Competitive Markovian pricing

Abstract

Dynamic pricing is often complicated by strategic customer behavior. One tactic utilized by retailers to manage strategic customer behavior, known as Markovian pricing, is to offer price discounts at random intervals to prevent customers from predicting when the next discount will occur, thereby simplifying their strategic waiting behavior. In this article, we study Markovian pricing in competitive settings. We show that retailers can effectively adopt Markovian pricing in competitive environments, establish the optimality of flash discounts under competitive Markovian pricing, and find surprisingly that increased levels of competition may benefit both retailers. We confirm the robustness of these insights and also establish their limits of applicability in two model extensions. Our findings suggest that retailers engaging in competitive Markovian pricing should refrain from naïvely applying common wisdom toward third-party price-monitoring and comparison services and reconsider the efforts in growing their loyal customer base, and more broadly highlight the unique properties of competitive Markovian pricing.

1. Introduction

Retail is a vast and important industry of the global economy. In the United States alone, the total retail sales were forecast to amount to more than 5 trillion US dollars in 2024 (Sabanoglu, 2024). Pricing is one of the most important operational decisions in retail. As The Economist (2013) notes, boosting sales volumes and cutting costs may not be viable for businesses in the “age of austerity” and “prices are all that is left.” Retailers regularly experiment with different pricing strategies to improve their profits (Grewal Levy Marketing News, 2016). Constrained by anti-price-discrimination regulations such as the Robinson-Patman Act ¹ which generally outlaw offering different prices to different customers simultaneously, retailers often resort to dynamic pricing, namely offering different prices to all customers over time, to potentially price-discriminate heterogenous customers. For example, Besbes and Lobel (2015) show that retailers may vary prices to capture low-value and patient customers while still being able to extract revenue from high-value and impatient customers.

Retailers’ efforts to improve revenues with dynamic pricing are however undermined by their customers’ strategic waiting behavior: if a retailer’s dynamic pricing strategy exhibits a pattern (e.g., discounting on the last day of each month), savvy customers may exploit the pattern and strategically postpone purchases, compromising the retailer’s dynamic pricing efforts. This insight dates back to Coase (1972) who famously asserts that, if all customers wait strategically, even a monopolist earns no profit. More recently, Aviv and Pazgal (2008) show that dynamic pricing cannot extract as much revenue from strategic customers as from non-strategic customers, as some revenue must be forfeited in exchange for customer compliance. Besbes and Lobel (2015) find that optimal dynamic pricing policies for strategic customers can be highly complex (e.g., cyclic pricing with non-monotone prices within each cycle). The practical effectiveness of such pricing strategies is however questionable given their complexities and that real-life customers may not possess the level of sophistication based on which these pricing strategies are derived.

Customers’ strategic waiting behavior hinges upon their ability to (at least partially) predict future price trajectories. In response, many retailers begin to offer discounts at random intervals (notwithstanding popular annual discounting events such as the Black Friday in the United States and the Singles’ Day in China). Wu and Zhao (2023) collect price trajectories of various products on JD.com (a major Chinese e-commerce retailer) and find that for many products (e.g., 62% of trolley cases and 50% of computers), the price trajectories contain apparently random discounts. Figure 1 (reproduced from Wu and Zhao, 2023) presents several such examples. In these examples, the product is mostly sold at the regular price but occasionally sold at a fixed discount. The discounts tend to be brief and their timing bears no obvious pattern. The apparently random discounts make it extremely difficult—if not impossible—for customers to predict when the next discount will occur. Chen et al. (2023) and Wu et al. (2025) provide additional evidence of random discounts.

Figure 1.

Online price trajectories from JD.com: (a) smart toilet, (b) watch, (c) spinning bike, and (d) trolley case.

The intuition behind retailers adopting random discounts is that customers facing random discounts cannot predict when future discounts would occur and thus cannot employ sophisticated waiting strategies; this can potentially improve the effectiveness of retailers’ dynamic pricing strategies. Empirical evidence supports this intuition. Février and Wilner (2016) analyze pricing and purchasing data and find that purchase quantities do not drop significantly prior to discounts, suggesting that most customers are unable to predict future discounts. Moon et al. (2018) collaborate with a North American specialty retailer to implement randomized markdowns and find the random discount policy to yield an 81% improvement in profit compared with conventional dynamic pricing practices.

This intuition is formalized in the Markovian-pricing model which assumes that the price of the next period/instant only depends stochastically on the current price and not on how long the price has been sustained (satisfying the Markov property) (Chen et al., 2023; Gökgür and Karabatı, 2023; Wu et al., 2025). The memorylessness of the Markovian-pricing model means that customers cannot predict future discounts, thus capturing actual customer behavior as evidenced by Février and Wilner (2016). When limited to explicit regular and discounted prices, Markovian pricing leads to geometric (discrete-time) or exponential (continuous-time) distributions of discount periods and inter-discount intervals. Wu et al. (2025) study Markovian pricing in a continuous-time single-retailer setting and find that the retailer’s dynamic (i.e., non-constant) optimal Markovian-pricing strategy implements flash discounts, namely discounts that last only for an instant. Similarly, Chen et al. (2023) and Gökgür and Karabatı (2023) show in discrete-time single-retailer Markovian-pricing models that the optimal discounts last only for a period. Such flash discounts achieve a good level of price discrimination by immediately selling full-price products to high-value and impatient customers, and inducing low-value and patient customers to wait for the next random discount to make purchases. Pricing practices reminiscent of flash discounts are widely seen. For example, Cui et al. (2019) study Amazon’s “lightning deals” that are deeply discounted for a short period of time. Figure 1(a) also depicts brief discounts in comparison to inter-discount intervals.

The existing Markovian-pricing literature (e.g., Chen et al., 2023; Wu et al., 2025) sheds light on its management and value when the policy is implemented by one retailer. By contrast, most practical retail environments involve competing retailers. Springboard (2021) notes that “fierce competition is the hallmark of modern retail.” Tuttle (2013) documents how a local grocery chain, WinCo, became “Walmart’s worst nightmare,” despite the latter being one of the most powerful players in the retail industry. Retailers often strive to soften competition by fostering customer loyalty (Narciso, 2023) because loyal customers may ignore cheaper options from competitors ( Zinrelo, 2023). On the other hand, notwithstanding common sense, there are also documented cases of retailers referring customers to competitors (Zhang et al., 2018, 2023).

As we have shown thus far, Markovian pricing and competition are both prevalent occurrences in the retail industry, yet there have not been studies of Markovian pricing in competitive settings. Several questions remain open in competitive Markovian pricing. Do equilibrium Markovian-pricing strategies still implement flash discounts? Do retailers offer more and/or deeper discounts with increased competition? And how does increased competition (both bilaterally and unilaterally) affect the competing retailers’ profits?

To investigate these questions, we consider Markovian pricing in a competitive setting. Specifically, we model two competing retailers implementing random price discounts for one product. Each retailer has a Poisson stream of incoming strategic customers with heterogenous (high or low) value and patience and exponentially distributed lifetimes. Each customer stream consists of loyal and opportunistic customers; the former would only make purchases from the host retailer whereas the latter are open to making purchases from both retailers. The proportion of opportunistic customers reflects the level of competition between the retailers. The retailers choose their discount depths and (random) frequencies and durations to maximize their respective profits anticipating the strategic customers’ optimal purchasing behavior (including waiting). Because of the model’s memorylessness, a strategic customer’s decision at any moment must only depend on the current price(s) at the retailer(s). More specifically, a strategic customer would either make an immediate purchase, immediately leave the market without a purchase, or wait indefinitely for the next discount until their lifetime expires.

We first analyze the model with all loyal customers (the no-competition case). In this case, the model is effectively reduced to a single-retailer Markovian-pricing model, and the optimality of flash discounts (Wu et al., 2025) is recovered. Next, we analyze the model with mixed loyal and opportunistic customers and find that, any equilibrium dynamic (non-constant) Markovian-pricing strategies must still implement flash discounts, thus extending the single-retailer Markovian-pricing insight to competitive settings.

We then investigate the impact of retailer competition under Markovian pricing. Surprisingly, we find that increased competition, both bilaterally (more opportunistic customers at both retailers) and unilaterally (more opportunistic customers at one retailer), may lead to increased profits of both retailers. The reason is that more opportunistic customers may drive both retailers to implement more frequent flash discounts, and because the opportunistic customers have access to more frequent discounts, much shallower discounts are sufficient to induce low-value and patient customers to wait and high-value and impatient customers to make immediate purchases, achieving price discrimination more efficiently. In addition, more customers make a purchase under increased competition, leading to an overall efficiency increase in capturing customers. Therefore, when adopting Markovian pricing, competing retailers may benefit from increased competition, both bilaterally and unilaterally.

We further check our results’ robustness in two model extensions. First, we consider two competing retailers of different customer arrival rates (capturing retailers of different sizes). We confirm that any equilibrium dynamic Markovian-pricing strategies must still implement flash discounts. We further observe that both retailers may still benefit from increased competition when their customer arrival rates do not differ too much. However, when their customer arrival rates differ significantly, increased competition tends to benefit the retailer with fewer customers at the expense of the retailer with more customers. Second, real-life customers may lack the sophistication to infer the retailers’ pricing policies; as such, we consider exogenous customer decisions (rather than rational decisions based on the retailers’ pricing policies). We confirm that any equilibrium dynamic Markovian-pricing strategies must still implement flash discounts, and partially recover the finding that both retailers may benefit from increased competition.

