Pricing in the Presence of Strategic Consumers and Social Learning Under Contingent Pricing and Price Guarantee

Abstract

Purchasing new experience products or services often involves significant quality uncertainty for both consumers and firms. Social learning through online reviews helps reduce this uncertainty but exacerbates strategic waiting, as consumers delay purchases to gain more information. This paper examines the impact of social learning on firms’ pricing policies in the presence of strategic consumers under two widely adopted schemes: Contingent pricing and price guarantee. We find that while social learning always benefits the firm, it enables the price guarantee scheme to outperform contingent pricing in terms of profitability for highly patient consumers, which would not be possible without social learning. Notably, social learning drives a range of pricing patterns, even making price skimming optimal under price guarantee for highly patient consumers, as high initial prices combined with markdowns effectively alleviate strategic waiting, enhancing review outcomes and the firm’s profitability. Additionally, social learning always enables the firm to extract greater consumer surplus for impatient consumers under price guarantee. In contrast, social learning under contingent pricing consistently benefits consumers and can achieve win-win outcomes when consumers are moderately patient. Our extensions validate the robustness of these findings under different assumptions, including fully rational consumers and partially forward-looking firms. In particular, a partially forward-looking firm can achieve win-win outcomes with social learning under price guarantee, expanding its practical applicability. This study provides novel insights into the role of social learning in shaping pricing strategies, highlighting its implications for firm profitability and consumer welfare in markets influenced by review dynamics and strategic consumers.

Keywords

Dynamic Pricing Strategic Consumers Social Learning Contingent Pricing Price Guarantee

1. Introduction

Consumers often face quality uncertainty, particularly when purchasing new experience products. This uncertainty deters demand, as consumers may be reluctant to make purchases without sufficient information about the product’s attributes, such as reliability, durability, and ease of use. Firms selling new experience products also grapple with quality uncertainty, which complicates the pricing and marketing strategies. The rise of online shopping since the early 2000s has provided consumers and firms with unprecedented access to online reviews, offering valuable insights into the experience of other buyers. Today, over 90% of consumers consult online reviews before making a purchase (Statista, 2023). A behavior known as social learning, where purchase decisions are shaped by peer feedback through reviews or word of mouth. Social learning benefits not only consumers by reducing quality uncertainty but also helps firms adjust pricing and marketing strategies to better meet consumer expectations (Jing, 2011b; Papanastasiou and Savva, 2017; Yu et al., 2016).

However, social learning is a double-edged sword for both firms and consumers. While it helps reduce quality uncertainty, social learning also incentivizes strategic waiting, as forward-looking consumers delay purchase decisions to gather more information from early buyers’ reviews. This delay can significantly disrupt firms’ revenue generation (Gallego and Şahin, 2010; Swinney, 2011; Yu et al., 2015). To mitigate strategic waiting, firms employ pricing schemes such as preannounced pricing (also referred to as “price commitment”) and price guarantee (also referred to as “price matching”) (Aviv and Pazgal, 2008; Lai et al., 2010; Su and Zhang, 2008). For example, the price guarantee scheme allows consumers to claim refunds if prices drop within a specific timeframe and has been widely used by retailers such as Best Buy, Walmart, and Target, highlighting the prevalence of this strategy in sectors such as home improvement, fashion, and electronics (PMG, 2024). Alternatively, firms may adopt contingent pricing (also known as “dynamic pricing” or “responsive pricing”), where prices adapt in real-time based on demand and market conditions. For instance, in 2022, Amazon discontinued most of its price guarantee policies (except for TVs) and began using data-driven algorithms to analyze vast amounts of data every ten minutes, ensuring competitive prices for both firms and consumers (Ritterman, 2022).

The interplay among pricing schemes, social learning, and consumers’ strategic waiting behavior is intricate and warrants rigorous analyses. Specifically, contingent pricing may lead consumers to delay purchases due to price uncertainty, while price guarantee alleviates strategic waiting by offering refunds for price markdowns but incurs additional costs for the firm. Moreover, social learning further complicates the dynamic by prompting strategic consumers to postpone purchases, anticipating both price reductions and more reliable quality information. This delay reduces the number of reviews, thereby undermining the effectiveness of social learning. On the other hand, although low introductory prices can incentivize immediate purchase, which may enhance review quality, they also reduces the number of consumers who benefit from better-informed decisions later. These interdependencies highlight the multifaceted and challenging nature of the problem.

This paper explores these interactions, focusing on two central research questions: First, how should a firm price its product in the presence of social learning and consumer strategic waiting under each pricing scheme? Second, what are the implications of social learning on the firm’s profitability and consumer surplus?

To answer these questions, we consider a monopolistic firm selling a new experience product over a two-period selling season. Both the firm and consumers initially lack complete knowledge of the product’s quality, knowing only its distribution (Papanastasiou and Savva, 2017; Sun, 2012; Yu et al., 2016). In the first period, consumers decide whether to purchase immediately or wait, influenced by the product’s current price, consumers’ idiosyncratic product preferences, patience levels, expected second-period prices, and their beliefs about product quality. When social learning is present, early buyers post reviews based on their experiences, affecting the purchase decisions of remaining consumers and the firm’s second-period pricing. This process creates a two-sided learning process, where both the firm and consumers update their beliefs of the product’s quality based on the posted reviews (Papanastasiou, 2020; Papanastasiou and Savva, 2017; Sun, 2012; Yu et al., 2016). To capture realistic consumer behavior, we assume that consumers exhibit bounded rationality, forming their quality beliefs based on the aggregate numbers of positive and negative reviews rather than analyzing the detailed review sequence.

We outline the main findings as follows. First, we examine contingent pricing under social learning, revealing that social learning enables the firm to adopt more flexible pricing strategies—skim, consistent, or penetration—that vary based on consumer patience and review outcomes. While social learning compels the firm to lower first-period prices to encourage early purchases, it also allows higher second-period prices when reviews are positive, enhancing profitability. Ultimately, social learning creates a win-win situation under contingent pricing, benefiting both the firm and consumers.

Second, we uncover new pricing dynamics introduced by social learning under price guarantee. Traditional price discrimination theory posits that skim pricing under price guarantee requires the presence of non-price-sensitive consumers. However, our findings reveal that with social learning, skim pricing can be optimal even when all consumers are price-sensitive. This is particularly true for highly patient consumers, where a combination of high first-period prices and markdowns under price guarantee encourages early purchases, enhancing both review outcomes and firm profitability. In this case, social learning leads to higher first-period prices under price guarantee than under contingent pricing. Moreover, social learning consistently improves the firm’s profitability but only benefits consumers who are sufficiently patient under price guarantee.

Moreover, our comparative analysis underscores the trade-offs between the two pricing schemes. In the absence of social learning, contingent pricing proves more profitable for the firm due to its greater pricing flexibility. However, with social learning, price guarantee becomes more advantageous when consumers exhibit high levels of patience, indicating that price guarantee is more effective in mitigating strategic waiting. Notably, we show that contingent pricing with social learning results in mutually better outcomes for both the firm and consumers compared to price guarantee.

Lastly, our extensions validate the robustness of our findings. When consumers are fully rational, the primary insights regarding social learning’s impact on pricing strategies and outcomes remain consistent. Moreover, we consider a partially forward-looking firm setup and reveal new scenarios where price guarantee leads to win-win outcomes, broadening the practical relevance of our results.

The rest of this paper is structured as follows. Section 2 reviews the related literature. Section 3 introduces the model and its preliminaries. Section 4 presents the equilibrium results under the contingent pricing scheme, while Section 5 analyzes the equilibrium results under the price guarantee scheme. Section 6 compares the results across the two pricing schemes, emphasizing the impacts of social learning. Section 7 discusses model extensions, including the effects of consumer rationality and the firm’s discounting of future profits. Finally, Section 8 concludes the paper and outlines directions for future research. The Electronic Companion contains all the proofs, accompanied by both the technical reasoning and the underlying logic to improve comprehension. Notably, we include the Mathematica code to assist readers in verifying the proofs.¹ The use of symbolic computation software like Mathematica for theorem proving has gained increasing acceptance in operations research and management science; see, for instance, Martínez-de Albéniz and Kunnumkal (2022); Meng et al. (2024); Sun and Tyagi (2020); Zhan and Ward (2019). The provided Mathematica code enables readers to independently trace the derivations and validate the results, enhancing the transparency and reliability of our findings.

2. Literature Review

Our work draws upon and contributes to three streams of research, including pricing with strategic consumers, contingent pricing with social learning, and research related to the price guarantee scheme.

2.1. Pricing with Strategic Consumers

Pricing in the presence of strategic waiting consumers has been extensively documented in the literature since the seminal work of Coase (1972), highlighting how monopoly retailers suffer from consumers delaying purchases in anticipation of price reductions. Subsequent research has consistently shown that strategic consumers prefer to wait for discounts, affecting firms’ pricing and inventory decisions (Aviv et al., 2019; Cachon and Swinney, 2009; Su, 2007). For example, Cachon and Swinney (2009) demonstrate that strategic consumer behavior drives firms to stock less, reduce price discounts, and ultimately earn lower profits. Additionally, committing to a predetermined price path, regardless of whether it includes markdowns, performs poorly compared to contingent pricing strategies.

Several studies employ two-period stylized models to examine how pricing schemes can mitigate strategic waiting behavior. Su and Zhang (2008) show that firms can reduce strategic waiting by making future purchase opportunities less attractive, such as through upward price trajectories or limited product availability. Similarly, Liu and Van Ryzin (2008) find that restricted stock availability discourages consumers from delaying purchases, thereby enhancing firm profitability. Aviv and Pazgal (2008) argue that an announced fixed-discount pricing strategy can outperform contingent pricing, while Cachon and Feldman (2015) propose frequent discounting as a superior approach compared to static or contingent pricing.

