The Impact of Optimism Bias on Strategic Decision-Making and Efficiency in Online Retail Supply Chains

Abstract

This paper examines the effects of optimism bias within online retail supply chains. Here, optimism refers to a cognitive bias wherein the third-party seller and the e-commerce platform underestimate the likelihood of facing low market potential. The analysis begins by exploring the impacts of each party’s respective biases. The results indicate that while seller optimism generally leads to self-detrimental outcomes, it can also benefit both the platform and the overall system. Conversely, platform optimism does not permanently harm the platform itself but consistently disadvantages the seller and negatively impacts the supply chain. This work then investigates the combined effects of seller and platform optimism on overall system performance, revealing that the entire channel can, in fact, gain from optimism bias. This research offers insights into strategic approaches that may enhance the efficiency of online retail supply chains.

1. Introduction

With advancements in payment technology and logistics services, online retailing has become a cornerstone of the modern economy. A vast number of direct transactions between buyers and third-party sellers are facilitated on e-commerce platforms such as Amazon.com and eBay in the U.S. and Tmall and JD.com in China. In 2020, global e-commerce retail sales exceeded $4 trillion, accounting for approximately 20% of total global retail sales, and are projected to reach $7.5 trillion by 2025 1. The rapid growth in online demand has spurred leading e-commerce platforms to invest heavily in their marketplaces, enabling third-party sellers to reach end consumers directly and forming online retail supply chains (ORSCs).

In ORSCs, independent businesses or individuals (third-party sellers) list their products on an e-commerce platform (such as Amazon). The e-commerce platform offers sellers a wide range of services, including advertising, fulfillment, payment processing, and customer service support. With these efforts, the e-commerce platform charges the seller fees by setting product-specific commission rates, which is impractical for the platform. For example, the commission rates for different product categories in Tmall range from 0.2% to 5%, and in eBay range from 0.2% to 12%. These sellers manage their inventory and pricing but must comply with the platform’s commission policies. An ORSC highlights the interplay between the seller and the platform, showcasing the complexities and dependencies unique to this supply chain structure. Therefore, the management of these systems has attracted increasing attention from both academia and industry [1].

Traditionally, much of the literature assumes that the key players in ORSCs are rational actors. However, recent advances in behavioral research suggest that cognitive biases can significantly influence decisions within ORSCs e.g., [2]. Many everyday decisions related to the production, distribution, and delivery of goods sold online require an estimation of future demand probabilities, or the likelihood of encountering favorable market conditions. In this context, decision-makers often exhibit an optimism bias—a cognitive tendency to overestimate the likelihood of positive outcomes while underestimating the likelihood of negative ones [3]. Optimism bias is not merely theoretical; it has been observed in financial markets [4], corporate finance [5], and insurance [6]. For example, ref. [5] show that optimistic CEOs are 65 percent more likely to pursue mergers, often overpaying for target companies and engaging in mergers that can reduce value.

In ORSCs, both e-commerce platforms and third-party sellers often exhibit optimism bias, assuming that negative events, such as downturns in the market, are less likely to affect them than others. Ref. [7] highlight that entrepreneurs make their decisions in circumstances of high uncertainty and tend to be overly optimistic about their chances of success, often persisting in unpromising business areas. For example, e-commerce platforms may invest heavily in new features or expansions based on overly optimistic projections, which can lead to financial strain if these investments fail to yield anticipated returns [8]. Similarly, a 2020 survey by Luohan Academy found that over 66% of nearly 24,000 merchants on Alibaba’s Taobao and Tmall marketplaces were optimistic about business prospects for the remainder of the year, with 14% expecting significant increases in sales or profits [9]. However, many of these sellers ultimately faced long-term profit losses due to challenges such as decreased orders, customer attrition, delivery delays, and tight cash flows resulting from the pandemic. Overall, this evidence suggests that optimism is a prevalent trait among all players in ORSCs.

Although optimism bias is widespread in ORSCs, its specific effects on decision-making and performance within these supply chains remain largely unexamined. Existing literature provides ample evidence that optimism bias can influence the decision-making of entrepreneurs and executives e.g., [10], often leading to poor decisions and suboptimal outcomes e.g., [11]. Against this backdrop, this work analyzes how optimism bias impacts the interactions and performance of the third-party seller and the e-commerce platform to advance theoretical understanding of the impacts of optimism bias in ORSCs. This study aims to assess the direct impact of optimism bias on ORSC efficiency, offering actionable insights for e-commerce platforms and third-party sellers to enhance supply chain efficiency and decision-making accuracy. To this end, I incorporate the notion of optimism bias into an ORSC consisting of a third-party seller and an e-commerce platform. This research primarily focuses on B2C (Business-to-Consumer) interactions, where one third-party seller offers products to individual consumers via an e-commerce platform that operates as a marketplace, such as Amazon, eBay, and Alibaba. Examples include independent merchants or small businesses selling goods like electronics, apparel, or home appliances to retail customers. The platform sets a commission rate and earns a fixed proportion of the seller’s revenue to facilitate each transaction while the seller determines order quantities and retail prices. The random market potential and price of the product determine consumer demand. When facing uncertain market potential, the decision-maker can be unrealistically optimistic by acting as if the likelihood of low market potential is lower than it is.