Overall, we extend the optimality of flash discounts established in the single-retailer Markovian-pricing literature to competitive Markovian-pricing settings, and show that increased competition, both bilaterally and unilaterally, may actually benefit the competing retailers under competitive Markovian pricing. Our findings support competing retailers’ adoption of Markovian pricing and flash discounts to manage strategic customers and suggest that retailers adopting Markovian pricing should refrain from naïvely applying common wisdom toward price-monitoring and comparison services and reconsider the efforts in growing their loyal customer base.

The remainder of the article is organized as follows. Section 2 reviews the related literature. Section 3 introduces the setting and model. Section 4 contains the equilibrium analyses, based on which Section 5 investigates the impact of competition. Section 6 treats two model extensions that confirm the robustness of our findings and also establish their limits of applicability. Section 7 concludes the article. All nontrivial proofs are found in e-Companion A.

2. Literature review

This article concerns pricing, which is a central topic in revenue management. Early work on revenue management focuses primarily on allocating limited inventory without considering strategic customer behavior (Federgruen and Heching, 1999; Feng and Gallego, 1995; Gallego and van Ryzin, 1994; Xu and Hopp, 2006); see Elmaghraby and Keskinocak (2003) for a comprehensive overview of this stream of literature. A large and growing literature on revenue management accounts for strategic customer behavior with capacity constraints (Dilmé and Li, 2019; Elmaghraby et al., 2008; Gallien, 2006; Liu and van Ryzin, 2008; Su, 2007). These papers generally aim to characterize inventory- and time-dependent optimal price trajectories that are predictable to certain extents. Inventory constraint creates scarcity; with limited inventories and/or capacities, pricing is utilized to maximize retailer profits by rationing scarce supplies.

There is another stream of literature on dynamic pricing without supply limitations that focuses on intertemporal price discrimination with strategic customer behavior. Stokey (1981) considers a fixed customer population and shows that, under customers’ rational expectations, the optimal price path is a decreasing one. Conlisk et al. (1984) extend Stokey’s model by considering incoming customers, and show that the optimal pricing policy is a constant regular price accompanied by periodic discounts. Besbes and Lobel (2015) generalize the model of Conlisk et al. (1984) and find the same basic cyclic optimal pricing pattern with more intricate structures within each cycle. Lin et al. (2018) and Yin et al. (2026) further show that the interactions of strategic customer behavior and the responses of supply chain members may yield counterintuitive outcomes. In all of these papers, the optimal pricing paths are once again predictable and must account for sophisticated strategic customer behavior, which on the one hand may significantly compromise the retailers’ dynamic pricing efforts, and on the other hand raises the question of whether the theoretically predicted behavior can be expected from real-life customers who may be of limited sophistication.

Chen et al. (2023), Wu et al. (2025), and Wu et al. (2014) attempt to address the aforementioned challenges of sophisticated strategic customer behavior by studying the type of dynamic pricing policies known as Markovian pricing, namely pricing policies in which the next period/instant’s price only depends stochastically on the current price and not on how long it has been sustained. The memoryless property, a feature of Markovian-pricing policies, means that customers cannot predict future prices, thus capturing similar real-life customer behavior as evidenced by Février and Wilner (2016). Wu et al. (2014) investigate the impact of Markovian pricing on a firm’s profitability, where high-valuation customers wait for at most one period and low-valuation customers wait for multiple periods. Chen et al. (2023) consider risk-neutral incoming customers with varying valuations and patience levels. They characterize conditions under which fully randomized pricing policies outperform cyclic pricing policies, and then discuss Markovian pricing as an extension of randomized pricing and show that it may in turn outperform randomized pricing. Wu et al. (2025) develop a continuous-time Markovian pricing framework and study the effect of offering price guarantees. Markovian pricing is found to also be an effective price-discrimination strategy when customers have heterogeneous valuations and stockpile-up-to levels (Gökgür and Karabatı, 2023) and in omni-channel retailing (Wu et al., 2020). All the aforementioned papers on Markovian pricing concern monopolist retailers. We contribute to the Markovian-pricing literature by extending the study to a competitive setting.

Dynamic pricing in competitive settings has been the subject of a number of studies. Some studies lead to randomized pricing policies being adopted. Sobel (1984) extends Conlisk et al. (1984)’s work to a competitive setting and shows that competing sellers follow a randomized cyclic pricing pattern where all sellers charge a regular price for a fixed period of time and then may offer a discount probabilistically. After a discount is offered, the cycle is reset and repeated. Sinitsyn (2017) considers prescheduled sales and shows that sellers may adopt mixed (randomized) pricing strategies in an equilibrium. Other studies lead to deterministic pricing strategies being adopted, including some cases of Sinitsyn (2017), and Liu and Zhang (2013) who consider competitive dynamic pricing for vertically differentiated products. Whereas these papers study general dynamic pricing, we focus on competitive Markovian pricing and reveal unique properties of this class of pricing policies.

A framework to investigate the impact of varying levels of competition is to assume some customers to be loyal (or captive) and others to be brand switchers (or contested). A stream of literature on retail competition adopting this framework stems from Varian (1980) and Narasimhan (1988). This literature generally concludes that reduced competition is associated with increased retailer profits (Kuksov and Zia, 2021; Narasimhan, 1988). Armstrong and Vickers (2022) consider a more general model under this framework with an arbitrary number of retailers and characterize the equilibrium pricing patterns. Kuksov et al. (2017) and Kuksov and Zia (2021) modify the framework to include a third segment of customers whose shopping behavior depends on their exposure to advertisements and search costs. Kuksov et al. (2017) provide an explanation for why retailers may allow their competitors to advertise in their own stores. Kuksov and Zia (2021) show that a retailer’s advantage in customer loyalty benefits the retailer by making its store a search hub for non-loyal customers. We however show that competition may increase retailer profits in the Markovian-pricing framework. Furthermore, with the notable exception of Gangwar et al. (2021), this stream of literature generally employs single-period models that do not capture customers’ strategic waiting behavior (Gangwar et al., 2021 consider a dynamic pricing model where all customers have unit demands per period and loyal customers can stockpile for future consumption). In static models of competition for brand switchers, random discounts typically arise from the lack of a pure-strategy equilibrium (Varian, 1980), where the retailer with the lowest realized price captures all brand switchers. In contrast, our dynamic random (Markovian) pricing serves an entirely different purpose, namely intertemporal price discrimination: it allows patient customers to potentially make purchases at a discount while capturing full revenues from impatient customers. As such, we contribute to the literature on market competition by highlighting the unique Markovian-pricing setting where random pricing is adopted for intertemporal price discrimination and competition may increase retailer profits.

3. Model

We consider a setting in which two profit-maximizing retailers sell an identical product over an infinite continuous-time horizon, with the product’s procurement cost normalized to 0. Each of the two retailers faces a Poisson stream of incoming strategic customers with a normalized arrival rate of 1 (an extension that allows the arrival rates to differ between the two retailers is discussed in Section 6.1). Customers are heterogenous in their valuations of the product as well as their patience. Higher-value customers are more likely to be less patient, for example due to them assigning higher values for their time. Wu et al. (2025) show in their single-retailer Markovian-pricing setting that positively correlated product valuations and waiting costs are necessary for alternating prices to occur, which we know is the case in practice (e.g., Figure 1). For tractability, we adopt the simplifying assumption that either stream of customers consists of two types where a high- (low-) type customer has product valuation $v_H$ ( $v_L$ ) and waiting cost $w_H$ ( $w_L$ ), with $v_H>v_L>0$ and $w_H>w_L\ge 0$ ; that is, perfectly correlated product valuations and waiting costs. We denote the proportion of high-type customers by $\alpha \in (0,1)$ . Consistent with the classic literature on retailer competition (Narasimhan, 1988; Sinitsyn, 2008, 2012), we assume that either stream of customers consists of a $\gamma \in [0,1]$ proportion of opportunistic customers who are open to purchasing the product from both retailers and a $1-\gamma$ proportion of loyal customers who would only consider purchasing the product from their host retailer, where $\gamma$ captures the level of competition. We allow the two retailers to have different levels of customer loyalty and use the subscripts ${}_1$ and ${}_2$ to indicate the retailer being referenced. Customer valuation/patience and loyalty are assumed independent. ² Each customer may exist in the market up to an exponentially distributed lifetime with parameter $\lambda$ (a similar assumption shared by Wu et al., 2025). We note that the lifetime is not a customer decision and is unrelated to customer patience; it is intended to capture the potential exogenous change of a customer’s need or situation when they wait for a discount. The exponential distribution assumption captures the unpredictability of such exogenous changes of a customer’s need or situation and also ensures the model’s Markov property (memorylessness).