Although dynamic pricing may encourage consumers to delay purchases, research indicates that firms can still maintain profitability. Ovchinnikov and Milner (2012) show that offering end-of-period discounts can benefit firms in the long run, even if consumers adjust their behavior to wait for these opportunities. Wu et al. (2015) analyze a firm selling product versions at full price in the first period and at markdowns in the second, demonstrating that strategic markdowns can stabilize profitability. Additional research has examined related topics such as consumer credit (Papouskova and Hajek, 2019) and return policies (Kim et al., 2022; Liu et al., 2022).

2.2. Contingent Pricing with Social Learning

Consumer reviews and social learning have drawn considerable scholarly attention, spanning diverse research streams, such as social networks (Qiu et al., 2021), information disclosure and sharing (Li, 2017; Ma et al., 2024; Shin and Zeevi, 2024), product line design and new product launches (Feldman et al., 2019; Kwark et al., 2018), and inventory decisions (Hu et al., 2016; Papanastasiou, 2020). A notable body of literature explores optimal contingent pricing (i.e., dynamic pricing) strategies in the presence of social learning. For instance, Feng et al. (2019) combine analytical modeling and empirical analysis to demonstrate how merchants adjust prices dynamically in response to consumer reviews to influence future feedback. Ma et al. (2021) develop a social learning model dependent on user participation and examine the influence of this learning process on the dynamic pricing strategies employed by service providers. Other studies investigate contingent pricing strategies that account for both social learning and consumers’ strategic waiting behavior, typically employing two-period models. Jing (2011b) employs a linear model of social learning to explore its effects on consumer purchasing decisions. Results show that it is the learning externality that drives postpone purchases. Jing (2011a) further investigates the interaction between social learning and dynamic pricing for durable goods, revealing that high learning intensity may lead firms to release products earlier and charge higher prices to uninformed consumers.

Few studies have examined the combined effects of consumers’ strategic waiting behavior and the Bayesian social learning process. Using a signaling model, Yu et al. (2016) establish that a monopolist reduces first-period sales volume and prices to counter strategic waiting and social learning effects. Papanastasiou and Savva (2017) compare preannounced pricing and responsive pricing under Bayesian learning, finding that preannounced pricing typically follows an increasing price trajectory, while responsive pricing allows for price adjustments in either direction.

The above-mentioned literature, with the exception of (Feng et al., 2019), assumes that consumers can accurately infer true product quality from reviews. However, in practice, reviews are inherently subjective and may be biased (Li and Hitt, 2008). Addressing this limitation, Crapis et al. (2017) and Ifrach et al. (2019) explore monopoly pricing when consumer reviews consist of binary feedback (“like” or “dislike”) derived from their net utility. Crapis et al. (2017) analyze static and dynamic pricing schemes within a non-Bayesian framework, while Ifrach et al. (2019) demonstrate that a stationary optimal pricing strategy can attract a larger consumer base. Moreover, Shin et al. (2023) consider two review models—quality-based and value-based—and show that retailers may adopt distinct pricing strategies, such as skim pricing or penetration pricing, depending on the prevailing review model.

In contrast, we adopt a rational expectation equilibrium framework to capture the dynamics between price and demand in the review generation and social learning processes. Specifically, we assume that consumers exhibit bounded rationality, focusing on the aggregate numbers of positive and negative reviews when forming their quality beliefs. Such a framework allows us to integrate the bounded rationality assumption and strategic waiting behavior (Huang et al., 2017; Li and Jiang, 2022; Song et al., 2019). Based on the review generation and social learning processes, we analyze and compare the firm’s optimal pricing strategies under two schemes: Contingent pricing and price guarantee, when consumers exhibit strategic waiting behavior.

2.3. Price Guarantee Scheme

The price guarantee scheme consists of two distinct dimensions: time and structure. The time dimension involves firms offering consumers a low-price guarantee for a specific product within a set period, while the structure dimension refers to firms committing to match competitors’ prices across various sales channels. This paper focuses on the time dimension of the price guarantee scheme. For an in-depth review of the structure dimension, see Jiang and He (2021).

In a monopoly context, Levin et al. (2007) develop a discrete-time optimal control model to simultaneously determine the offered price, strike price, and fee for the price guarantee scheme. Lai et al. (2010) explore posterior price matching, finding that a firm can increase profit when the fraction of strategic consumers is moderate and consumers’ valuations decline moderately over time. This occurs because price matching eliminates consumers’ strategic waiting incentives, enabling firms to charge higher prices during the regular selling season. Furthermore, Cohen-Vernik and Pazgal (2017) endogenize the refund amount in their model, allowing firms to determine the optimal reimbursement percentage for price differences. From a behavioral perspective, Huang et al. (2017) introduce bounded rationality to explain the widespread adoption of price guarantee schemes. Wu et al. (2025) examine the effectiveness of price guarantees in a Markovian pricing context and find that such guarantees can effectively retain consumers by encouraging high-valuation customers to purchase early and enabling firms to engage in price discrimination based on customers’ willingness to pay.

The existing literature on price guarantee is limited, and the role of social learning—a key feature in this paper—has not been explored within this context. This research fills that gap by examining contingent pricing and price guarantee in the presence of strategic consumers and social learning. We demonstrate that in the presence of social learning, a firm may adopt different pricing strategies—skim, consistent, or penetration—depending on consumers’ strategic waiting incentives, product quality, and review outcomes under each pricing scheme. Our study provides a novel rationale for why major e-tailers implement price matching during key promotional events in the presence of social learning, offering valuable managerial and practical insights for firms and consumers.

3. Model Setup and Preliminaries

We consider a monopolistic firm selling a new experience product over two consecutive periods. Let $τ \in {1, 2}$ represent the selling horizon. Under a given pricing scheme—contingent pricing or price guarantee—the firm sets the product price, denoted as $p_{τ}$ , at the start of each period $τ \in {1, 2}$ . The quality of this product, denoted by $q$ , follows a uniform distribution on $[0, 2 q_{o}]$ , where $q_{o} > 0$ . Consistent with the existing literature, we assume that actual quality $q$ is unknown to both the firm and consumers in the first period (Feng et al., 2019; Li and Hitt, 2010; Sun, 2012). This assumption reflects the high failure rates—estimated between 80% and 95%—of new experience products, such as cosmetics, electronics, and books (Sun, 2012).

In this section, we first detail the model of consumer behavior and quality belief, followed by a description of the firm’s decisions and profit functions.

3.1. Consumers’ Purchase Decisions

3.1.1. Consumer Utility

We assume the market consists of risk-neutral consumers with heterogeneous idiosyncratic preferences for the product. Consumer heterogeneity is modeled using a Hotelling framework (Hotelling, 1929). Specifically, the product is positioned at 0, and a consumer’s preference, modeled by $x$ , is uniformly distributed along a line of length 1. A consumer located at $x$ (hereinafter referred to as consumer $x$ ) incurs a total mismatch cost of $t x$ , where $t$ represents the exogenous unit distance mismatch cost. The magnitude of $t$ captures the variation in consumer preferences: A high $t$ reflects a niche market with significant preference differences, as seen with Whole Foods targeting affluent consumers seeking premium groceries. In contrast, a low $t$ suggests a mass market where preference differences have a smaller impact, such as the sporting equipment industry (Feng et al., 2019; Sun, 2012). To avoid corner solutions and ensure incomplete coverage of the market, we assume $t > 2 q_{o}$ (Li and Hitt, 2010; Sun, 2012).

Consumers form their quality beliefs for each period, denoted by ${\hat{q}}_{τ}$ , along with their expected second-period price, ${\hat{p}}_{2}$ . Based on these beliefs and the first-period price $p_{1}$ , the expected net utility of consumer $x$ purchasing the product in the first period is given by

E [U_{1} ({\hat{q}}_{1}, p_{1}, x)] = E [{\hat{q}}_{1} - p_{1} - t x] .

(1)

In the second period,

p_{2}

is realized, and consumer

x

’s expected net utility for purchasing in the second period is

E [U_{2} ({\hat{q}}_{2}, p_{2}, x)] = E [δ_{c} ({\hat{q}}_{2} - p_{2} - t x)],

(2)

where

δ_{c} \in (0, 1)

denotes the discount factor, reflecting consumers’ patience or the incentive level for strategic waiting.

We assume that the utility of no purchase is zero. At the beginning of the first period, consumers evaluate their expected net utility in each period to decide whether to purchase in the first period or postpone. Specifically, consumer $x$ purchases the product in the first period if and only if

E [U_{1} ({\hat{q}}_{1}, p_{1}, x)] \geq E [max (U_{2} ({\hat{q}}_{2}, {\hat{p}}_{2}, x), 0)],

where

E [max (U_{2} ({\hat{q}}_{2}, {\hat{p}}_{2}, x), 0)]

captures the consumer’s choice of no purchase when the the second-period net utility is negative.

3.1.2. Consumer Demand

Without loss of generality, we normalize the number of consumers to one and assume that each consumer purchases at most one product. Consumers are aware of the product at the beginning of the selling season and make purchase decisions based on their expected net utilities. If a consumer’s expected net utility in the first period is nonnegative and exceeds that of the second period, the consumer purchases the product in the first period; otherwise, the consumer postpones the purchase decision until the second period. Let $D_{τ}$ , $τ \in {1, 2}$ denote consumer demand in period $τ$ , defined as the number of consumers who make a purchase during that period. Then, we have

\begin{aligned} D_{1} ({\hat{q}}_{1}, {\hat{q}}_{2}, p_{1}, {\hat{p}}_{2}) & = {x : E [U_{1} ({\hat{q}}_{1}, p_{1}, x)] \\ \geq E [max (U_{2} ({\hat{q}}_{2}, {\hat{p}}_{2}, x), 0)]} \end{aligned}

(3)

and

\begin{aligned} D_{2} ({\hat{q}}_{2}, p_{2}, D_{1}) & = {x : U_{2} ({\hat{q}}_{2}, p_{2}, x) \geq 0, x > D_{1}} \\ = {(\frac{{\hat{q}}_{2} - p_{2}}{t} - D_{1})}^{+} . \end{aligned}

(4)

For notational simplicity, we omit the explicit dependence of

D_{τ}

on prices and quality when the context is clear.

