action at $\lambda$ is independent of her private signal: she always buys in $\mathcal{C}^{+}$ and never buys in $\mathcal{C}^{-}$. Hence the equilibrium purchase probabilities satisfy

[TABLE]

Let $\lambda^{+}(\lambda)$ and $\lambda^{-}(\lambda)$ be the successor beliefs after observing a purchase or a non-purchase at $\lambda$ under the equilibrium $(\sigma^{S},\sigma^{B},\Phi)$. By definition of the belief-update map $\Phi$, these successor beliefs do not depend on the seller’s period-$t$ quality choice, only on the realized action. Thus, conditional on the current belief $\lambda$, both $\theta_{t}=0$ and $\theta_{t}=1$ induce the same distribution over $\{\lambda^{+}(\lambda),\lambda^{-}(\lambda)\}$.

The Bellman equation (1) implies that the equilibrium continuation value from choosing $\theta_{t}=\theta$ at $\lambda$ is

[TABLE]

Because $\gamma(\lambda,1)=\gamma(\lambda,0)$ in cascades, the continuation part is identical under $\theta=0$ and $\theta=1$. The only difference is the current cost $c>0$ incurred under high quality. Hence high quality strictly lowers the seller’s payoff in period $t$, and it is strictly dominated by low quality. Sequential rationality of the seller in equilibrium then implies $\sigma^{S}(\lambda)=0$ for all $\lambda\in\mathcal{C}^{-}\cup\mathcal{C}^{+}$. ∎

C.2 Inverse-U investment

Proof of Proposition 2.

Throughout the proof, work within the learning region $\mathcal{L}=(\underline{\lambda},\bar{\lambda})$. In this region, Lemma 1 implies that the buyer buys if and only if her signal is $H$, so

[TABLE]

for all $\lambda\in\mathcal{L}$.

Recall the dynamic marginal value of a purchase,

[TABLE]

and the quality-decision condition specialized to $\mathcal{L}$,

[TABLE]

Define

[TABLE]

Then high quality is optimal at $\lambda$ if and only if $F(\lambda)\geq c$.

Under Assumption 1, the value function $V$ is continuous on $[0,1]$ and concave in log-odds on $\mathcal{L}$. Within $\mathcal{L}$, belief transitions correspond to adding and subtracting a fixed increment in log-odds, and as $\lambda\downarrow\underline{\lambda}$ or $\lambda\uparrow\bar{\lambda}$ the successor beliefs $\lambda^{+}(\lambda)$ and $\lambda^{-}(\lambda)$ converge to the corresponding cascade thresholds.

By continuity of $V$ and $\Phi$ we therefore have

[TABLE]

Moreover, $\Delta(\lambda)\geq 0$ for $\lambda\in\mathcal{L}$: a purchase moves beliefs closer to the optimistic cascade than a non-purchase, and $V$ is nondecreasing in $\lambda$ by Theorem 1. Thus $\Delta$ is nonnegative, vanishes at the boundaries of $\mathcal{L}$, and is continuous on $\mathcal{L}$.

Assumption 1 states that the function

[TABLE]

is concave in the log-odds variable $\ell$ on the subset corresponding to $\lambda\in\mathcal{L}$. In the learning region, observing a purchase or a non-purchase shifts log-odds from $\ell$ to $\ell\pm\log z$, where $z>1$ is the likelihood ratio. Hence

[TABLE]

For any concave $W$ and any fixed increment $d>0$, the function $\ell\mapsto W(\ell+d)-W(\ell-d)$ is continuous, nonnegative (when $W$ is nondecreasing) and single-peaked: it increases up to some point and then decreases.111111This is a standard property of concave functions: the discrete derivative $W(\ell+d)-W(\ell)$ is nonincreasing in $\ell$. Thus $W(\ell+d)-W(\ell-d)$ is the difference of a nonincreasing function and its left-shifted version, which yields a single-peaked profile.

It follows that $\Delta(\lambda)$, viewed as a function of belief via $\ell(\lambda)$, is continuous and single-peaked on $\mathcal{L}$, vanishing at the boundaries and attaining a positive maximum at some interior belief $\lambda^{\ast}\in\mathcal{L}$.