Inspired by the practical prominence of random discounts (see Section 1), we adopt the Markovian-pricing framework in our competitive setting. Specifically, following Chen et al. (2023) and Wu et al. (2014, 2025), we assume that, instead of directly setting prices at every instant, each retailer chooses continuous transition rates between a regular and a discounted price and allows the retail price to randomly transition between the regular and discounted prices as a continuous-time Markov chain defined by these transition rates. The price Markov chains, alongside the customers’ exponential lifetimes, ensure that the model is a Markovian decision-making environment; that is, all customers’ optimal decisions are solely based on the system’s current state, namely the current prices at both retailers. While it is extremely challenging to know with certainty whether real-life retailers truly randomize the timing of discounts, we do know that many price trajectories are apparently highly random (Figure 1); that some retailers are actively and successfully experimenting with truly randomized discounts (Moon et al., 2018); and that discounts are often perceived as random and unpredictable by customers (Février and Wilner, 2016). We note that our assumption of truly randomized timing of discounts yields a rigorous and tractable model where the price trajectory is unpredictable to fully strategic customers—an outcome that is consistent with anecdotes, experimental exercises, and empirical evidence.

The assumption that the price trajectory alternates between a regular and a discounted level is commonly adopted by the literature (Chen et al., 2023; Wu et al., 2014, 2025). Whereas this setup helps simplify the model and maintain its tractability, it is nonetheless also consistent with common retail practice. Figure 1 clearly illustrates the prices of several goods alternating between regular and discounted levels. Similarly, Hosken et al. (2000) document the practice that “price is usually at a ‘high’ level, and then periodically declines to a lower level for a short period of time.” We assume the regular price at both retailers to be set exogenously at $p_r=v_H$ so that low-type customers never make purchases at the regular price (the specific value of $p_r$ is an inconsequential convenience; all analyses can be generalized to any $p_r\in (v_L,v_H]$ ). An exogenous regular price is consistent with the prevalent practice that competing retailers usually settle on a consensus regular price with a reasonable margin (Ross, 2022). It is also a common assumption in studies of retailers’ competition through discounts (e.g., Sinitsyn, 2017). On the other hand, each retailer $i$ can set its discounted price $p_{di}$ , as well as the discount rate ${\mu }_i$ and revert rate ${\mu }_i^{\prime }$ , respectively, defined as the transition rates from $p_r$ to $p_{di}$ and in reverse; that is, the random price trajectory at retailer $i$ , $i=1,2$ forms a continuous-time Markov chain where the regular (discounted) price $p_r$ ( $p_{di}$ ) persists for an exponentially distributed period of time with parameter ${\mu }_i$ ( ${\mu }_i^{\prime }$ ) before transitioning to the discounted (regular) price $p_{di}$ ( $p_r$ ). We require $p_{di}\le v_L$ so that low-type customers may make purchases at the discounted price. Finally, following Wu et al. (2025), we assume that each price change incurs a cost of $c/2$ on the retailer (and thus the roundtrip cost of each price discount is $c$ ). For brick-and-mortar retailers, $c$ may represent the labor cost of physically changing a price tag. For online retailers, $c$ primarily captures the administrative and managerial overheads associated with price changes. The cost also captures the risk of too many price changes potentially confusing customers (Dholakia, 2015).

The sequence of our game goes as follows: at the beginning of the game, both retailers simultaneously set their own discounted prices $p_{di}$ and the two transition rates ${\mu }_i$ and ${\mu }_i^{\prime }$ , $i=1,2$ , and then allow their price trajectories to stochastically transition between the regular and discounted prices following continuous-time Markov chains defined by these transition rates while incurring the resulting price-changing costs. After the retailers set their Markovian-pricing strategies, customers arrive stochastically and make decisions to maximize their expected utilities. We solve the game by backward induction. First, in Section 3.1, we characterize the customers’ optimal decisions given any pair of Markovian-pricing strategies chosen by the two retailers. Then, in Section 4, we analyze the simultaneous pricing game between the two retailers, where each retailer determines its own Markovian-pricing strategy while anticipating both the other retailer’s pricing strategy and the customers’ resulting optimal responses.

3.1. Customer decision

In order to solve for the retailers’ Markovian-pricing equilibria, we first characterize different types of customers’ decisions facing competitive Markovian pricing. Recall that at this time the retailers’ pricing strategies $(p_{di},{\mu }_i,{\mu }_i^{\prime })$ , $i=1,2$ have already been determined, and we assume that the customers are aware of the retailers’ pricing strategies (e.g., inferred through third-party price-monitoring and tracking services such as Keepa and CamelCamelCamel) and accordingly make their optimal decisions (in Section 6.2, we explore an alternative model setup where the customers lack the sophistication to infer the retailers’ pricing policies and instead exhibit exogenous behavior, and show that our main insights are preserved). Specifically, at every instant, each customer can choose whether to make an immediate purchase, wait in the market, or leave the market to maximize their expected utility. However, as explained earlier, due to the Markovian model’s memorylessness, all customers’ optimal decisions must only depend on the system’s current state (i.e., the current prices at both retailers) and do not depend on how long these prices have been sustained. As a result, customers only need to make/update their decisions when they arrive and when a price change occurs without loss of generality.

We first consider loyal customers who would only make purchases from the host retailer (to which they arrive). As such, their decisions are essentially identical to those of the customers in the single-retailer Markovian setting. To simplify exposition, we codify a loyal customer’s optimal decision as either to buy (from the host retailer at the current price and leave the market), to wait (in the market without making a purchase), or to leave (the market without making a purchase). The following lemma characterizes loyal customers’ optimal decisions, where $R$ ( $D$ ) denotes the state of the host retailer currently offering the regular (discounted) price. This lemma is consistent with Lemmas 2 and 3 of Wu et al. (2025) (all nontrivial proofs here and henceforth can be found in e-Companion A).

Lemma 1 Loyal Customer Decision

At anytime, a loyal customer’s optimal decision depends on the current state of their host retailer $i$ as follows:

State $D$ : A loyal customer buys.

State $R$ :

–

A high-type loyal customer waits if $w_H\le {\mu }_i(v_H-p_{di})$ . Otherwise, s/he buys.

–

A low-type loyal customer waits if $w_L\le {\mu }_i(v_L-p_{di})$ . Otherwise, s/he leaves.

Lemma 1 is quite intuitive. When loyal customers are offered a discounted price by their host retailer, they will make a purchase (as we required $p_{di}\le v_L$ , ensuring the discounted price is affordable to all). When facing a regular price, however, loyal customers will wait if their waiting cost is sufficiently low; otherwise, they will make a spot decision of purchase or leaving, which only depends on their valuation of the product. The waiting threshold, $\mu (v-p_d)$ , depends critically on the retailer’s Markovian-pricing strategy. Specifically, loyal customers are more likely to wait if (1) the retailer offers discounts more frequently on average (i.e., a higher discount rate $\mu$ ), or (2) the retailer offers a deeper discount (i.e., a lower discounted price $p_d$ ). e-Companion B contains more sensitivity analysis based on Lemma 1.

We next consider opportunistic customers who would make purchases from either retailer. Accordingly, we re-codify an opportunistic customer’s decision as to buy (from the retailer with the strictly lower price or from the host retailer if both retailers offer the same prices, and leave the market). The decisions to wait and to leave remain unchanged. Without loss of generality, we denote the retailer charging a weakly higher discounted price as retailer 1; in other words, we assume $p_{d2}\le p_{d1}(\le v_L)$ . For an opportunistic customer, the system consisting of two retailers may be in one of four states: both retailers offering regular prices; both retailers offering discounted prices; and two states where one offering a regular price and the other offering a discounted price. We, respectively, denote these four states by $RR$ , $DD$ , $R{D}_1$ , and $R{D}_2$ , where $R{D}_i$ refers to only retailer $i$ offering a discounted price. We denote an opportunistic customer’s optimal value function for state $s$ by $J(s)$ . It is immediate that $J(DD)=J(R{D}_2)=v-p_{d2}$ because $p_{d2}$ is the lowest price possible. Therefore, we only need to determine $J(RR)$ and $J(R{D}_1)$ .

We apply the standard uniformization technique (Whitt, 2006, p. 14) to the continuous-time Markov chain. Specifically, we add appropriate fictitious Poisson transitions to the same states such that, including the fictitious transitions, the transition rate from each state always equals $\zeta :=\lambda +{\mu }_1+{\mu }_2+{\mu }_1^{\prime }+{\mu }_2^{\prime }$ . This yields a continuous-time Markov chain with a uniform transition rate from all states that is easier to present and analyze. With uniformization, the optimality equations that determine a high-/low-type opportunistic customer’s optimal values in states $RR$ and $R{D}_1$ can be presented as follows (recall that $p_r=v_H$ ):

$$\begin{array}{rl}J(RR)=& max\{v_{H/L}-p_r,\frac{{\mu }_1^{\prime }+{\mu }_2^{\prime }}{\zeta }J(RR)+\frac{{\mu }_1}{\zeta }J(R{D}_1)\\ & +\frac{{\mu }_2}{\zeta }J(R{D}_2)-\frac{w_{H/L}}{\zeta },0\},\\ J(R{D}_1)=& max\{v_{H/L}-p_{d1},\frac{{\mu }_1^{\prime }}{\zeta }J(RR)+\frac{{\mu }_2^{\prime }+{\mu }_1}{\zeta }J(R{D}_1)\\ & +\frac{{\mu }_2}{\zeta }J(DD)-\frac{w_{H/L}}{\zeta },0\}.\end{array}$$

In these equations, the three terms within each maximization, respectively, represent the values of buying, waiting, and leaving. The second terms that represent the value of waiting consist of the opportunistic customer’s optimal values in all states multiplied by the uniformized transition probabilities (including fictitious transitions to the same states) less the expected waiting cost until the next uniformized transition, $w_{H/L}/\zeta$ . Solving these optimality equations yields the following lemma to characterize the opportunistic customers’ optimal decisions:

Lemma 2 Opportunistic Customer Decision

At anytime, an opportunistic customer’s optimal decision depends on the current state of the two retailers as follows:

States $DD$ , $R{D}_2$ : An opportunistic customer buys.