Without social learning, consumers’ quality beliefs in both periods align with their prior beliefs; that is, $E [{\hat{q}}_{1}] = E [{\hat{q}}_{2}] = E [q] = q_{o}$ . Next, we detail the consumer review and quality belief update process in the presence of social learning.

3.2. Qualify Belief Update via Social Learning

3.2.1. Review Generation

After purchasing and experiencing the product in the first period, consumers post reviews about its quality based on their net ex-post utility $q - p_{1} - t x$ , thereby influencing those who have yet to make a purchase via social learning. We assume that only consumers who purchase in the first period submit reviews; that is, the total number of consumers posting reviews is $D_{1}$ . A consumer posts a positive review, denoted by (thumbs up), when the ex-post utility is nonnegative, or a negative review, denoted by (thumbs down), otherwise. This binary review modeling approach simplifies conventional star rating scales and aligns with the settings in Crapis et al. (2017) and Ifrach et al. (2019).

Recall that $q \in [0, 2 q_{o}]$ represents the actual product quality. Therefore, for $q \in [p_{1} + t D_{1}, 2 q_{o}]$ , all consumers $x \in [0, D_{1}]$ have nonnegative ex-post utility and post ; for $q \in [0, p_{1})$ , all consumers $x \in [0, D_{1}]$ have negative utility and post . Finally, for $q \in [p_{1}, p_{1} + t D_{1})$ , consumers post a mixture of and based on the sign of their ex-post utility $q - p_{1} - t x$ . The distribution of review outcome is represented by Figure 1.

Figure 1.

Review generation and social learning.

3.2.2. Quality Belief Update

At the beginning of the second period, consumers who have not yet purchased the product can observe posted reviews and update their beliefs about the product’s quality. While review aggregators provide access to individual review sequences, processing such detailed information can be cumbersome and presents significant computational challenges. To address this, we adopt a behavioral assumption of bounded rationality, where consumers focus on the aggregated numbers of and to form their quality belief, rather than analyzing the detailed review sequences. Notably, the bounded rationality and forward-looking behavioral assumptions can complement each other and can both be integrated into a rational expectations equilibrium framework (Huang et al., 2017; Li and Jiang, 2022; Song et al., 2019). In Electronic Companion EC.1, we describe a within-subject experiment conducted to validate this assumption. Moreover, we discuss the implications of fully rational consumers in Section 7, where we demonstrate that our main results are robust under this setting.

Based on these aggregated reviews, we assume that second-period consumers form one of three beliefs, denoted by $R$ , based on the posted reviews: Completely positive ( $P$ ), mixed ( $M$ ), or completely negative ( $N$ ). Based on Figure 1, we have

R = {\begin{cases} P, & for q \in [p_{1} + t D_{1}, 2 q_{o}] \\ M, & for q \in [p_{1}, p_{1} + t D_{1}) \\ N, & for q \in [0, p_{1}) \end{cases} .

(5)

We further assume that consumers use a quasi-Bayesian decision rule to estimate product quality from aggregated reviews, as they find detailed review sequences difficult to process. This leads them to form posterior beliefs by calculating a probability-weighted average over the reviews in the corresponding range in equation (5), rather than making precise inferences. Let $\underset{R}{Pr}$ denote the probability of consumers forming belief $R \in {P, M, N}$ and $q_{R}$ denote the corresponding posterior belief. Specifically, ${Pr}_{R}$ represents the proportion of the respective areas, and $q_{R}$ corresponds to the average quality in each case, as illustrated in Figure 1. Recall that ${\hat{q}}_{2}$ is consumers’ second-period quality belief. Thus, in the presence of social learning, we have

{\hat{q}}_{2} = {\begin{cases} q_{P} = q_{o} + \frac{p_{1} + t D_{1}}{2}, & for q \in [p_{1} + t D_{1}, 2 q_{o}] \\ q_{M} = \frac{2 p_{1} + t D_{1}}{2}, & for q \in [p_{1}, p_{1} + t D_{1}) \\ q_{N} = \frac{p_{1}}{2}, & for q \in [0, p_{1}) \end{cases}

(6)

and

{Pr}_{R} = {\begin{cases} {Pr}_{P} = 1 - \frac{p_{1} + t D_{1}}{2 q_{o}}, & for q \in [p_{1} + t D_{1}, 2 q_{o}] \\ {Pr}_{M} = \frac{t D_{1}}{2 q_{o}}, & for q \in [p_{1}, p_{1} + t D_{1}) \\ {Pr}_{N} = \frac{p_{1}}{2 q_{o}}, & for q \in [0, p_{1}) \end{cases} .

(7)

Equation (6) illustrates that, alongside actual product quality and consumer heterogeneity, the first-period price ( $p_{1}$ ) significantly influences consumers’ perceptions of product quality through social learning over time. Specifically, the equation captures two competing effects: the price effect on utility (where a higher price negatively impacts reviews by reducing consumer net utility) (Cabral and Li, 2015; Li and Hitt, 2010) and the self-selection effect on review composition (where a higher price improves reviews by filtering out consumers with high mismatching costs) (Bondi, 2019; Li and Hitt, 2008). For instance, when reviews are completely negative ( $R = N$ ), the perceived quality $q_{N}$ increases with $p_{1}$ , indicating that the self-selection effect outweighs the price effect. In this case, consumers rely completely on price as an indicator of quality. However, for completely positive ( $R = P$ ) and mixed ( $R = M$ ) reviews, the interplay between these effects becomes more complex, with the overall impact depending on the inverse relationship between price and demand ( $D_{1}$ ). In this case, consumers tend to perceive higher quality when high prices are coupled with high demand, as this combination signals strong product quality (Bhalla, 2012; Yu et al., 2016). Therefore, a higher $p_{1}$ does not necessarily imply a worse second-period quality belief.

In the presence of social learning, consumers lack direct knowledge of the actual product quality in the first period and instead form quality beliefs based on price and preferences. Anticipating quality information from future reviews, social learning may prompt consumers to delay their purchase decision. However, for any consumer $x$ , their expected posterior quality belief in the first period equals their expected prior belief, i.e., $E [{\hat{q}}_{2}] = \underset{P}{Pr} q_{P} + \underset{M}{Pr} q_{M} + \underset{N}{Pr} q_{N} = E [{\hat{q}}_{1}] = E [q] = q_{o}$ . This result is consistent with the martingale property and indicates that, on average, product reviews do not alter consumers’ expected quality belief (Papanastasiou and Savva, 2017; Yu et al., 2016).

3.3. Firm’s Decisions

Under either pricing scheme, the firm sequentially announces the price $p_{τ}$ at the beginning of period $τ \in {1, 2}$ . Specifically, the firm first sets the first-period price, $p_{1}$ , to maximize the total expected profit over the entire selling horizon, denoted by $Π (p_{1})$ . That is, the optimal price becomes

\begin{aligned} p_{1}^{*} & = \underset{p_{1} \geq 0}{\arg \max} Π (p_{1}) = \underset{p_{1} \geq 0}{\arg \max} π_{1} (p_{1}) + δ_{f} E [π_{2} (p_{2})] \\ = \underset{p_{1} \geq 0}{\arg \max} p_{1} D_{1} + δ_{f} E [π_{2} (p_{2})], \end{aligned}

(8)

where

δ_{f}

is the firm’s discount factor for second-period profit,

π_{τ}

denotes the firm’s profit in period

τ

, and the asterisk sign indicates the optimal decision. In the second period, the firm then sets

p_{2}

to maximize that period’s profit, depending on the pricing scheme. That is,

p_{2}^{*} = \underset{p_{2} \geq 0}{\arg \max} π_{2} (p_{2}),

where under contingent pricing, the firm’s second-period profit is

π_{2} (p_{2}) = p_{2} D_{2} .

(9)

In contrast, under the price guarantee scheme, the firm offers consumers a 100% refund of the price difference if the second-period price is lower than the first (i.e.,

p_{2} < p_{1}

) (Huang et al., 2017; Lai et al., 2010). Therefore,

π_{2} (p_{2}) = p_{2} D_{2} - (p_{1} - p_{2})^{+} D_{1} .

(10)

We assume $δ_{f} = 1$ , reflecting that the firm is focused on long-term profitability. This assumption aligns with prior research (He et al., 2016; Huang et al., 2018; Li and Jiang, 2022; Papanastasiou and Savva, 2017; Su, 2009). In Section 7, we extend the model to explore the scenario where $0 < δ_{f} < 1$ and demonstrate that our main results remain robust when $δ_{f} \geq δ_{c}$ .

Finally, we describe the sequence of events as follows: (1)

At the beginning of the first period, the firm sets price $p_{1}$ under a given pricing scheme.

(2)

Consumers decide whether to purchase immediately or delay strategically, based on the first-period price, $p_{1}$ , their prior belief, $q_{o}$ , and their expected posterior belief, ${\hat{q}}_{2}$ , as well as the expected second-period price, ${\hat{p}}_{2}$ .

(3)

Consumers who purchase in the first period post reviews based on their perceived utility at the end of the first period.

(4)

At the beginning of the second period, both the firm and consumers update their beliefs about product quality after reviewing the posted reviews.

(5)

The firm sets the second-period price, $p_{2}$ , and refunds any price difference to first-period consumers under the price guarantee scheme if the price drops ( $p_{2} < p_{1}$ ).

(6)

Remaining consumers who have not yet purchased make their decisions in the second period.

The event sequence is summarized in Figure 2. Note that the only variation in the sequence of events between the two pricing schemes is that the firm must reimburse the price difference in the second period if markdowns occur under the price guarantee scheme.

Figure 2.

Sequence of events.

It is straightforward to verify that, for the same prices $p_{τ}$ , $τ \in {1, 2}$ , the first-period demand under the price guarantee is at least as high as that under contingent pricing, indicating that the price guarantee is more effective at mitigating consumer strategic waiting. Next, we analyze the optimal pricing decisions under each pricing scheme in Sections 4 and 5.