The function $F(\lambda)=(2q-1)[p+\delta\Delta(\lambda)]$ is continuous and inherits the single-peakedness of $\Delta$. Let

[TABLE]

We have $\underline{F}<\overline{F}$ because $\Delta(\lambda^{\ast})>0$.

If $c\geq\overline{F}$, then $F(\lambda)<c$ for all $\lambda\in\mathcal{L}$ and high quality is never optimal: $\sigma^{S}(\lambda)=0$ on $\mathcal{L}$. If $c\leq\underline{F}$, then $F(\lambda)\geq c$ for all $\lambda\in\mathcal{L}$ and high quality is (weakly) optimal on the entire learning region: $\sigma^{S}(\lambda)\in[0,1]$ everywhere and can be chosen as $1$ almost everywhere.

For $c$ in the intermediate range $(\underline{F},\overline{F})$, the equation $F(\lambda)=c$ has two distinct solutions

[TABLE]

by continuity and single-peakedness of $F$. Moreover, $F(\lambda)>c$ for all $\lambda\in(\lambda_{1}(c),\lambda_{2}(c))$ and $F(\lambda)<c$ for all $\lambda\in\mathcal{L}\setminus[\lambda_{1}(c),\lambda_{2}(c)]$. Using (3), this implies that high quality is strictly optimal on $(\lambda_{1}(c),\lambda_{2}(c))$ and strictly suboptimal on $\mathcal{L}\setminus[\lambda_{1}(c),\lambda_{2}(c)]$. Allowing for mixed strategies at the boundary yields

[TABLE]

Combining this with Proposition 1, which gives $\sigma^{S}(\lambda)=0$ on $\mathcal{C}^{-}\cup\mathcal{C}^{+}$, yields the inverse-U pattern stated in Proposition 2. ∎

C.3 Effect of quality cost

Proof of Proposition 4.

Fix $(p,v,q,\delta)$ and consider two costs $c<c^{\prime}$. Let $\mathcal{I}(c)$ and $\mathcal{I}(c^{\prime})$ denote the corresponding equilibrium investment regions under a fixed equilibrium selection rule.

On the learning region $\mathcal{L}$, the quality decision for a given value function $V$ is governed by

[TABLE]

For a given $(\sigma^{B},\Phi)$, and hence for a given $V$ and $\Delta$, the set $\{\lambda:F(\lambda)\geq c\}$ is nonincreasing in $c$ in the sense of set inclusion: if $c^{\prime}>c$, then $\{\lambda:F(\lambda)\geq c^{\prime}\}\subseteq\{\lambda:F(\lambda)\geq c\}$. Equilibrium value functions depend on $c$, but for any two costs with a fixed equilibrium selection we can compare the induced sets pointwise: whenever high quality is optimal at some belief under cost $c^{\prime}$, it is also optimal at the same belief under cost $c$. This yields $\mathcal{I}(c^{\prime})\subseteq\mathcal{I}(c)$, establishing the monotonicity claim.

For the threshold statements, note that for sufficiently small $c$ the function $F(\lambda)$ exceeds $c$ on a large connected subset of $\mathcal{L}$ (by the argument in the proof of Proposition 2), yielding an early-resolution policy with a large investment region. Conversely, for $c$ sufficiently large, $F(\lambda)<c$ for all $\lambda\in\mathcal{L}$, so high quality is never optimal and $\mathcal{I}(c)=\emptyset$. ∎

C.4 Effect of patience

Proof of Proposition 5.

Fix $(p,v,q,c)$ and an equilibrium selection rule. For each $\delta\in(0,1)$, let $V_{\delta}$ denote the equilibrium value function and $\mathcal{I}(\delta)$ the corresponding investment region.

In the learning region, the quality decision is governed by

[TABLE]

where $\Delta_{\delta}(\lambda)=V_{\delta}(\lambda^{+}(\lambda))-V_{\delta}(\lambda^{-}(\lambda))$. The static part $(2q-1)p$ does not depend on $\delta$. Because $V_{\delta}$ is nondecreasing in $\lambda$ and $\lambda^{+}(\lambda)>\lambda^{-}(\lambda)$ in $\mathcal{L}$, we have $\Delta_{\delta}(\lambda)\geq 0$ for all $\lambda,\delta$. Increasing $\delta$ scales up the dynamic component of $F_{\delta}(\lambda)$.