State $R{D}_1$ : An opportunistic customer waits if $w_{H/L}\le {\mu }_2(p_{d1}-p_{d2})-\lambda (v_{H/L}-p_{d1})$ . Otherwise, s/he buys.

State $RR$ :

–

A high-type opportunistic customer waits if $w_H\le {\mu }_1(v_H-p_{d1})+{\mu }_2(v_H-p_{d2})$ . Otherwise, s/he buys.

–

A low-type opportunistic customer waits if $w_L\le {\mu }_1(v_L-p_{d1})+{\mu }_2(v_L-p_{d2})$ . Otherwise, s/he leaves.

The opportunistic customers’ optimal decisions are structurally similar to those of loyal customers: they will buy at the best possible price $p_{d2}$ when it is available (states $DD$ and $R{D}_2$ ); and when facing regular prices (state $RR$ ), high-type customers may buy or wait and low-type customers may wait or leave depending on their waiting costs (their waiting thresholds are also structurally comparable to those of the loyal customers). A unique strategy of the opportunistic customers is that they may forgo buying at the discounted price $p_{d1}$ and opt to wait for the even better price $p_{d2}$ from the other retailer if their waiting costs are sufficiently low (state $R{D}_1$ ). The following corollary characterizes the conditions under which this strategy is more likely to occur (e-Companion B contains more sensitivity analysis based on Lemma 2).

Corollary 1

An opportunistic customer is more likely to forgo buying at a discounted price and opt to wait for the better discounted price when one or more of the following conditions holds: (1) the retailer with the better discounted price offers discounts more frequently on average (i.e., a higher discount rate ${\mu }_2$ ); (2) the better discounted price is even better (i.e., a smaller $p_{d2}$ ); (3) the other discounted price becomes even less attractive (i.e., a higher $p_{d1}$ ); or (4) the customer’s expected lifetime is longer (i.e., a smaller $\lambda$ ).

Finally, we note that it is possible for a customer executing optimal decisions to still realize a negative utility. For example, suppose a loyal customer’s optimal decision is to wait for the next discount which never arrives before the customer’s lifetime expires in the realized price trajectory. In this case, the customer’s realized utility consists only of waiting costs and is negative. However, the decision to wait is optimal in expectation, which does not contradict with negative realized utilities along certain sample paths.

4. Equilibrium analysis

With optimal customer decisions characterized in Section 3.1, we next analyze the two retailers’ Markovian-pricing equilibria; that is, their choices of discounted price $p_{di}$ , discount rate ${\mu }_i$ , and revert rate ${\mu }_i^{\prime }$ . We first analyze the special cases of no competition ( $\gamma =0$ ) and full competition ( $\gamma =1$ ) and then analyze partial competition ( $0<{\gamma }_i<1$ ) allowing different proportions of opportunistic customers at each retailer $i$ .

4.1. No competition

In the special case where all customers are loyal to and only purchase from their host retailers ( $\gamma =0$ ), each retailer effectively operates in its own market without competition. In this sense, our no-competition case resembles the single-retailer model of Wu et al. (2025), and the following proposition is reminiscent of their Proposition 4. For expositional simplicity, we define $\beta :=1-\alpha$ as the proportion of low-type customers. The superscript ${}^n$ denotes no competition. All descriptions of customer decisions in this article are almost surely; that is, notwithstanding zero-probability exceptions.

Proposition 1 No Competition

With $\gamma =0$ , each retailer’s optimal Markovian-pricing strategy may be of three types:

Flat regular price: Each retailer sets a flat regular price $p_r=v_H$ . High-type customers immediately buy from their host retailers. Low-type customers immediately leave. Each retailer’s profit rate is $\alpha v_H$ .

Flat discounted price: Each retailer sets a flat discounted price $p_{di}=v_L$ . All customers immediately buy from their host retailers. Each retailer’s profit rate is $v_L$ .

Flash discounts: If and only if

$$\begin{array}{rl}& \lambda <\frac{\beta v_L}{c},w_L<\frac{(\beta v_L-c\lambda {)}^2}{4\beta c}\end{array}$$

$$\begin{array}{rl}{v}_H>& \frac{\alpha v_L-c\lambda +2\sqrt{\beta c(\lambda v_L+w_L)}}{\alpha },\\ w_H>& w_L-\lambda (v_H-v_L)\\ & +\frac{(v_H-v_L)(\sqrt{\beta c(\lambda v_L+w_L)})}{c},\end{array}$$

each retailer sets the discounted price

$$p_d^n=v_L-\frac{c{w}_L}{\sqrt{\beta c(\lambda v_L+w_L)}-c\lambda }$$

and the discount and revert rates

$${\mu }^n=-\lambda +\frac{\sqrt{\beta c(\lambda v_L+w_L)}}{c},{\mu }^{\prime n}=\mathrm{\infty }.$$

High-type customers buy immediately from their host retailers at price $p_r=v_H$ . Low-type customers wait for the next discount from their host retailers to buy at price $p_d^n$ within their lifetimes. The fraction of satisfied low-type customers is

$${\varphi }^n=1-\frac{c\lambda }{\sqrt{\beta c(\lambda v_L+w_L)}}.$$

Each retailer’s profit rate is

$${\pi }^n=c\lambda +\alpha v_H+\beta v_L-2\sqrt{\beta c(\lambda v_L+w_L)}.$$

We note that the optimal revert rate of ${\mu }^{\prime n}=\mathrm{\infty }$ in the flash-discount strategy is a slightly informal way to express the result that the retailers’ profits monotonically increase in ${\mu }^{\prime n}$ . Its practical implication is however clear: the retailers should mostly charge the regular price for high-type customers while randomly implement discounts that last only for an instant (i.e., flash discounts) to extract additional revenue from accumulated low-type customers waiting for discounts. Flash discounts are actually widely documented in practice (e.g., Cui et al., 2019 and Figure 1(a)). Flash discounts are the most interesting strategy in Proposition 1. On the one hand, this strategy ensures that high-type customers almost surely make purchases at full prices because the chance that they arrive during a price discount is effectively zero. On the other hand, the flash discounts allow the retailer to randomly extract additional revenues from accumulated low-type customers waiting for discounts. One can see that flash discounts achieve a good level of intertemporal price discrimination between the two types of customers, although not a perfect one. There are three sources of inefficiency: (a) the retailers need to incur price-changing costs to implement discounts; (b) low-type customers need to incur waiting costs; and (c) some low-type customers may leave without making purchases when their lifetimes expire. Inefficiency (b) determines the necessary discount depth, which alongside the trade-off between (a) and (c) in turn determines the optimal discount frequency, leading to the flash-discount solution in Proposition 1. The proposition also shows that such intertemporal price discrimination through flash discounts is possible when the customers have (stochastically) sufficiently long lifetimes and when their two types are sufficiently distinct in terms of valuations and waiting costs.

We note that, for a retailer to implement the flash-discount policy, it is implied that the retailer has the power to commit to Markovian-pricing policies; for example, as an increasing number of low-type customers accumulate to wait for a discount, we assume that the retailer can resist the growing temptation to immediately offer a discount and extract additional revenue from the accumulated customers and instead strictly depend on the stochastic pricing policy (Poisson process) to generate the next discount. Commitment power is a standard assumption in the pricing literature. For example, Borgs et al. (2014), Besbes and Lobel (2015), and Golrezaei et al. (2020) assume that a retailer commits to a pre-determined price sequence at the outset; Chen (2007) and Duenyas et al. (2013) assume that a procurer commits to a contracting mechanism; and Chen et al. (2023) and Wu et al. (2025) assume that a retailer commits to a stochastic Markovian-pricing policy similar to this article. Such commitment power is essential for a firm to design and implement effective pricing policies. Empirically, Moon et al. (2018) collaborate with a retail brand to design and implement “randomized markdown policy, which combines price commitment with the exploitation of consumers’ heterogeneous monitoring costs” to great effect; their commitment to random-discount policies yields an 81% improvement in profit compared with subgame-perfect (non-committed) state-dependent pricing policies.

The following corollary provides additional insights on how the optimal discounted price and discount rate change with the composition of customers (e-Companion B contains more sensitivity analysis based on Proposition 1).

Corollary 2

Suppose the retailer implements flash discounts. When $\beta$ increases (proportionally more low-type customers), both the optimal discounted price and the optimal discount rate increase, and more low-type customers are satisfied.

Corollary 2 suggests that proportionally more low-type customers actually lead to higher optimal discounted prices. To see why, recall Lemma 1 which suggests that discount rate and depth are substitutes. Proportionally more low-type customers waiting for discounts justify more frequent discounts to generate revenue from them. Given that discount rate and depth are substitutes, shallower discounts would suffice, thus the result.