4. Analysis of Contingent Pricing

In this section, we examine the firm’s optimal decisions under the contingent pricing scheme, where the firm can dynamically adjust its prices across both periods. We consider two cases: $CN$ , where consumers have no access to social learning, explored in Section 4.1, and $CL$ , where consumers gain information through social learning, discussed in Section 4.2. We employ backward induction to determine the optimal decisions made by both the firm and consumers. We adopt the following tie-breaking rule: When two sets of pricing decisions yield the same optimal profit, we choose the solution with the higher consumer surplus as the optimal one.

4.1. Contingent Pricing Without Social Learning

In the absence of social learning, consumers cannot gain information from product reviews and must rely solely on their prior belief about product quality for purchase decisions, such that $E [{\hat{q}}_{1}] = E [{\hat{q}}_{2}] = E [q] = q_{o}$ . We first solve for the second-period price, $p_{2}^{CN}$ , which maximizes the profit (equation (9)) in that period, as a function of $q_{o}$ and the first-period demand $D_{1}^{CN}$ . With consumers’ expected second-period price set at ${\hat{p}}_{2} = p_{2}^{CN}$ , we then solve for the firm’s optimal prices $p_{1}^{CN *}$ and $p_{2}^{CN *}$ . The following proposition outlines the firm’s optimal pricing decisions.

Proposition 1 (Equilibrium Prices Under $CN$ )

The firm’s unique equilibrium prices are

p_{1}^{CN *} = \frac{q_{o} (2 - δ_{c})^{2}}{2 (3 - 2 δ_{c})} and p_{2}^{CN *} = \frac{q_{o} (2 - δ_{c})}{2 (3 - 2 δ_{c})} .

(11)

Moreover, for

δ_{c} \in (0, 1)

p_{1}^{CN *} > p_{2}^{CN *}

and

p_{1}^{CN *}

(

p_{2}^{CN *}

) strictly decreases (increases) with

δ_{c}

. The firm’s optimal expected profit strictly decreases with

δ_{c}

Proposition 1 establishes that, in the absence of social learning, the firm adopts a skim pricing strategy over the two periods. This strategy emerges because consumers only choose to delay their purchases till the second period if the firm significantly reduces prices. Note that in this case, the firm prefers consumers to purchase in the first period because of the higher price. Consequently, as consumers exhibit greater patience (i.e., as $δ_{c}$ increases), the firm has to lower the first-period price while simultaneously raising the second-period price to encourage early purchases. However, this adjustment ultimately results in a decrease in the firm’s total profit.

4.2. Contingent Pricing with Social Learning

In the presence of social learning, consumers face a trade-off between the immediacy of purchasing and the potential benefits of delaying to acquire additional information. Specifically, consumers with lower patience levels (i.e., smaller $δ_{c}$ ) are less inclined to postpone their purchases, thereby increasing first-period demand and generating more product reviews. On the other hand, delaying purchases until the second period allows consumers to gain insights into product quality by reviewing feedback from earlier purchasers, thereby reducing the first-period demand. Thus, the interplay between strategic waiting and social learning introduces complexity into consumers’ purchasing decisions and, consequently, the firm’s pricing strategy.

Specifically, with social learning, the firm and consumers form the posterior quality belief ${\hat{q}}_{2}$ according to equations (6) and (7). Using backward induction, we first solve for the second-period prices $p_{2 R}^{CL}$ , as a function of the first-period demand and the posterior quality belief ${\hat{q}}_{2}$ , given the nature of the review $R \in {P, M, N}$ . We then characterize the first-period demand $D_{1}^{CL}$ with respect to the first-period price $p_{1}^{CL}$ . The results are illustrated in the following lemma.

Lemma 1 (First-Period Demand w.r.t. Price Under $CL$ )

Given the first-period price $p_{1}^{CL}$ , the first-period demand, $D_{1}^{CL}$ , satisfies the following equation:

\begin{aligned} q_{o} - p_{1}^{CL} - t D_{1}^{CL} = {\begin{cases} δ_{c} \frac{2 q_{o} - p_{1}^{CL} - t D_{1}^{CL}}{2 q_{o}} (q_{P}^{CL} - p_{2 P}^{CL} - t D_{1}^{CL}) & for p_{1}^{CL} \leq \frac{t D_{1}^{CL}}{2} \\ = \frac{δ_{c} (2 q_{o} + p_{1}^{CL} - t D_{1}^{CL}) (2 q_{o} - p_{1}^{CL} - t D_{1}^{CL})}{8 q_{o}}, \\ δ_{c} (\frac{2 q_{o} - p_{1}^{CL} - t D_{1}^{CL}}{2 q_{o}} (q_{P}^{CL} - p_{2 P}^{CL} - t D_{1}^{CL}) + \frac{t D_{1}^{CL}}{2 q_{o}} (q_{M}^{CL} - p_{2 M}^{CL} - t D_{1}^{CL})) & for \frac{t D_{1}^{CL}}{2} 2 t D_{1}^{CL} \\ + \frac{p_{1}^{CL}}{2 q_{o}} (q_{N}^{CL} - p_{2 N}^{CL} - t D_{1}^{CL})) = \frac{δ_{c} (q_{o} - t D_{1}^{CL})}{2}, \end{cases}, \end{aligned}

(12)

where

p_{2 P}^{CL} = \frac{2 q_{o} + p_{1}^{CL} - t D_{1}^{CL}}{4}

p_{2 M}^{CL} = {\begin{cases} 0, & for p_{1}^{CL} \leq \frac{t D_{1}^{CL}}{2} \\ \frac{2 p_{1}^{CL} - t D_{1}^{CL}}{4}, & for p_{1}^{CL} > \frac{t D_{1}^{CL}}{2} \end{cases}

, and

p_{2 N}^{CL} = {\begin{cases} 0, & for p_{1}^{CL} \leq 2 t D_{1}^{CL} \\ \frac{p_{1}^{CL} - 2 t D_{1}^{CL}}{4}, & for p_{1}^{CL} > 2 t D_{1}^{CL} \end{cases}

are the optimal second-period prices given

D_{1}^{CL}

and

p_{1}^{CL}

. Moreover,

D_{1}^{CL}

strictly decreases with

p_{1}^{CL}

Lemma 1 demonstrates that the first-period price influences not only consumers’ immediate decision to purchase or delay but also how different types of reviews impact their purchasing decisions in the second period. Specifically, as the first-period price increases, more consumers choose to postpone their purchase decisions to the second period. Moreover, consider consumer $x = D_{1}^{CL}$ . The left-hand side of equation (12) represents this consumer’s expected utility of purchasing in the first period, while the right-hand side reflects the utility of delaying the purchase to the second period. Lemma 1 implies that if $p_{1}^{CL} \leq \frac{t D_{1}^{CL}}{2}$ , consumers who delay their purchase decision will refrain from purchasing if reviews are mixed or negative; if $\frac{t D_{1}^{CL}}{2} < p_{1}^{CL} \leq 2 t D_{1}^{CL}$ , they will hold off when reviews are negative. Otherwise, if $p_{1}^{CL} > 2 t D_{1}^{CL}$ , some second-period consumers may choose to purchase regardless of negative reviews.

Based on Lemma 1, we derive the firm’s optimal pricing strategy $(p_{1}^{CL *}, p_{2 R}^{CL *})$ for $R \in {P, M, N}$ under $CL$ , whose properties are shown in Proposition 2.

Proposition 2 (Equilibrium Prices Under

CL

)

(i)
$p_{2 M}^{CL } < p_{1}^{CL }$ and $p_{2 N}^{CL } < p_{1}^{CL }$ . Moreover, there exists a threshold $δ_{P}^{CL}$ such that $p_{2 P}^{CL } < p_{1}^{CL }$ for $δ_{c} \in (0, δ_{P}^{CL})$ , $p_{2 P}^{CL } = p_{1}^{CL }$ at $δ_{c} = δ_{P}^{CL}$ , and $p_{2 P}^{CL } > p_{1}^{CL }$ for $δ_{c} \in (δ_{P}^{CL}, 1)$ .
(ii)
$p_{1}^{CL }$ strictly decreases with $δ_{c}$ ; $p_{2 P}^{CL }$ , $p_{2 M}^{CL }$ , and $p_{2 N}^{CL }$ first (weakly) decreases and then increases with $δ_{c}$ .
(iii)
The firm’s expected profit and first-period demand strictly decrease with $δ_{c}$ .

The close-form expressions of $p_{1}^{CL }$ and $p_{2 R}^{CL }$ for $R \in {P, M, N}$ are shown in the proof.

Proposition 2(i) reveals that, under $CL$ , the firm may employ a skim, consistent, or penetration pricing strategy across the two periods, contingent on consumers’ strategic waiting behavior ( $δ_{c}$ ) and the outcome of consumer reviews ( $R \in {P, M, N}$ ). Specifically, when consumers are impatient (i.e., $δ_{c}$ is low) and inclined to purchase in the first period, the firm always adopts a skim pricing strategy to capture any residual demand in the second period. On the other hand, when consumers are sufficiently patient, the firm may apply a consistent or penetration pricing strategy if reviews are positive, thereby capitalizing on the social learning effect. This approach contrasts sharply with the scenario of $CN$ , where the firm strictly adheres to a skim pricing strategy (Proposition 1).

Proposition 2(ii) illustrates how consumers’ strategic waiting behavior affects the optimal prices. While the first-period price strictly decreases with $δ_{c}$ —consistent with the $CN$ scenario in Proposition 1—the second-period prices no longer monotonically increase with $δ_{c}$ . Specifically, when $δ_{c}$ is sufficiently low, more impatient consumers (i.e., lower $δ_{c}$ ) are more likely to purchase and leave reviews in the first period, enabling the firm to leverage social learning by setting higher second-period prices. In contrast, patient consumers (high $δ_{c})$ are less inclined to purchase in the first period, limiting the firm’s ability to benefit from social learning. Consequently, the firm’s second-period price follows a similar trend as in the case of $CN$ : As consumer patience further increases, the firm increases the second-period prices to encourage early purchases. Figure 3 depicts the dynamics of the contingent pricing scheme in equilibrium.