Let $0<\delta<\delta^{\prime}<1$. Consider any belief $\lambda$ at which high quality is optimal under $\delta$; then

[TABLE]

Since $\Delta_{\delta^{\prime}}(\lambda)\geq 0$ and $\delta^{\prime}>\delta$, the corresponding term satisfies

[TABLE]

Moreover, under a fixed equilibrium selection and continuity of the Bellman operator in $\delta$, the difference $\Delta_{\delta^{\prime}}(\lambda)$ is weakly larger than $\Delta_{\delta}(\lambda)$ in the relevant region. Hence $F_{\delta^{\prime}}(\lambda)\geq F_{\delta}(\lambda)\geq c$, so any belief at which high quality is optimal under $\delta$ remains a belief at which high quality is (weakly) optimal under $\delta^{\prime}$. This implies $\mathcal{I}(\delta)\subseteq\mathcal{I}(\delta^{\prime})$, establishing the monotonicity claim. ∎

C.5 Outcome disclosure

Proof of Proposition 6.

We prove parts (a)–(c) in turn. Throughout, fix primitives $(p,v,c,q,\delta)$ and an initial belief $\lambda_{1}$.

Let $T$ and $T^{\mathrm{out}}$ denote the Bellman operators in the benchmark and outcome-disclosure environments, respectively. For any bounded continuous function $f:[0,1]\to\mathbb{R}$ we have

[TABLE]

and an analogous expression for $T^{\mathrm{out}}$ with successor beliefs $\lambda^{+}_{\mathrm{out}}(\lambda)$ and $\lambda^{-}_{\mathrm{out}}(\lambda)$. In the learning region, outcome disclosure strictly increases the informativeness of a purchase: observing a purchase reveals realized quality rather than just a high signal. This implies that, conditional on the current belief and quality policy, the distribution of successor beliefs under outcome disclosure is a mean-preserving spread of the distribution under the benchmark. Because the seller’s continuation value $V$ is nondecreasing and convex in the relevant direction (moving closer to the optimistic cascade and away from the pessimistic one raises continuation value), Jensen’s inequality yields

[TABLE]

for any $f$ that is bounded and nondecreasing in $\lambda$. Iterating both operators from the same initial function (e.g. $f_{0}\equiv 0$) and using the contraction property then implies that the fixed point of $T^{\mathrm{out}}$ dominates the fixed point of $T$ pointwise:

[TABLE]

Under outcome disclosure, the quality-decision condition in the learning region becomes

[TABLE]

where

[TABLE]

Compared to the benchmark, outcome disclosure separates successor beliefs more and increases continuation values: $\lambda^{+}_{\mathrm{out}}(\lambda)\geq\lambda^{+}(\lambda)$, $\lambda^{-}_{\mathrm{out}}(\lambda)\leq\lambda^{-}(\lambda)$ and $V^{\mathrm{out}}(\cdot)\geq V(\cdot)$. Combining these inequalities with the monotonicity of $V$ in $\lambda$ yields

[TABLE]

Thus, for any belief at which the benchmark inequality

[TABLE]

holds, the corresponding inequality under outcome disclosure also holds. This implies that any belief at which high quality is optimal in the benchmark lies in the interior of the investment region under outcome disclosure, establishing the containment claim.

Define welfare in each environment as

[TABLE]

We have shown that the seller’s value is weakly higher under outcome disclosure for any initial belief. Buyers are myopic and their decision problem is unchanged in terms of their own information: at any belief $\lambda_{t}$ and given their signal, they compare the same expected payoffs in both environments. Outcome disclosure affects buyer payoffs only through changes in the equilibrium quality policy and the resulting distribution of realized qualities along the path.

Furthermore, outcome disclosure weakly expands the region in which the seller invests in quality, leading to a higher frequency of high-quality trades in the learning region and shorter expected time in states where low-quality purchases are common. Assumption 2 ensures that faster and more informative belief changes raise the discounted sum of social surplus: in particular, postponing the resolution of uncertainty in a region with low quality cannot be offset by concentrating learning later. Combining these observations implies

[TABLE]

with strict inequality whenever the benchmark equilibrium induces an interior learning region with low-quality purchases that outcome disclosure replaces by more frequent high-quality trades. ∎