4.2. Full competition

In the other special case where all customers are opportunistic and open to making purchases from both retailers ( $\gamma =1$ ), the two retailers engage in full competition. In this section, we analyze this full competition case. The following two propositions show that, under full competition, the Markovian-pricing equilibria remain structurally similar to the no-competition case with either flat prices or flash discounts being offered. The superscript ${}^f$ denotes full competition.

Proposition 2 Full Competition–Symmetric Equilibria

With $\gamma =1$ , the two retailers’ symmetric Markovian-pricing equilibria may be of three types:

Flat regular price: Each retailer sets a flat regular price $p_r=v_H$ . High-type customers immediately buy from their host retailers. Low-type customers immediately leave. Each retailer’s profit rate is $\alpha v_H$ .

Flat discounted price: Each retailer sets a flat discounted price $p_{di}=0$ . All customers immediately buy from their host retailers. Each retailer’s profit rate is 0.

Flash discounts: Let $\underset{\_}{w}$ be a certain function of $\lambda$ , $\beta$ , $c$ , and $v_L$ (see the proof for its characterization). If and only if

$$\begin{array}{rl}& 0<\lambda <\frac{2\beta v_L}{c},\underset{\_}{w}<w_L<\frac{(2\beta v_L-c\lambda {)}^2}{9\beta c},\end{array}$$

$$\begin{array}{rl}& v_H>\frac{-2c\lambda +8\alpha v_L+3\beta v_L+3\sqrt{{\beta }^2{v}_L^2+4\beta c(\lambda v_L+w_L)}}{4\alpha },\end{array}$$

$$\begin{array}{rl}& w_H>w_L+\frac{\begin{array}{l}(v_H-v_L)(-2c\lambda +\beta v_L\\ +\sqrt{{\beta }^2{v}_L^2+4\beta c(\lambda v_L+w_L)})\end{array}}{2c},\end{array}$$

each retailer sets the discounted price

$$p_d^f=v_L-\frac{2c{w}_L}{-2c\lambda +\beta v_L+\sqrt{{\beta }^2{v}_L^2+4\beta c(\lambda v_L+w_L)}}$$

and the discount and revert rates

$$\begin{array}{rl}{\mu }^f=& \frac{1}{4}(-2\lambda +\frac{\beta v_L+\sqrt{{\beta }^2{v}_L^2+4\beta c(\lambda v_L+w_L)}}{c}),\\ {\mu }^{\prime f}=& \mathrm{\infty }.\end{array}$$

High-type customers buy immediately from their host retailers at price $p_r=v_H$ . Low-type customers wait for the next discount from either retailer to buy at price $p_d^f$ within their lifetimes. The fraction of satisfied low-type customers is

$${\varphi }^f=\frac{3\beta \lambda v_L+2\beta w_L-\lambda \sqrt{{\beta }^2{v}_L^2+4\beta c(\lambda v_L+w_L)}}{2\beta (\lambda v_L+w_L)}.$$

Each retailer’s profit rate is

$$\begin{array}{rl}{\pi }^f& =\alpha v_H+\frac{1}{4}\\ & \times (2c\lambda +5\beta v_L-3\sqrt{{\beta }^2{v}_L^2+4\beta c(\lambda v_L+w_L)}).\end{array}$$

Proposition 2 characterizing symmetric Markovian-pricing equilibria is largely reminiscent of Proposition 1. In the flat-regular-price equilibria, the retailers forgo all low-type customers. In the flat-discounted-price equilibria, the retailers engage in Bertrand competition which drives their prices down to the costs (zero in our model). Technically, the flat-discounted-price equilibrium always exists, but is clearly dominated by the other two types of equilibria when they also exist. Finally, both retailers implementing flash discounts can still be equilibria. The flash-discount equilibria again achieve a good level of intertemporal price discrimination and are possible when the two types of customers are sufficiently long-lived and disctinct. A unique requirement for the full-competition flash-discount equilibria absent in Proposition 1 is for the low-type customers’ waiting cost to surpass a threshold. Otherwise, the competing retailers would keep lowering their discounted prices to attract each others’ low-type customers, eventually rendering flash discounts inferior to flat prices and unraveling the flash-discount equilibria.

Similar to Corollary 2, under symmetric flash-discount equilibria, when a larger fraction of customers are low-value customers, both the equilibrium discounted price and the discount rate increase, and more low-type customers are satisfied. e-Companion B contains more sensitivity analysis based on Proposition 2.

Proposition 3 Full Competition–Asymmetric Equilibria

With $\gamma =1$ , any asymmetric Markovian-pricing equilibria must have both retailers implementing flash discounts, with one retailer setting a higher discounted price, a lower discount rate, and achieving a higher expected profit rate than the other. In such equilibria, all high-type customers buy immediately from their host retailers, and all low-type customers wait to buy at the next discount from either retailer. For any asymmetric flash-discount equilibrium, a symmetric flash-discount equilibrium must exist with the total profit rate of the two retailers and the fraction of satisfied low-type customers being equal between both equilibria.

Proposition 3 shows that asymmetric flash-discount equilibria may exist, but there always exists a symmetric flash-discount equilibrium yielding the same total profit rate and proportion of satisfied low-type customers as any asymmetric flash-discount equilibrium. For this reason, we only consider symmetric equilibria when later studying the impact of changing competition.

Overall, Propositions 2 and 3 show that the full-competition case preserves the key structure of the optimal Markovian-pricing strategies in the no-competition case (Proposition 1), namely that dynamic (i.e., non-constant) optimal Markovian-pricing strategies must implement flash discounts. Flash discounts are widely seen in practice (Cui et al., 2019 and Figure 1) and are likely to offer richer managerial insights than flat prices. Therefore, we henceforth focus on the dynamic flash-discount optimal/equilibrium strategies in the rest of this article.

4.3. Partial competition

Having analyzed the cases of no and full competition, we next consider partial competition where each retailer faces mixed loyal and opportunistic customers with potentially different mixture proportions between the retailers ( $0<{\gamma }_i<1$ , $i=1,2$ ). The partial-competition case is expectedly much more complex, and the following proposition shows that it may yield additional equilibria in addition to those in the full-competition case.

Proposition 4 Partial Competition

With $0<{\gamma }_i<1$ , $i=1,2$ , the retailers’ Markovian-pricing equilibria may be of seven types:

Flat regular price: Each retailer sets a flat regular price $p_r=v_H$ . High-type customers immediately buy from their host retailers. Low-type customers immediately leave.

Flat regular and discounted prices: The retailer with strictly more loyal customers sets a flat regular price $p_r=v_H$ and the other retailer sets a flat discounted price $p_d=v_L$ .

Unilateral flash discounts: Retailer $i\in \{1,2\}$ sets a flat regular price $p_r=v_H$ and the other retailer $j$ offers flash discounts. High-type customers immediately buy from their host retailers. Retailer $i$ ’s low-type loyal customers leave immediately. All other low-type customers wait for the next discount from the discount-offering retailer $j$ .

Flash discounts for low-type customers: Both retailers offer flash discounts. High-type customers immediately buy from their host retailers. Low-type loyal customers wait for the next discount from their host retailers. Low-type opportunistic customers wait for the next discount from either retailer.

Flash discounts for opportunistic customers: Both retailers offer flash discounts. High-type loyal customers immediately buy from their host retailers. Low-type loyal customers leave immediately. All opportunistic customers wait for the next discount from either retailer. Multiple equilibria that yield equal total profit rates of the two retailers may coexist and must include a symmetric equilibrium.

Flash discounts for low-type customers and opportunistic customers: Both retailers offer flash discounts. High-type loyal customers immediately buy from their host retailers. Low-type loyal customers wait for the next discount from their host retailers. All opportunistic customers wait for the next discount from either retailer.

Flash discounts for low-type opportunistic customers: Both retailers offer flash discounts. High-type customers immediately buy from their host retailers. Low-type loyal customers leave immediately. Low-type opportunistic customers wait for the next discount from either retailer. Multiple equilibria that yield equal total profit rates of the two retailers may coexist and must include a symmetric equilibrium.

Compared with the full-competition case (Propositions 2 and 3), partial competition brings about significant complexity. First, flat discounted prices are no longer sustainable as an equilibrium, whereas flat regular and discounted prices may be an equilibrium. Second, a new unilateral flash-discount equilibrium arises, whereas all asymmetric equilibria under full competition have both retailers offering flash discounts. Among bilateral flash-discount equilibria, low-type and/or opportunistic customers may wait for discounts, which is reasonable. These bilateral flash-discount equilibria can be further categorized into four subtypes based on whether all low-type customers, all opportunistic customers, or their union or intersection may wait for flash discounts. Notably, when mixed with loyal customers, high-type opportunistic customers may also wait for discounts, whereas no high-type customers in the no- and full-competition cases wait for discounts. Due to the overwhelming complexity of the partial-competition model, analytically listing all equilibrium conditions is intractable. However, the key result that the retailers’ dynamic (i.e., non-constant) equilibrium Markovian-pricing strategies must implement flash discounts remains robust.

5. Impact of changing competition

Having characterized competitive Markovian-pricing equilibria under no, full, and partial competition in Section 4, we are ready to investigate the impact of changing competition in Markovian pricing. In this investigation, we focus on flash discounts which are always implemented in dynamic (i.e., non-constant) optimal Markovian-pricing strategies and are widely seen in practice (Cui et al., 2019 and Figure 1).