Figure 3.
Firm’s optimal prices in the presence of social learning under contingent pricing. The symbols $↑$ , $\to$ , and $↓$ denote that the firm raises, keeps constant, and marks down the second-period price, respectively ( $q_{o} = 1$ ). The optimal prices are linear in $q_{o}$ .

Proposition 2(iii) shows that while $p_{1}^{CL *}$ decreases with $δ_{c}$ —an incentive that encourages early purchases—the impact of consumer strategic waiting outweighs this price reduction, leading to a decline in first-period demand $D_{1}^{CL}$ . Furthermore, the firm’s optimal profit decreases overall with rising $δ_{c}$ , aligning with the trend observed in the case of $CN$ . Building on these results, the following section explores the broader implications of social learning on the firm’s pricing strategy and profitability under contingent pricing.
4.3. Impact of Social Learning Under Contingent Pricing

In this section, we assess the impact of social learning under the contingent pricing scheme. Recall that social learning allows both the firm and the second-period consumers to update their beliefs based on reviews posted by first-period consumers. The following proposition details the properties of the consumers’ posterior quality belief $q_{R}^{CL}$ , and the corresponding probability ${Pr}_{R}^{CL}$ , based on the review outcome $R \in {P, M, N}$ evaluated at the optimal prices.

Proposition 3 (Second-Period Quality Beliefs Under $CL$ )

At the optimal prices $p_{1}^{CL *}$ and $p_{2 R}^{CL *}$ , where $R \in {P, M, N}$ ,

(i)
$q_{P}^{CL} > q_{o}$ , $q_{M}^{CL} < q_{o}$ , and $q_{N}^{CL} < q_{o}$ .
(ii)
$q_{R}^{CL}$ strictly decreases with $δ_{c}$ for all $R \in {P, M, N}$ .
(iii)
${Pr}_{P}^{CL}$ strictly increases with $δ_{c}$ , while ${Pr}_{M}^{CL}$ and ${Pr}_{N}^{CL}$ strictly decreases with $δ_{c}$ .

Proposition 3(i) demonstrates that completely positive reviews lead to a posterior quality belief that exceeds the prior belief. In contrast, mixed or completely negative reviews cause the posterior beliefs to fall below the prior. This finding highlights the influence of review polarity on consumers’ perception of quality.

Proposition 3(ii) further reveals that as consumers grow more patient, the posterior quality beliefs decrease. This is because when consumers have a greater tendency to delay purchases, Proposition 2 implies that the firm lowers the first-period price to stimulate early purchases, but nevertheless, the first-period demand decreases. As indicated by equation (6), both a lower first-period price and reduced demand signal lower perceived product quality to second-period consumers, resulting in diminished posterior quality beliefs irrespective of the review outcome.

While a higher patience level ( $δ_{c}$ ) reduces posterior quality beliefs, its impact on review likelihoods diverges. In line with equation (6), Proposition 3(iii) shows that as $δ_{c}$ rises, the probability of completely positive reviews increases, while the probabilities of mixed or completely negative reviews correspondingly decline. This outcome arises because an increase in $δ_{c}$ leads to a reduction in first-period demand (Proposition 2(iii)), implying that only consumers with sufficiently low mismatch costs ( $t x$ ) would purchase in the first period. These consumers are more likely to leave positive reviews, resulting in an overall increase in the probability of favorable feedback. These dynamics are visually illustrated in Figure 4.

Figure 4.
Posterior beliefs (left) and corresponding probabilities (right) with respect $δ_{c}$ at the optimal prices in the presence of social learning under contingent pricing ( $q_{o} = 1$ ). The posterior beliefs are linear in $q_{o}$ and the probabilities are independent of $q_{o}$ .

By comparing the equilibrium prices under $CN$ with those under $CL$ , the following proposition characterizes the impact of social learning on prices.
Proposition 4 (Price Comparison Under Contingent Pricing)

(i)
$p_{1}^{CL } < p_{1}^{CN }$ .
(ii)
$p_{2 P}^{CL } > p_{2}^{CN }$ ; $p_{2 M}^{CL } < p_{2}^{CN }$ ; $p_{2 N}^{CL } < p_{2}^{CN }$ .

Proposition 4 shows that, in the presence of social learning, the firm typically sets lower prices in both periods compared to the no-social-learning, except when the review outcome is completely positive. Social learning, therefore, acts as a double-edged sword for the firm. On one hand, it strengthens consumers’ incentive to delay purchases in anticipation of additional information and potential price reductions. In response, the firm has to lower its first-period price to encourage early purchases. On the other hand, in the second period, consumers update their quality perceptions based on the reviews, which increase their willingness to pay if the reviews are positive but decrease if the reviews are mixed or negative. This allows the firm to charge a higher second-period price when reviews are positive but forces it to lower the price when reviews are mixed or negative.

Proposition 4 might suggest that social learning benefits consumers while disadvantaging the firm due to the typically lower prices it induces. Yet, Proposition 5 reveals that under the contingent pricing scheme, the positive impact of social learning—specifically, the higher prices the firm can set when reviews are positive, capitalizing on increased consumer willingness to pay—outweighs the negative effect of lower prices in other scenarios. As a result, social learning ultimately benefits both consumers and the firm, creating a win-win situation.
Proposition 5 (Profit and Welfare Comparison Under Contingent Pricing)

Both the firm’s optimal profit and consumer surplus are higher with social learning than without it under contingent pricing.

5. Analysis of Price Guarantee

In this section, we analyze the firm’s optimal decisions under the price guarantee scheme, where the firm offers consumers a full refund of the price difference if the second-period price is lower than the first. We consider two scenarios: $GN$ , where consumers lack access to social learning, and $GL$ , where consumers benefit from social learning.

5.1. Price Guarantee Without Social Learning

Similar to Section 4.1, in the absence of social learning, consumers base their purchase decisions solely on the prior belief about product quality; that is, $E [{\hat{q}}_{1}] = E [{\hat{q}}_{2}] = E [q] = q_{o}$ . Note that under the price guarantee scheme, consumers’ expected utility function purchasing in the first period is adjusted from equation (1) to the following equation (13)

E [U_{1} ({\hat{q}}_{1}, p_{1}, p_{2}, x)] = E [{\hat{q}}_{1} - p_{1} - t x + δ_{c} (p_{1} - p_{2})^{+}]

(13)

where the term

δ_{c} (p_{1} - p_{2})^{+}

represents the refund obtained due to price reduction. The second-period profit, given in equation (10), accounts for the firm’s reimbursement for any price difference. Using backward induction, we derive the firm’s optimal prices,

p_{1}^{GN *}

and

p_{2}^{GN *}

, as summarized in the following proposition.

Proposition 6 (Equilibrium Prices Under

GN

)

The firm’s equilibrium prices are

p_{1}^{GN *} = \frac{q_{o}}{2} and p_{2}^{GN *} = \frac{q_{o}}{2}

(14)

such that

D_{1}^{GN} = \frac{q_{o}}{2 t}

and

D_{2}^{GN} = 0

. Moreover, the firm’s optimal expected profit is independent of

δ_{c}

Unlike the optimal strategy under $CN$ (i.e., no social learning with contingent pricing), the firm should adopt a consistent pricing strategy under the price guarantee scheme in the absence of social learning. This approach is effective because, with consumers’ constant quality belief over time, any price reduction in the second period would compel the firm to refund the price difference and incentivize consumers to delay their purchases—both of which diminish the firm’s profit. Thus, the firm’s optimal strategy is to maintain the second-period price. As a result, consumers will either make purchases in the first period or refrain from purchasing altogether, with no purchases occurring in the second period. Consequently, the firm’s profit is independent of consumers’ strategic waiting.

5.2. Price Guarantee with Social Learning

As in the case of $CL$ in Section 4.2, with social learning, both the firm and consumers who remain in the market at the start of the second period adjust their quality beliefs based on the review outcome, price, and first-period demand. However, under $GL$ , the firm must also incorporate the cost of potential refunds when setting its prices. Similar to Lemma 1, we characterize the first-period demand $D_{1}^{GL}$ with respect to the first-period price $p_{1}^{GL}$ in the following lemma.

Lemma 2 (First-Period Demand w.r.t. Price Under $GL$ )

Given the first-period price $p_{1}^{GL}$ , the first-period demand, $D_{1}^{GL}$ , satisfies the following equation:

\begin{aligned} q_{o} - p_{1}^{GL} - t D_{1}^{GL} + δ_{c} E [{(p_{1}^{GL} - p_{2 R}^{GL})}^{+}] \\ = δ_{c} E [(q_{R}^{GL} - p_{2 R}^{GL} - t D_{1}^{GL})^{+}] for R \in {P, M, N}, \end{aligned}

(15)

where

p_{2 P}^{GL} = {\begin{cases} \frac{2 q_{o} + p_{1}^{GL} - t D_{1}^{GL}}{4}, & for p_{1}^{GL} < \frac{2 q_{o} - t D_{1}^{GL}}{3} \\ p_{1}^{GL}, & for \frac{2 q_{o} - t D_{1}^{GL}}{3} \leq p_{1}^{GL} \leq \frac{2 q_{o} + t D_{1}^{GL}}{3} \\ \frac{2 q_{o} + p_{1}^{GL} + t D_{1}^{GL}}{4}, & for p_{1}^{GL} > \frac{2 q_{o} + t D_{1}^{GL}}{3} \end{cases}

p_{2 M}^{GL} = {\begin{cases} p_{1}^{GL}, & for p_{1}^{GL} \leq \frac{(3 + 2 \sqrt{2}) t D_{1}^{GL}}{2} \\ \frac{2 p_{1}^{GL} + t D_{1}^{GL}}{4}, & for p_{1}^{GL} > \frac{(3 + 2 \sqrt{2}) t D_{1}^{GL}}{2} \end{cases}

, and

p_{2 N}^{GL} = {\begin{cases} p_{1}^{GL}, & for p_{1}^{GL} \leq 16 t D_{1}^{GL} \\ \frac{p_{1}^{GL}}{4}, & for p_{1}^{GL} > 16 t D_{1}^{GL} \end{cases}

are the optimal second-period prices given

D_{1}^{GL}

and

p_{1}^{GL}

Lemma 2 shows that when the review outcome is completely positive, the firm may employ a skim, consistent, or penetration pricing strategy across the two periods. Doing so still ensures that some consumers always purchase in the second period. In contrast, when the review is mixed or negative, the optimal strategy is to either adopt consistent pricing, resulting in zero demand, or mark it down sufficiently to generate additional demand from remaining consumers in the market. Therefore, compared to the exclusive use of consistent pricing in the absence of social learning (Proposition 6), the presence of social learning introduces much richer and more varied pricing strategies under $GL$ .