This research employs a game-theoretic modeling approach to analyze the effects of optimism bias in an online retail supply chain. This methodology is primarily descriptive and analytical as I develop mathematical models to characterize equilibrium strategies and examine how optimism bias influences the decisions of the seller and the platform. Below, I highlight the main findings of this paper (see

Table 1. Summary of main results. (“+” = benefits, “−” = hurts, “(+)” = potentially benefits, “(−)” = potentially hurts).

	Seller	Platform	ORSC
Seller Optimism	−	+	(+)
Platform Optimism	−	(+)	−
Combined Optimism	(−)	(+)	(+)

for summaries).

I first examine a scenario where the third-party seller is unrealistically optimistic while the e-commerce platform remains unbiased. The findings indicate that, although the optimistic seller earns less than an unbiased seller, seller optimism ultimately benefits the platform. Specifically, as the seller’s optimism increases, so does the platform’s revenue. This occurs because optimism bias distorts the seller’s ordering decisions, leading them to stock more than the optimal level. While this overstocking harms the seller’s profitability, the increased inventory boosts product sales revenue, thereby benefiting the platform.

This research further suggests that seller optimism could enhance the overall performance of ORSC. To illustrate, I consider a hypothetical centralized system comprising a third-party seller and an online platform coordinated by an unbiased central planner. In this setup, removing the central planner would worsen the ORSC’s performance if the seller and platform were left to make decisions independently. However, if an optimistic (biased) seller replaced the central planner, the ORSC could potentially benefit, depending on the degree of optimism. This is because the optimistic seller would increase the stocking level, counteracting the effects of decentralization, which tends to reduce stocking levels below the optimal benchmark. Ultimately, this stocking level boost could enhance the entire channel’s performance.

This work then examines the impact of platform optimism bias. Interestingly, the findings reveal that platform optimism can increase the platform’s profit when production costs are moderate. Specifically, the more biased commission rate decision that arises from heightened optimism benefits the platform financially. However, this profit gain comes at the expense of the third-party seller. In fact, platform optimism consistently harms both the seller and the overall supply chain. Intuitively, the increased commission rate resulting from platform optimism has two main effects: it allows the platform to capture a larger share of the seller’s revenue, thereby disadvantaging the seller, and it reduces the stocking level, pushing it further from the first-best benchmark. This misalignment ultimately damages the performance of the entire channel.

I analyze the combined effects of seller and platform optimism to understand the underlying interactions better. The findings suggest that the advantages of seller optimism for the online retail supply chain (ORSC) can outweigh the negative impacts of platform optimism. In other words, optimism bias can enhance supply chain performance, even when both the seller and the platform are optimistic. I also examine the broader managerial implications of seller optimism for ORSCs. Traditional operations literature and practice view decentralization as detrimental, often advocating for contracts to coordinate decentralized channels effectively [12,13]. However, the findings of this paper challenge this conventional wisdom. Even without coordination, I demonstrate that a decentralized ORSC can achieve performance comparable to its centralized benchmark when the seller is optimistic.

The rest of this paper is organized as follows. Section 2 reviews the relevant literature and highlights the main contributions. Section 3 presents the basic analytical model for ORSC. Section 4 further investigates the impacts of seller and platform optimism and then extends the analysis to the case where both the platform and the seller are biased. Section 5 explores the case of an AI-driven platform. Section 6 concludes the paper with managerial insights and further directions. Proofs are in the Appendix A.

2. Literature Review

This work relates to a broad literature on behavioral operations. Behavioral operations, a branch of operations management (OM), explores how individuals behave and make decisions in operational contexts [14]. This literature has identified a wide range of biases that can influence the decision-making process and explored the impacts of biases on decision-makers’ performance [15]. Among these biases, overconfidence is one of the most consistent, powerful, and widespread behavioral factors in decision-making [16], which has been extensively studied in recent research [17,18]. Overconfidence is an individual overestimating their abilities, knowledge, or control over events. It reflects a misjudgment of one’s competence or capability. In this paper, optimism bias refers to a cognitive bias where individuals systematically overestimate the likelihood of positive outcomes or underestimate the likelihood of negative ones. It pertains to external events and probabilities, often independent of personal ability or competence. While optimism bias focuses on forecasting external outcomes, overconfidence centers on the self-assessment of one’s ability to predict or influence those outcomes.