5.1. Bilateral change of competition

The general level of competition (i.e., proportion of opportunistic customers) in a market may be influenced by third-party services such as Google Shopping and PriceGrabber that facilitate price monitoring and comparisons. Such services make it easier for customers to discover prices at competing retailers and thus may increase the proportion of opportunistic customers for both retailers. Such services are usually perceived as detrimental to retailer profits (Wolfe, 2023) and retailers often attempt to block them with technical tools (Krunic, 2021; Pakay, 2017). However, for the many competing retailers adopting Markovian pricing, the impact of changing competition can be non-straightforward. More opportunistic customers in the market indeed drive retailers to compete more intensely, but in this setting, there are two means to attract customers from the competitor—deeper discounts and more frequent discounts—and the adoption of one may lead to the relaxation of the other (see Corollary 2 and the ensuing discussion). Therefore, the impact of increased general level of competition in the market on retailer profits can be ambiguous.

To inform the impact of bilateral changes of competition under Markovian pricing, in this section, we consider the case of both retailers facing equal proportions of opportunistic customers ${\gamma }_1={\gamma }_2=\gamma$ and investigate how the retailers’ equilibrium profits change with $\gamma$ . Our characterizations of the symmetric equilibria under no competition and full competition (resp. Propositions 1 and 2) allow us to fully compare these two cases, and the results are presented in the following proposition.

Proposition 5

When the competitive Markovian-pricing equilibria under no and full competition both implement symmetric flash discounts, each retailer’s equilibrium discount rate, discounted price, and expected profit rate, and the fraction of satisfied low-type customers are larger under full competition than under no competition; that is, $p_d^f>p_d^n$ , ${\mu }^f>{\mu }^n$ , ${\pi }^f>{\pi }^n$ , and ${\varphi }^f>{\varphi }^n$ .

Proposition 5 presents interesting results: compared with the no-competition case, competitive Markovian pricing under full competition yields more frequent yet shallower discounts from and higher profits for both retailers. To see why, recall that under competitive Markovian pricing, the retailers have two means to attract opportunistic customers from the competitor: deeper discounts and more frequent discounts. A retailer offering deeper flash discounts can convince opportunistic customers to skip the shallower discounts from the other retailer and wait exclusively for the deeper discounts, thus attracting them from the competitor. However, this approach to attract opportunistic customers is inefficient as it increases their average waiting time and cost. By contrast, a retailer offering more frequent discounts is a more efficient approach to attract opportunistic customers because doing so reduces their average waiting time and cost. In addition, because opportunistic customers are open to making purchases from both retailers who offer independent random discounts, their average waiting time and cost are further reduced compared with loyal customers. Recall that the discount depth is calibrated to offset the average waiting cost, thereby incentivizing the targeted customers to wait. As a result, the offered discounts to all opportunistic customers are much shallower than to all loyal customers. Furthermore, the waiting low-value customers are more likely to see a discount within their lifetime and make a purchase with all opportunistic customers, given more frequent discounts being offered and their willingness to consider both retailers; therefore, the efficiency in capturing the waiting customers is also increased. Together, the retailer profits are actually increased despite more discounts being offered. In summary, under full competition, retailers primarily compete for low-value customers by offering more frequent discounts. This helps the retailers satisfy more low-type customers and allows shallower discounts to be offered, both of which contribute to improving retailer profits.

Given that full competition yields higher profits for both retailers than no competition, it is natural to ask if the same trend holds for partial competition (i.e., some but not all customers are opportunistic). The following proposition sheds light on the question. Note in Proposition 4 that symmetric and asymmetric equilibria that yield equal total retailer profits may coexist. Consistent with our treatment of the similar case under full competition (see the discussion following Proposition 3), we restrict our attention to symmetric equilibria and henceforth suppress the subscript $i$ when doing so does not introduce ambiguity.

Proposition 6

Among all flash-discount equilibria of Proposition 4, increasing the proportion of opportunistic customers $\gamma$ at both retailers leads to increased equilibrium discounted price $p_d$ , discount rate $\mu$ , and the fraction of satisfied low-type customers as long as the equilibrium type remains unchanged. Among the “unilateral flash discounts,” “flash discounts for opportunistic customers,” and “flash discounts for low-type opportunistic customers” equilibria, increasing $\gamma$ at both retailers increases their profits as long as the equilibrium type remains unchanged.

Proposition 6 is consistent with Proposition 5: under partial competition, a larger proportion of opportunistic customers at both retailers (i.e., bilaterally increased competition) leads to more frequent yet shallower discounts within the same equilibrium type. However, the mixed opportunistic and loyal customers and the resulting multiple equilibrium types complicate the retailer profit comparison. First, it is challenging to compare retailer profits across equilibrium types when the customers may change their purchasing behavior. Furthermore, low-type loyal customers waiting for discounts only at their host retailers do not generate as much additional sales as opportunistic customers when both retailers offer more frequent discounts. As a result, Proposition 6 shows that proportionally more opportunistic customers increase retailer profits in unilateral flash-discount equilibria where the retailers do not truly compete to attract opportunistic customers with discounts, as well as in two of the four types of bilateral flash-discount equilibria where low-type loyal customers do not wait for discounts. In the two other types of bilateral flash-discount equilibria, the profit impact of bilaterally increased competition is analytically ambiguous.

In what follows, we complement the analysis with extensive numerical experiments which yield relatively consistent results. Figure 2 provides a representative example, generated with $c=3$ , $\beta =0.8$ , $v_L=1$ , $\lambda =0.007$ , $w_L=0.0623$ , and sufficiently high $v_H$ and $w_H$ to yield flash-discount equilibria. Because the revenue from high-type customers is constant and does not impact retailer profit comparisons, we only need to consider retailer profits from low-type customers. In Figure 2(a), one can see that as the proportion of opportunistic customers $\gamma$ increases from 0 to 1 (more competition from left to right), the equilibrium retailer profit from low-type customers hovers close to zero for small $\gamma$ values and monotonically increases for large $\gamma$ values, with a gap between $\sim 0.35<\gamma <0.7$ where no pure-strategy equilibrium exists. Figure 2(b) zooms into the $0<\gamma <0.35$ region of Figure 2(a) to show that the equilibrium retailer profit from low-type customers is actually non-monotonic (increases then decreases) for small $\gamma$ values.

Figure 2.

Equilibrium retailer profit from low-type customers with mixed loyal and opportunistic customers.

There are multiple observations worth explaining. First, under partial competition, no pure-strategy equilibrium exists for a range of intermediate $\gamma$ values. This is a new phenomenon absent under full competition; in the latter case, the two retailers compete to attract low-type opportunistic customers and always reach an equilibrium. Under partial competition, each retailer’s discounts cater to its loyal customers as well as attract opportunistic customers from the other retailer. The dual purposes of discounts mean that, with sufficiently many loyal and opportunistic customers (intermediate $\gamma$ ), a retailer’s best response to the competitor’s pricing strategy may contain discontinuities when it switches between accommodating and forgoing low-type loyal customers, leading to no pure-strategy equilibrium. This would not be an issue with relatively small/large $\gamma$ values where the sufficiently many/few loyal customers are always accommodated/forgone, yielding continuous best responses and pure-strategy equilibria. The lack of pure-strategy equilibrium for intermediate $\gamma$ values is indicative of the strategic complexity of competitive Markovian pricing under partial competition.

Second, the equilibrium retailer profit from low-type customers is non-monotonic for small $\gamma$ values (low levels of competition). As explained earlier, with many loyal customers, the retailers tend to adopt flash-discount strategies that accommodate low-type loyal customers while competing for low-type opportunistic customers, and the dual purposes of the discounts mean that the profit impact of more opportunistic customers is ambiguous. However, compared with the equilibrium retailer profits from low-type customers for large $\gamma$ values, those for small $\gamma$ values are relatively invariant and significantly smaller; the fact that large $\gamma$ values yield significantly higher retailer profits than small $\gamma$ values is consistent with Proposition 5. Technically, the non-monotonic retailer profits for small $\gamma$ values is again indicative of the strategic complexity of competitive Markovian pricing under general levels of competition; practically, though, the minuscule variations of retailer profits for small $\gamma$ values are negligible in the big picture.

Finally, note that the equilibrium retailer profit from low-type customers monotonically increases for $\gamma >0.7$ . It stands to reason that loyal customers impervious to discounts offered by competing retailers are a minority, a notion supported by anecdotal evidence. According to a survey by AYTM (2016), 79% of respondents considered themselves to be bargain shoppers. In another survey, Loria (2019) finds that 90%, 84%, 78%, and 71% of respondents would compare prices before buying electronics, appliances, food, and clothing, respectively. Therefore, $\gamma >0.7$ (<30% loyal customers) is likely to capture practical scenarios. In this range, the result that bilaterally increased competition actually improves retailer profits under competitive Markovian pricing recovers the similar result from Proposition 5’s comparison of the no- and full-competition cases, and the explanation is also similar to the one thereafter: the competition for more opportunistic customers leads to more frequent yet shallower flash discounts, such that the retailer profits are actually increased despite more discounts being offered. The extension of this result to the practically relevant cases under partial competition confirms its robustness.