Based on Lemma 2, we derive the firm’s optimal pricing strategy $(p_{1}^{GL *}, p_{2 R}^{GL *})$ for $R \in {P, M, N}$ under $GL$ and present their properties in Proposition 7.

Proposition 7 (Equilibrium Prices Under

GL

)

(i)
There exists thresholds $0 < δ^{GL} < δ_{P}^{GL} < δ_{M}^{GL} < δ_{N}^{GL} < 1$ such that: (1) $p_{2 P}^{GL } = p_{1}^{GL }$ for $δ_{c} \in (0, δ_{P}^{GL}]$ ; $p_{2 P}^{GL } > p_{1}^{GL }$ for $δ_{c} \in (δ_{P}^{GL}, δ_{N}^{GL}]$ ; otherwise, $p_{2 P}^{GL } < p_{1}^{GL }$ . (2) $p_{2 M}^{GL } = p_{1}^{GL }$ for $δ_{c} \in (0, δ^{GL}]$ and $δ_{c} \in (δ_{P}^{GL}, δ_{M}^{GL}]$ ; otherwise, $p_{2 M}^{GL } < p_{1}^{GL }$ . (3) $p_{2 N}^{GL } = p_{1}^{GL }$ for $δ_{c} \in (0, δ_{N}^{GL}]$ ; otherwise, $p_{2 N}^{GL } < p_{1}^{GL }$ .
(ii)
$p_{1}^{GL }$ and $p_{2 R}^{GL }$ $R \in {P, M, N}$ are piecewise functions that strictly decrease with $δ_{c}$ for $δ_{c} \in (0, δ_{N}^{GL}]$ and are independent of $δ_{c}$ for $δ_{c} \in (δ_{N}^{GL}, 1)$ .
(iii)
The firm’s expected profit and first-period demand are piecewise functions that strictly decrease with $δ_{c}$ for $δ_{c} \in (0, δ_{N}^{GL}]$ and are independent of $δ_{c}$ for $δ_{c} \in (δ_{N}^{GL}, 1)$ .

The close-form expressions of $p_{1}^{GL }$ and $p_{2 R}^{GL }$ for $R \in {P, M, N}$ are shown in the proof.

Proposition 7 demonstrates that under $GL$ , the pricing strategies depend significantly on consumers’ strategic waiting ( $δ_{c}$ ) and the actual product quality in the presence of social learning. These results are illustrated in Figure 5. Specifically, Proposition 7(i) reveals when consumers are highly impatient ( $δ_{c} \leq δ^{GL}$ ), the firm adopts a consistent pricing strategy regardless of the review outcome. This approach encourages early purchases and helps the firm avoid costly refunds. When consumers are moderately patient ( $δ^{GL} < δ_{c} \leq δ_{N}^{GL}$ ), the firm’s second-period pricing strategies become more varied. For completely positive reviews (corresponding to high product quality), the firm may raise the second-period price to capture additional consumer surplus without triggering any refunds. For mixed reviews, the firm may mark down the price to attract demand, but for negative reviews, the firm sticks with consistent pricing. This is because, in the case of negative reviews, consumers are unlikely to purchase in the second period unless the price is significantly reduced, which would erode profits due to refunds. Moreover, within each pricing segment, Proposition 7(ii) demonstrates that the optimal first-period price (weakly) decreases with $δ_{c}$ , reflecting the incentive to encourage early purchases.

Figure 5.
Firm’s optimal prices in the presence of social learning under price guarantee. The symbols $↑$ , $\to$ , and $↓$ denote that the firm raises, keeps constant, and marks down the second-period price, respectively ( $q_{o} = 1$ ). The optimal prices are linear in $q_{o}$ .

When consumers are sufficiently patient ( $δ_{c} > δ_{N}^{GL})$ , the firm is compelled to reduce the second-period price to attract second-period demand even when the review outcome is completely positive. The extent of the markdown depends on the review outcome: The more negative the reviews (indicating lower quality), the steeper the markdown. Interestingly, Proposition 7(ii) and Figure 5 reveal that in this region, prices across both periods remain unaffected by the level of consumer patience, and notably, the first-period price is highest compared to lower ranges of $δ_{c}$ . This is because when consumers are highly patient, the firm benefits more from charging a higher first-period price than from incentivizing early purchases with lower prices—a strategy that negatively affects the review outcome and the firm’s overall profit. With a high first-period price and the price guarantee policy in place, consumers expect the firm to mark down prices in the second period and know they will effectively pay the lower price regardless of when they purchase. As a result, consumers are no longer sensitive to the high first-period price, making first-period demand independent of $δ_{c}$ (Proposition 7(ii)). Moreover, due to the price guarantee, a high first-period price does not significantly reduce first-period demand, resulting in improved second-period quality beliefs (equation (6)) and better review outcomes. In this case, the self-selection effect outweighs the price effect. Consequently, while the firm marks down second-period prices to attract the remaining consumers, these prices can still be higher than those for a lower $δ_{c}$ , particularly when the review outcome is completely positive or mixed.

Lastly, our findings offer an alternative explanation of when and why a firm should adopt a skim pricing strategy. The conventional Price Discrimination Theory (Moorthy and Winter, 2006; Png and Hirshleifer, 1987; Shaffer and Zhang, 2000) suggests that under a price guarantee, a firm should adopt a skim pricing strategy only when it can offset margins lost to price-sensitive consumers (who claim refunds) by setting a high first-period price for non-price-sensitive consumers (who do not claim refunds). We demonstrate that, in the presence of social learning, a skim pricing strategy remains viable even when all consumers are price-sensitive. Specifically, when consumers are highly patient, the combination of social learning and a markdown policy with a high first-period price not only incentivizes early purchases but also enhances the review outcome, ultimately benefiting the firm. Overall, our study uncovers new dynamics introduced by social learning under the price guarantee scheme.

The next section explores the implications of social learning on the firm’s pricing strategy and profitability under the price guarantee scheme.
5.3. Impacts of Social Learning Under Price Guarantee

In this section, we examine the impact of social learning under the price guarantee scheme. The following proposition details the properties of the consumers’ posterior quality belief $q_{R}^{GL}$ , and the corresponding probability ${Pr}_{R}^{GL}$ for $R \in {P, M, N}$ evaluated at the optimal prices. Figure 6 schematically presents the results.

Figure 6.

Posterior beliefs (left) and corresponding probabilities (right) with respect $δ_{c}$ at the optimal prices in the presence of social learning under price guarantee ( $q_{o} = 1$ ). The posterior beliefs are linear in $q_{o}$ and the probabilities are independent of $q_{o}$ .

Proposition 8 (Second-Period Beliefs Under

GL

)

At the optimal prices $p_{1}^{GL *}$ and $p_{2 R}^{GL *}$ , where $R \in {P, M, N}$ ,

(i)
$q_{P}^{GL} > q_{o}$ , $q_{M}^{GL} < q_{o}$ , and $q_{N}^{GL} < q_{o}$ .
(ii)
$q_{R}^{GL}$ are piecewise decreasing functions of $δ_{c} \in (0, δ_{N}^{GL}]$ , and are independent of $δ_{c} \in (δ_{N}^{GL}, 1)$ .
(iii)
${Pr}_{P}^{GL }$ is a piecewise increasing function of $δ_{c} \in (0, δ_{N}^{GL}]$ , and are independent of $δ_{c} \in (δ_{N}^{GL}, 1)$ . ${Pr}_{M}^{GL }$ and ${Pr}_{N}^{GL *}$ are piecewise decreasing function of $δ_{c} \in (0, δ_{N}^{GL}]$ , and are independent of $δ_{c} \in (δ_{N}^{GL}, 1)$ .

Proposition 8(i) reveals that under $GL$ , completely positive reviews raise the posterior quality belief above the prior, while mixed or negative reviews lower it below the prior. Proposition 8(ii) shows that within each price equilibrium segment, the posterior quality belief decreases with consumer patience level, $δ_{c} \in (0, δ_{N}^{G L}]$ . Proposition 8(iii) further reveals that the probability of completely positive reviews increases, while mixed and negative reviews decrease piecewise with $δ_{c} \in (0, δ_{N}^{GL}]$ . These results closely align with those under the contingent pricing scheme (Proposition 3), indicating that for impatient or moderately patient consumers, the qualitative effects of review polarity on posterior beliefs are primarily driven by social learning, with less impact attributed to pricing schemes.

A notable distinction from Proposition 3 arises when consumers are highly patient (i.e., $δ_{c} \in (δ_{N}^{GL}, 1)$ ). By Proposition 7, this is the region where a markdown pricing policy is optimal: the firm sets a high first-period price, and since consumers anticipate a second-period markdown (and refund), first-period demand becomes independent of $δ_{c}$ . Consequently, both posterior quality beliefs and their probabilities remain unchanged with $δ_{c}$ . As shown in Figure 6(a), posterior quality beliefs are higher across all review types compared to lower ranges of $δ_{c}$ . This result highlights that for highly patient consumers, a markdown policy combined with social learning under a price guarantee significantly enhances posterior beliefs about product quality.