The finding that competing retailers implementing random flash discounts can earn higher profits under bilaterally increased competition (more opportunistic customers at both retailers) highlights the unique properties of competitive Markovian pricing. As we explained, the common perception is that third-party services facilitating price monitoring and comparisons, such as Google Shopping and PriceGrabber, generally increase competition and reduce retailer profits (Wolfe, 2023), and there is anecdotal evidence of cat-and-mouse games between retailers and price-monitoring and comparison services going on where the former attempt to block the latter from tracking real-time prices and the latter attempt to bypass such blockades (Krunic, 2021; Pakay, 2017). Our results challenge such common wisdom under competitive Markovian pricing and show that competing retailers implementing random flash discounts may actually benefit from increased market competition brought forth by price-monitoring and comparison services. Given the prevalence of random flash discounts in retailing, this finding has significant practical implications and suggests that the many retailers adopting Markovian pricing should refrain from naïvely applying common wisdom toward price-monitoring and comparison services.

5.2. Unilateral change of competition

Whereas price-monitoring and comparison services influence the general proportion of opportunistic customers in the market, a retailer can grow its own loyal customer base, for example, through improving customer experience, building brand affinity, and loyalty programs, and thereby unilaterally reduce competition (Sinitsyn, 2008). Both practitioners (Narciso, 2023; Zinrelo, 2023) and scholars (Kuksov and Zia, 2021; Narasimhan, 1988) note that retailers generally benefit from more loyal customers. Given that bilaterally increased competition can benefit retailers engaging in competitive Markovian pricing, it begs to ask whether unilaterally increased competition (increased proportion of opportunistic customers at one retailer) may also benefit the retailer(s) in the same setting. The next proposition confirms that this is indeed possible. Similarly to the discussion before Proposition 6, note in Proposition 4 that symmetric and asymmetric equilibria that yield equal total retailer profits may coexist. Consistent with our treatment of the similar case under full competition (see the discussion following Proposition 3), we restrict our attention to symmetric equilibria in the next proposition ³ and suppress the subscript $i$ when doing so does not introduce ambiguity.

Proposition 7

Among the “flash discounts for low-type opportunistic customers” equilibria of Proposition 4, increasing the proportion of opportunistic customers ${\gamma }_i$ at either retailer $i$ leads to increased equilibrium discounted price $p_d$ , discount rate $\mu$ , fraction of satisfied low-type customers, and profit at both retailers, as long as the equilibrium type remains unchanged.

Proposition 7 shows that, in the equilibria where only low-type opportunistic customers wait for flash discounts, a larger proportion of opportunistic customers at one retailer (i.e., unilaterally increased competition) may increase both retailers’ profits. To see why, note that all low-type loyal customers are lost whereas all high-type customers buy at the regular price from their host retailers in these equilibria (Proposition 4). If one retailer’s loyal customers become opportunistic customers, no high-type customers are lost, whereas the originally lost low-type loyal customers become low-type opportunistic customers who may now be captured by both retailers’ flash discounts, thereby increasing their profits. Therefore, the surprising finding that bilaterally increased competition may benefit both retailers is recovered for unilaterally increased competition as well.

Although Proposition 7 establishes this result only for one type of equilibria, we have numerically observed that the “flash discounts for low-type opportunistic customers” equilibria actually correspond to the scenario with a dominant majority of opportunistic customers (e.g., ${\gamma }_1,{\gamma }_2>0.7$ with the same parameters as Figure 2) where the retailers’ flash-discount strategies forgo the relatively few low-type loyal customers and compete for low-type opportunistic customers. Recall the anecdotal evidence that more than 70% of real-life customers would compare prices across retailers before making purchases (AYTM, 2016; Loria, 2019); therefore, the scenario with a dominant majority of opportunistic customers (highly competitive retail environment) is reflective of the actual retail landscape, and thus the result of Proposition 7 has wide applicability in practice.

Similar to Propositions 5 and 6, Proposition 7 shows that despite the common wisdom that a retailer generally benefits from the resulting reduced competition (Kuksov and Zia, 2021; Narasimhan, 1988) and should grow its loyal customer base (Narciso, 2023; Zinrelo, 2023), in the competitive Markovian-pricing setting with a dominant majority of opportunistic customers, unilaterally increased competition (increased proportion of opportunistic customers at one retailer) may actually improve both competing retailers’ profits. Once again, this finding highlights the unique properties of competitive Markovian pricing and suggests that retailers adopting Markovian pricing should reconsider the efforts in growing their loyal customer base. Incidentally, Zhang et al. (2018, 2023) document sellers of insurance, vitamin supplements, cars, and retail products posting competitors’ real-time prices and/or links on their own websites—a practice known as competitor referral. Although Zhang et al. (2018, 2023) provide their own explanations (entry deterrence and double marginalization mitigation, respectively), we offer an alternative motivation for why a seller may voluntarily reduce its own customer loyalty under competitive Markovian pricing.

6. Model extensions

In this section, we treat two model extensions to confirm the robustness of our findings and provide additional insights. For concision, we focus on the model setups and main results while relegating most analyses to e-Companion C and D.

6.1. Asymmetric retailer sizes

In our main model, we assume that the two retailers face Poisson streams of incoming customers with the same rate (normalized to 1). In practice, retailers of different sizes can have different abilities to attract customers. In this section, we capture this factor by assuming two retailers with different customer arrival rates—one designated as the “small” retailer who faces an arrival rate of $1-b$ , while the other, referred to as the “large” retailer, faces an arrival rate of $1+b$ , where $b\in (0,1)$ is a proxy for the size differentiation between the two retailers. We maintain all other assumptions, including the same $\alpha$ proportion of high-type customers and each customer’s lifetime being exponentially distributed with parameter $\lambda$ .

It is algebraically straightforward to see that, with asymmetric retailer sizes, the three types of optimal strategies in Proposition 1 under no competition, the three types of equilibria in Proposition 2 under full competition, and the seven types of equilibria in Proposition 4 under partial competition are all preserved in the new setting, albeit with different conditions and optimal/equilibrium discount frequencies and depths. ⁴ This is because our analysis throughout Section 4 did not critically depend on the two retailers facing equal customer arrival rates. e-Companion C provides the optimal/equilibrium discount frequencies and depths under no/full competition, whereas explicit expressions for the equilibria and their respective conditions in the partial-competition cases are unavailable.

We next investigate the impact of changing competition. The additional complexity with asymmetric retailer sizes prohibits analytical comparisons. We conduct extensive numerical investigations and find highly consistent results. A representative example is presented in Figure 3, generated with $c=3$ , $\beta =0.8$ , $v_L=1$ , $\lambda =0.005$ , $w_L=0.0631$ , and sufficiently high $v_H$ and $w_H$ to yield flash-discount equilibria under full competition.

Figure 3.

Retailer profits from low-type customers for varying size differentiations.

Figure 3 illustrates the profits of two retailers of different sizes from low-type customers under no versus full competition for varying size differentiations. Similar to Figure 2, we focus on profits from low-type customers because profits from high-type customers remain identical under changing competition and do not affect the comparison. When the two retailers’ size differentiation increases, their respective profits from low-type customers under full competition remain equal and unchanged because their combined customer arrival rate remains unchanged at $(1-b)+(1+b)=2$ and they always share profits from all low-type customers. By contrast, under no competition, the small retailer’s profit from low-type customers shrinks to zero as the arrival rate drops out of the flash-discount parameter region and the small retailer switches to charging a flat regular price and forgoes low-type customers; whereas the large retailer’s profit from low-type customers increases with the arrival rate until exceeding its full-competition equilibrium profit from low-type customers. Beyond this point, competition is no longer beneficial for the large retailer.

This exercise reveals that, for retailers of different sizes, our key insight that more competition may benefit retailers under Markovian pricing holds true when the two retailers are moderately asymmetric in size; whereas when their size differentiation is substantial, more competition may no longer benefit the large retailer. This is because, with high levels of competition, the small retailer has much to gain by accessing the large retailer’s customers whereas the opposite is not true. Interestingly, this observation is consistent with the anecdote that a small local grocery chain, WinCo, became “Walmart’s worst nightmare” (Tuttle, 2013). Overall, this exercise demonstrates the robustness of our key insight that more competition may benefit retailers under Markovian pricing to a certain extent while also establishing its limit of applicability.

6.2. Exogenous customer behavior

We derived the base model in Section 3 assuming that the customers are aware of the retailers’ Markovian pricing strategies, for example inferred through price-monitoring and comparison services such as Keepa and CamelCamelCamel. The customers then make informed buy, wait, or leave decisions (Section 3.1) that lead to the model’s equilibria (Section 4). Nevertheless, one may argue that real-life customers may not always have the sophistication to infer the retailers’ pricing strategies and behave accordingly. In this section, we analyze an alternative model assuming that the customers exhibit exogenous purchase behavior and the retailers charge exogenous market prices to check if our main results are robust when customers do not make fully informed purchase decisions. Most technical details are relegated to e-Companion D.