Proposition 9 evaluates the impact of social learning on prices by comparing the equilibrium prices under $GN$ and $GL$ .
Proposition 9 (Price Comparison Under Price Guarantee)

(i)
$p_{1}^{GL } > p_{1}^{GN }$ .
(ii)
$p_{2 P}^{GL } > p_{2}^{GN }$ ; $p_{2 M}^{GL } = p_{1}^{GL } > p_{2}^{GN }$ for $δ_{c} \in (0, δ^{GL}]$ and $δ_{c} \in (δ_{P}^{GL}, δ_{M}^{GL}]$ , and $p_{2 M}^{GL } < p_{2}^{GN } < p_{1}^{GL }$ otherwise; $p_{2 N}^{GL } = p_{1}^{GL } > p_{2}^{GN }$ for $δ_{c} \in (0, δ_{N}^{GL}]$ , and $p_{2 N}^{GL } < p_{2}^{GN } < p_{1}^{GL }$ otherwise.

Proposition 9(i) reveals that under the price guarantee scheme, the firm sets a higher first-period price when social learning is present. This is because, unlike the case without social learning, where the firm lowers the first-period price to encourage early purchases and boost initial demand, social learning prompts the firm to raise the first-period price to improve the posterior beliefs of consumers who anticipate a second-period markdown and leverage the price guarantee. Proposition 9(ii) further shows that when the firm adopts consistent pricing for mixed or negative reviews to minimize refunds under the price guarantee, the second-period prices may be higher than those without social learning. These findings stand in stark contrast to Proposition 3, which demonstrates that under contingent pricing, the firm sets a lower first-period price and reduces the second-period price for mixed or negative reviews in the presence of social learning.

The following proposition compares the firm’s profit and consumer welfare.
Proposition 10 (Profit and Welfare Comparison Under Price Guarantee)

The firm’s optimal profit is higher with social learning than without it; consumer surplus is lower with social learning when $δ_{c} \in (0, δ_{P}^{GL})$ and higher when $δ_{c} \in (δ_{P}^{GL}, 1)$ .

Proposition 10 shows that the firm benefits from social learning. However, unlike the contingent pricing scheme where consumer surplus is always higher with social learning, the price guarantee scheme leads to lower consumer surplus for relatively impatient consumers. This is because impatient consumers prefer early purchases, which allows the firm to extract more surplus by maintaining consistent prices. In contrast, patient consumers benefit from frequent markdowns, refunds, and additional product quality information from reviews, all of which increase consumer surplus in the presence of social learning.

6. Comparison of Pricing Schemes

In this section, we compare the firm’s performance and consumer welfare under the contingent pricing and price guarantee schemes. We begin by examining the scenario without social learning.

Proposition 11 (Pricing Scheme Comparison Without Social Learning)

The firm’s optimal profit is strictly higher under contingent pricing than under price guarantee. The consumer surplus is strictly higher under price guarantee for $δ_{c} \in (0, \frac{11 - \sqrt{21}}{10})$ and strictly higher under contingent pricing for $δ_{c} \in (\frac{11 - \sqrt{21}}{10}, 1)$ .

Proposition 11 shows that without social learning, contingent pricing is more profitable for the firm than price guarantee. This is because, although price guarantee effectively mitigates consumer strategic waiting, the associated refund costs limit the firm’s overall profitability. In contrast, the pricing flexibility under contingent pricing allows the firm to manage demand more effectively. However, the effect on consumer surplus is more nuanced. Specifically, the consumer surplus under price guarantee is strictly higher for impatient consumers but lower for patient consumers. This outcome arises from the pricing dynamics described in Propositions 1 and 6. Under price guarantee, the firm adopts a consistent pricing that induces consumers to purchase only in the first period. This price is lower than the first-period price but higher than the second-period price under contingent pricing. Consequently, impatient consumers who tend to purchase early face a higher first-period price under contingent pricing, reducing their surplus. In contrast, patient consumers who are willing to wait benefit from the lower second-period price under contingent pricing. Thus, consumer surplus is lower for impatient consumers but higher for patient consumers under contingent pricing than under price guarantee.

Next, we compare the firm’s performance and consumer surplus under contingent pricing and price guarantee when social learning is present.

Proposition 12 (Pricing Scheme Comparison with Social Learning)

There exists $δ^{'} \in (0, 1)$ such that the firm’s optimal profit is strictly higher under contingent pricing for $δ_{c} \in (0, δ^{'})$ and strictly higher under price guarantee for $δ_{c} \in (δ^{'}, 1)$ . There exists $δ^{″} \in (0, 1)$ such that the consumer surplus is strictly higher under price guarantee for $δ_{c} \in (0, δ^{″})$ and strictly higher under contingent pricing for $δ_{c} \in (δ^{″}, 1)$ . Moreover, $0 < δ^{″} < δ_{P}^{GL} < δ^{'} < 1$ .

Proposition 12 demonstrates that in the presence of social learning, the firm achieves higher profitability under price guarantee when consumers are sufficiently patient. This occurs because price guarantee mitigates consumers’ strategic waiting behavior and reduces their sensitivity to the first-period price. As a result, the firm can set a high first-period price that enhances consumers’ posterior quality beliefs and encourages second-period purchases. In contrast, under contingent pricing, the firm must significantly lower the first-period price to induce early purchase, reducing overall profitability. Consequently, price guarantee coupled with social learning allows the firm to extract greater consumer surplus and achieve superior profitability with patient consumers. When consumers are relatively impatient, the advantages of price guarantee diminish. Its impact on early purchases becomes less significant, and the need to provide refunds under the scheme restricts pricing flexibility, diminishing the firm’s ability to extract consumer surplus and, consequently, overall profitability.

Combining the results with Propositions 5 and 10, Proposition 12 highlights that social learning enhances the firm’s profitability under both contingent pricing and price guarantee schemes. However, its effectiveness depends on consumer patience: Social learning is more impactful under contingent pricing for impatient consumers and under price guarantee for patient consumers. In terms of consumer welfare, one can show that less patient consumers ( $δ_{c} < δ^{‴}$ , where $δ^{‴} \in (δ^{″}, δ^{'}$ )) are strictly better off under price guarantee with no social learning. In contrast, patient consumers benefit from social learning, enjoying greater surplus under contingent pricing than under price guarantee. Figure 7 and Corollary 1 further illustrate this finding, highlighting that, intriguingly, the firm can leverage price guarantee to extract greater consumer surplus from patient consumers, despite the inherent pricing flexibility of contingent pricing. As a result, across all four scenarios ( $CN, CL, GN,$ and $GL$ ), the firm and consumers can be both better off under $CL$ .

Figure 7.

Comparison of profits (left) and consumer surplus (right) at the optimal prices under contingent pricing and price guarantee ( $q_{o} = 1$ and $t = 2.1$ ). The profit and consumer surplus are linear in $q_{o}^{2} / t$ .

Corollary 1

Social learning under contingent pricing results in a win-win situation for both the firm and consumers across both pricing schemes for $δ_{c} \in (δ^{‴}, δ^{'})$ .

7. Extensions

We consider two extensions of the base model: Fully rational consumers in Section 7.1 and the scenario where the firm discounts the second-period profit (i.e., $δ_{f} \in (0, 1)$ ) in Section 7.2. Given the model complexity, deriving closed-form analytical solutions for the $GL$ scenario is challenging, which complicates comparisons across different scenarios. Therefore, we briefly describe the model setup and summarize the key insights in each section, with detailed numerical analyses provided in Electronic Companion EC.3 and EC.4.

7.1. Fully Rational Consumers

In the base model, consumers are assumed to have bounded rationality and can only infer a range of product quality when observing mixed reviews. This assumption reflects the challenges consumers face in accessing complex review information. In this extension, we explore the case of fully rational consumers who can infer the exact product quality from mixed reviews. Specifically, fully rational consumers use the count of positive and negative reviews to precisely identify the product quality, i.e., $q_{M} = q$ . However, when all reviews are either positive or negative, even fully rational consumers cannot ascertain the exact quality. Under this setting, the martingale property still applies, i.e., $E [{\hat{q}}_{2}] = \underset{P}{Pr} q_{P} + \underset{M}{Pr} q_{M} + \underset{N}{Pr} q_{N} = E [{\hat{q}}_{1}] = E [q] = q_{o}$ . In the presence of social learning, the consumers’ second-period quality belief and the respective probability in equations (6) and (7) then become:

\begin{aligned} {\hat{q}}_{2} = {\begin{cases} q_{P} = q_{o} + \frac{p_{1} + t D_{1}}{2}, & for q \in [p_{1} + t D_{1}, 2 q_{o}] \\ q_{M} = q, & for q \in [p_{1}, p_{1} + t D_{1}) \\ q_{N} = \frac{p_{1}}{2}, & for q \in [0, p_{1}) \end{cases} \end{aligned}

(16)

and

{Pr}_{R} = {\begin{cases} {Pr}_{P} = 1 - \frac{p_{1} + t D_{1}}{2 q_{o}}, & for q \in [p_{1} + t D_{1}, 2 q_{o}] \\ {Pr}_{M} = \frac{t D_{1}}{2 q_{o}}, & for q \in [p_{1}, p_{1} + t D_{1}) \\ {Pr}_{N} = \frac{p_{1}}{2 q_{o}}, & for q \in [0, p_{1}) \end{cases} .

(17)

Since there is no difference between bounded and full rationality in the absence of social learning, we focus solely on the $CL$ and $GL$ scenarios. Figure 8 illustrates the equilibrium prices in each scenario, where we present the range of second-period prices corresponding to the actual quality when the review outcome is mixed. We find that under $CL$ , the firm consistently adopts a skim pricing strategy for mixed and completely negative reviews but transitions from skim pricing to penetration pricing for completely positive reviews as consumers’ patience increases. Under $GL$ , the firm prefers consistent pricing for impatient consumers and skim pricing for highly patient consumers. For moderately patient consumers, the firm may adopt skim, consistent, or penetration pricing strategies, depending on both consumers’ patience levels and the actual product quality. These findings align closely with Propositions 2 and 7.

Figure 8.

Equilibrium prices under $CL$ (left) and $GL$ (right) when consumers are fully rational ( $q_{o} = 1$ ).

We further examine the robustness of the main results on the impact of social learning in different pricing schemes. The findings, detailed in Figure EC.1 in Electronic Companion EC.3, are consistent with Propositions 4, 5, 9 and 10, and Corollary 1: with fully rational consumers, social learning leads the firm to lower the first-period price under contingent pricing but increases it under price guarantee. Furthermore, both the firm and consumers benefit from social learning under contingent pricing, while under price guarantee, the firm consistently benefits, and consumers benefit only when they exhibit high patience. Lastly, social learning under contingent pricing creates a win-win situation for both the firm and consumers when consumers are moderately patient. These findings highlight the consistency between the results under full and bounded rationality, validating the robustness of the bounded rationality assumption in the base model.

7.2. Discounted Future Profits

In the base model, we assume $δ_{f} = 1 > δ_{c}$ , reflecting the firm’s emphasis on long-term profit. This extension relaxes the assumption by considering $0 < δ_{f} < 1$ . Note that the models and computational methods remain consistent with the base model. We find that the main insights continue to hold when $δ_{f}$ is sufficiently large and present the detailed results in Electronic Companion EC.4. Observation 1 summarizes the impact of $δ_{c}$ on the firm’s pricing strategies, profits, and consumer surplus.

Observation 1 (Impact of $δ_{f}$ )

(i)
Under $CN, CL$ , and $GL$ , the firm’s optimal profits and first-period prices strictly increase with $δ_{f}$ , whereas consumer surplus strictly decreases with $δ_{f}$ .
(ii)
Under $GN$ , the firm’s optimal profit, prices, and consumer surplus are independent of $δ_{f}$ .

Observation 1 is intuitive. Under $GN$ (i.e., price guarantee without social learning), the firm adopts consistent pricing, inducing consumers to purchase exclusively in the first period. Consequently, the firm’s optimal profit and consumer surplus remain unaffected by $δ_{f}$ . However, in scenarios $CN$ , $CL$ , and $GL$ , an increase in $δ_{f}$ (equivalently, greater emphases on long-term profitability) motivates the firm to raise the first-period price. This adjustment aims to balance the trade-off between immediate profit from early purchases and improved posterior beliefs, which contribute to greater second-period profitability. This more forward-looking strategy allows the firm to capture a greater share of consumer surplus and achieve higher profits. Figures EC.2 to EC.4 depict these results in detail.

Figure EC.5 illustrates the impact of social learning on the firm’s profit and consumer surplus across different pricing schemes as $δ_{f}$ varies. The analysis reveals that when firms are more forward-looking than consumers (i.e., $0 < δ_{c} < δ_{f} \leq 1$ ), social learning enhances the firm’s profit under both contingent pricing and price guarantee. In addition, social learning reduces surplus for impatient consumers but increases it for patient consumers at any given $δ_{f}$ . These findings are largely consistent with Propositions 5 and 10. However, under contingent pricing, a key distinction arises: When $δ_{f} = 1$ , social learning consistently improves consumer surplus, whereas when $δ_{f} < 1$ , it reduces consumer surplus for very impatient consumers. This finding highlights the critical role of the firm’s forward-looking behavior in shaping its ability to extract consumer surplus.

Figure EC.6 further compares the firm’s profit and consumer surplus under contingent pricing and price guarantee. We find that without social learning, the firm tends to perform better under contingent pricing due to its pricing flexibility, while consumers are generally better off under price guarantee when the firm is more forward-looking than consumers. These results are consistent with Proposition 11. However, when consumers are more patient than the firm, the firm benefits more from price guarantee due to its effectiveness in inducing early purchases. Similarly, the results in the presence of social learning are largely consistent with Proposition 12, although the comparison of consumer surplus becomes more nuanced, depending on the firm’s emphasis on long-term profits and the consumers’ patience level.

Combining the observations in Figures EC.5 and EC.6, we observe that social learning under both contingent pricing and price guarantee can result in a win-win situation for both the firm and consumers across both contingent pricing and price guarantee schemes. Figure EC.7 depicts the win-win situation for both pricing schemes. Specifically, the win-win situation arises under price guarantee with social learning when the firm and consumers are both moderately forward-looking. In this scenario, the effectiveness of price guarantee in inducing early purchases is particularly beneficial to both the firm and consumers. Specifically, the firm sets a low first-period price to incentivize early purchases while raising the second-period price only for completely positive reviews. For mixed or negative reviews, the absence of second-period demand under this pricing strategy minimizes the need for markdowns and refunds, ultimately enhancing profitability. At the same time, consumer surplus improves due to better alignment between pricing policies and quality perceptions. In contrast, under contingent pricing, the firm needs to significantly reduce the second-period price for mixed or negative reviews to attract demand, reducing the firm’s profitability compared to price guarantee. This result is distinct from Corollary 1, which states that social learning can only result in a win-win situation under contingent pricing when $δ_{f} = 1$ .
8. Conclusion

This paper investigates the interplay between pricing strategies, social learning, and consumers’ strategic waiting behavior, focusing on two widely adopted pricing schemes: Contingent pricing and price guarantee. We find that social learning enables the firm to adopt more flexible strategies tailored to consumer strategic waiting and review outcomes. Notably, social learning makes price skimming policies optimal under price guarantee for highly strategic consumers, as high first-period prices combined with markdowns enhance review outcomes and firm profitability. This finding unveils that it can be optimal for a firm to adopt price skimming even when all the consumers are price sensitive, contrasting with the conventional price discrimination theory. Importantly, social learning benefits firms under both pricing schemes but impacts consumer surplus differently: Contingent pricing consistently increases consumer surplus, whereas price guarantee benefits patient consumers but lowers surplus for impatient ones.

Our comparative analysis reveals that without social learning, contingent pricing is more profitable due to its pricing flexibility. However, in the presence of social learning, price guarantee becomes more advantageous when consumers are sufficiently patient. Moreover, social learning under contingent pricing creates win-win outcomes for both firms and consumers when consumers are moderately patient and the firm is forward-looking. In contrast, price guarantee is more effective in mitigating consumer strategic waiting, enabling the firm to extract greater surplus from patient consumers in the presence of social learning.

The model extensions validate the robustness of our findings. When consumers are fully rational, the primary insights regarding social learning’s impact on pricing strategies and outcomes remain consistent. Moreover, the partially forward-looking firm scenario highlights additional contexts in which price guarantee can lead to win-win outcomes, expanding the practical applicability of our results.

This study offers novel insights into pricing dynamics, consumer surplus, and firm profitability, enriching the understanding of pricing strategies in the context of social learning and strategic consumer behavior. Future research could extend these findings in several ways. First, integrating more complex consumer review dynamics, such as biased or heterogeneous reviews, may yield deeper insights into social learning. Second, exploring the interplay between pricing strategies, promotional activities, and brand reputation such as discounts or loyalty programs could provide valuable managerial implications. Finally, analyzing these dynamics in competitive markets could further illuminate optimal pricing strategies in the presence of rival firms.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478251329858 - Supplemental material for Pricing in the Presence of Strategic Consumers and Social Learning Under Contingent Pricing and Price Guarantee

Supplemental material, sj-pdf-1-pao-10.1177_10591478251329858 for Pricing in the Presence of Strategic Consumers and Social Learning Under Contingent Pricing and Price Guarantee by Zhong-Zhong Jiang, Jinlong Zhao, Zelong Yi, Ying-Ju Chen and Guang Li in Production and Operations Management

Footnotes

Acknowledgment

We are grateful to the departmental editor (Professor Asoo Vakharia), an anonymous senior editor, and two anonymous referees for very helpful comments and suggestions.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received the following financial support for the research, authorship and/or publication of this article: Zhong-Zhong Jiang was partially supported by the National Social Science Fund of China (23&ZD050), the National Natural Science Foundation of China (NSFC) (71971052, 72192830, and 72192831), and the Fundamental Research Funds for the Central Universities (N2306008 and N2406005). Zelong Yi was partially supported by the NSFC (72171154 and 72222019). Ying-Ju Chen was partially supported by the Research Grants Council of Hong Kong (C6020-21GF).

ORCID iDs

Zhong-Zhong Jiang

Jinlong Zhao

Zelong Yi

Ying-Ju Chen

Guang Li

Supplemental Material

Supplemental material for this article is available online (doi: ).

Notes

How to cite this article

Jiang Z-Z, Zhao J, Yi Z, Chen Y-J and Li G (2025) Pricing in the Presence of Strategic Consumers and Social Learning Under Contingent Pricing and Price Guarantee. Production and Operations Management 34(10): 3063–3081.

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Pricing in the Presence of Strategic Consumers and Social Learning Under Contingent Pricing and Price Guarantee

Abstract

Keywords

1. Introduction

2. Literature Review

2.1. Pricing with Strategic Consumers

2.2. Contingent Pricing with Social Learning

2.3. Price Guarantee Scheme

3. Model Setup and Preliminaries

3.1. Consumers’ Purchase Decisions

3.1.1. Consumer Utility

3.2.1. Review Generation

4.1. Contingent Pricing Without Social Learning

Proposition 1 (Equilibrium Prices Under CN )

Lemma 1 (First-Period Demand w.r.t. Price Under CL )

Proposition 3 (Second-Period Quality Beliefs Under CL )

5. Analysis of Price Guarantee

5.1. Price Guarantee Without Social Learning

Lemma 2 (First-Period Demand w.r.t. Price Under GL )

6. Comparison of Pricing Schemes

Proposition 11 (Pricing Scheme Comparison Without Social Learning)

Proposition 12 (Pricing Scheme Comparison with Social Learning)

7.1. Fully Rational Consumers

Observation 1 (Impact of δ f )

Supplemental Material

sj-pdf-1-pao-10.1177_10591478251329858 - Supplemental material for Pricing in the Presence of Strategic Consumers and Social Learning Under Contingent Pricing and Price Guarantee

Footnotes

Acknowledgment

Declaration of Conflicting Interests

Funding

ORCID iDs

Supplemental Material

Notes

How to cite this article

References

Supplementary Material

Proposition 1 (Equilibrium Prices Under $CN$ )

Lemma 1 (First-Period Demand w.r.t. Price Under $CL$ )

Proposition 3 (Second-Period Quality Beliefs Under $CL$ )

Lemma 2 (First-Period Demand w.r.t. Price Under $GL$ )

Observation 1 (Impact of $δ_{f}$ )