Specifically, we take inspiration from Lemmas 1 and 2, which show that a rational customer would either buy (immediately at the current price), wait (indefinitely for a discount until their lifetime expires), or leave (without a purchase) upon arrival. Accordingly, we now assume that an exogenous $\tilde{\alpha }\in (0,1)$ proportion of all incoming customers (independent of being loyal or opportunistic) are impatient customers who would buy immediately at the current price, and a $\tilde{\beta }:=1-\tilde{\alpha }$ proportion are bargain hunters who would wait indefinitely for a discount (from either the host retailer or both retailers for a loyal or opportunistic customer) until their lifetime expires (the customers who leave immediately are excluded from our consideration). Paired with the exogenous customer behavior, we now assume that the two retailers also charge equal exogenous market regular and discounted prices ${\tilde{p}}_r>{\tilde{p}}_d>0$ . Technically, given the exogenous customer behavior, the retailers’ optimal discount prices would become equal to the regular price which is unreasonable. Economically, the customers would only behave exogenously when there is a reasonable gap between the regular and discount market prices; no or extreme price differences would induce the customers to reconsider their behavior. Similar exogenous prices and customer behavior have been adopted in studies of retailers’ competition through discounts; for example, Sinitsyn (2017). All other assumptions remain unchanged, and each retailer $i$ determines the discount rate ${\tilde{\mu }}_i$ and the revert rate ${\tilde{\mu }}_i^{\prime }$ , $i=1,2$ .

Similar to the base model, we first establish that the retailers’ symmetric optimal/equilibrium pricing strategies can be of three types: flat regular price, flat discounted price, and flash discounts. The formal results under no, full, and partial competition are, respectively, presented in Propositions D.1, D.2, and D.3 in e-Companion D. Propositions D.1 and D.2 are, respectively, parallel to Propositions 1 and 2 in structure. Proposition D.3 is simpler in structure than Proposition 4. Nevertheless, the base model’s key finding that any equilibrium dynamic (non-constant) Markovian-pricing strategies must implement flash discounts is preserved with exogenous customer behavior. We henceforth once again focus on flash-discount equilibria.

We then investigate the impact of changing competition. The following proposition compares the retailers’ profits under no versus full competition in parallel with Proposition 5.

Proposition 8

When the competitive Markovian-pricing equilibria under no and full competition both implement flash discounts, each retailer’s equilibrium discount rate and the fraction of satisfied bargain hunters are larger under full competition than under no competition. Each retailer’s expected profit rate is higher under full competition than under no competition if and only if $\lambda >\hat{\lambda }$ , where $\hat{\lambda }$ is the first root to $c^4{x}^4-52\tilde{\beta }{c}^3{\tilde{p}}_d{x}^3+96{\tilde{\beta }}^2{c}^2{\tilde{p}}_d^2{x}^2-40{\tilde{\beta }}^{3}c{\tilde{p}}_d^{3}x+4{\tilde{\beta }}^4{\tilde{p}}_d^4=0$ .

Proposition 8 partially recovers the findings in Proposition 5 with exogenous customer behavior: compared with the no-competition case, competitive Markovian pricing under full competition yields more frequent discounts from both retailers; when the bargain hunters’ lifetimes are not too long ( $\lambda >\hat{\lambda }$ ), both retailers also earn more profits under full competition than no competition. Similar to the base model, under full competition, both retailers offer more frequent discounts to attract bargain hunters from each other and collectively satisfy more bargain hunters; the difference is that now customer behavior and prices are exogenous and the retailers would not offer shallower discounts in conjunction with more frequent discounts which always lead to increased retailer profits in the base model. Nevertheless, the increased efficiency in extracting revenue from bargain hunters may still dominate the retailers’ increased discounting costs, leading to overall improved retailer profits. Specifically, bargain hunters with shorter lifetimes (larger $\lambda$ ) are more likely to expire before making purchases, therefore the increased efficiency in extracting revenue from bargain hunters under competition is greater when the bargain hunters have shorter lifetimes. This explains the condition $\lambda >\hat{\lambda }$ for increased retailer profits under full versus no competition.

Having analytically compared full versus no competition, we numerically investigate gradually changing competition; this approach is necessitated by the lack of explicit solutions of the partial competition model. Figure 4 offers a representative set of examples, generated with $c=2$ , $\tilde{\beta }=0.4$ , ${\tilde{p}}_d=1$ , and sufficiently high ${\tilde{p}}_r$ to yield flash-discount equilibria. Because the revenue from impatient customers is constant and does not impact retailer profit comparisons, we only need to consider retailer profits from bargain hunters.

Figure 4.

Equilibrium retailer profit from bargain hunters with mixed loyal and opportunistic customers under different average bargain hunter lifetimes.

The first observation is that Figure 4 is consistent with Proposition 8. By comparing the two ends of each curve, one can see that the retailers’ full-competition profits may exceed no-competition profits when the bargain hunters have short lifetimes (large $\lambda$ ). Furthermore, the retailers’ profits are concave in the proportion of opportunistic customers $\gamma$ . This means that the retailers’ profits are more likely to increase with low levels of competition but may decrease with high levels of competition. This observation shares some similarity with Figure 2 with endogenous discounted prices and customer decisions where the equilibrium retailer profit can be concave in the proportion of opportunistic customers, albeit only when the proportion is low in the latter case. Most importantly, our analysis shows that the key insight from the base model that competing retailers implementing random flash discounts can earn higher profits under more competition is preserved with exogenous prices and customer behavior. This exercise demonstrates the robustness of this key insight to a certain extent while also establishing its limit of applicability.

7. Conclusion

Pricing is a critical decision in retail, and many retailers adopt dynamic pricing to achieve intertemporal price discrimination and improve revenues. However, dynamic pricing is undermined by customers’ strategic waiting behavior. One tactic utilized by retailers to counter strategic customer waiting is to offer price discounts at random intervals. This idea has been anecdotally documented in practice, experimented with in the field, and modeled and analyzed in the Markovian-pricing literature. A notable missing element in the research of random discounts is however competition.

Filling in this gap, this article studies two competing retailers adopting Markovian-pricing strategies facing respective streams of mixed loyal and opportunistic strategic customers, where the loyal customers only consider buying from their host retailers and the opportunistic customers are open to buying from both retailers. We show in our competitive setting that any equilibrium dynamic (non-constant) Markovian-pricing strategies must implement flash discounts that last only for an instant, recovering a key finding of Markovian pricing in single-retailer settings and conforming to anecdotal evidence. We then investigate the impact of changing competition between retailers adopting Markovian-pricing strategies. Surprisingly, we find that increased levels of competition (increased proportions of opportunistic customers), both bilaterally and unilaterally, may lead to increased profits of both retailers, especially when a dominant majority of the customers are opportunistic (highly competitive retail environment). The general explanation is that, under increased levels of competition, the retailers offer more frequent discounts. On the one hand, when the retailers compete with more frequent discounts, much shallower discounts are sufficient to entice low-value and patient customers into waiting and potentially making purchases, leading to overall less revenue loss due to discounting. On the other hand, more frequent discounts mean fewer lost low-value and patient customers, leading to an increased efficiency in capturing customers. Both effects conspire to cause this counterintuitive phenomenon.

We then analyzed two model extensions, one with asymmetric retailer sizes (asymmetric arrival rates of incoming customers) and the other with uninformed customers exhibiting exogenous purchase behavior, to check the robustness of our main results. In both extensions, we find that any equilibrium dynamic (non-constant) Markovian-pricing strategies must still implement flash discounts, and that increased levels of competition (increased proportions of opportunistic customers) may still lead to increased profits of both retailers, demonstrating the robustness of these results. In both cases, we also establish the limit of applicability of the latter result: with highly asymmetric retailer sizes, the large retailer may not benefit from competition; and with exogenous customer behavior, the retailers may not benefit from competition when the bargain hunters can wait for a long time.

Overall, we show that retailers can effectively adopt Markovian pricing in competitive environments, establish the optimality of flash discounts under competitive Markovian pricing, and find in this setting that increased levels of competition may benefit both retailers. Our findings suggest that retailers engaging in competitive Markovian pricing should refrain from naïvely applying common wisdom toward price-monitoring and comparison services and reconsider the efforts in growing their loyal customer base. More broadly, these findings highlight the unique properties of competitive Markovian pricing and call for further research into this prevalent yet understudied setting.

Finally, we acknowledge two model simplifications that may warrant further exploration in future research. First, we adopt a two-type customer value formulation and show that, in most equilibria, low-type customers wait for discounts whereas high-type customers do not. In practice, customer values are continuously distributed, and the subset of customers willing to wait for discounts may also continuously depend on the discount frequencies and depths. Studying such a model may reveal additional subtleties of competitive Markovian pricing. Second, we (as well as all the established Markovian-pricing literature) show that flash discounts lasting for an instant can efficiently capture waiting low-value customers without affecting revenue from high-value customers. This is a theoretically ideal scenario; in practice, such flash discounts last for relatively short yet non-zero periods of time, and some high-value customers will unintentionally make purchases at discounted prices, leading to revenue losses. The price-changing cost in our model can capture such revenue losses to some extent. A more rigorous way to model this phenomenon within the Markovian-pricing framework is to cap the revert rate; this will complicate the analysis but may reveal additional subtleties of competitive Markovian pricing. We leave these extensions to future research.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478261447023 - Supplemental material for Competitive Markovian pricing

Supplemental material, sj-pdf-1-pao-10.1177_10591478261447023 for Competitive Markovian pricing by Haokun Du, Bin Hu and Elena Katok in Production and Operations Management