Optimism bias influences behaviors and decisions across diverse domains. In project management, optimism bias is widely recognized as a key factor behind project cost overruns and unrealistic timelines. Research shows that optimism bias impacts baseline schedule development, often leading project managers to underestimate the time and resources necessary for project completion [19,20]. In health contexts, optimism bias significantly shapes parental attitudes toward child safety and health behaviors, with studies indicating that parents frequently underestimate their children’s risk of unintentional injuries, resulting in insufficient safety measures [21]. Optimism bias also plays a critical role in financial decision-making, where it contributes to investor overconfidence in investment decisions and market predictions, further leading to market anomalies [22,23]. Additionally, research on e-government adoption highlights optimism bias as a factor shaping users’ attitudes toward digital platforms, with users displaying higher optimism levels more likely to engage with e-government services due to their belief in the benefits and effectiveness of these platforms [24]. This work contributes to the existing literature by theoretically examining the impacts of optimism bias within online retail supply chains.

This paper, therefore, also contributes to the growing body of literature on online retailing, or e-commerce supply chain management, an increasingly crucial field as global commerce transitions to digital platforms. Ref. [25] identifies three predominant e-commerce supply chain models, analyzing their operational processes and the common challenges they encounter. Ref. [26] highlight the productivity gains, cost savings, and economic growth potential enabled by integrating AI and automation into e-commerce operations. Additionally, ref. [27] explore how factors such as delivery speed, vehicle efficiency, and logistics network design impact the environmental footprint of e-commerce supply chains. Their findings suggest that optimizing these parameters can significantly reduce GHG emissions. Innovations in supply chain finance, particularly through blockchain technology, present new opportunities for e-commerce platforms to enhance efficiency. Ref. [28] argues that incorporating blockchain in supply chain finance can address issues like credit rationing and information asymmetry, providing a robust framework for financial integration within supply chains.

This work also relates to the literature examining strategic interactions between e-commerce platforms and third-party sellers. For example, ref. [29] study how an e-commerce platform’s encroachment decisions impact a seller’s sales effort. Ref. [30] explore the optimal bundling strategy for an e-commerce platform offering products from independent sellers. Ref. [31] analyze contract decisions between an e-commerce platform and a third-party seller with superior demand information. Recently, this discussion has expanded to topics such as information sharing [28,32,33], logistics service sharing [34,35], and platform-based financing for sellers [36,37,38].

Note that some research focuses on the impacts of risk appetite, which refers to the behavioral and strategic preference regarding the level of uncertainty or risk a decision-maker is willing to accept to achieve specific objectives, in an e-commerce management context e.g., [39,40]. This paper examines a prevalent yet understudied cognitive bias in online retail supply chains, namely optimism bias. In the current study, optimism is defined as a cognitive bias where decision-makers overestimate the likelihood of favorable outcomes or underestimate the likelihood of unfavorable ones. Unlike optimism, risk appetite involves a more deliberate evaluation of the trade-offs between potential gains and losses, even if subjective preferences influence this evaluation. In this sense, while optimism and risk appetite can lead to risk-taking behavior, their mechanisms differ. Optimism influences behavior through misjudgment of probabilities, while risk appetite governs behavior through conscious trade-offs based on risk tolerance. As a result, this work contributes to the existing literature by demonstrating how the ORSC model in this paper adds to understanding a belief system influenced by psychological factors instead of simply reflecting the impacts of a rational assessment of risks and rewards.

3. Basic Model

This paper considers an online retail supply chain (ORSC) comprising one third-party seller (she) and one e-commerce platform (he), as illustrated in Figure 1. The seller orders a product and sells it to the end consumers via the platform. The per-unit cost of ordering the product is c (≥0). The third-party seller determines the order quantity q and the unit retail price p for consumers. For each unit sold through the platform, the platform charges a fixed commission rate, $\eta \in (0,1)$, representing a proportion of the revenue; the remaining revenue goes to the seller. Both parties are assumed to be risk-neutral.

3.1. Demand Uncertainty

The demand function from final consumers is assumed to be linear, represented as:

$$d=\alpha -\beta p,$$

(1)

where $\alpha$ denotes the market size (potential). To incorporate uncertainty, I assume that $\alpha$ is a random variable following a two-point (Bernoulli) distribution. Specifically, $\alpha$ can take one of two values: ${\alpha }_l\ge 0$ (representing a ‘bad’ market) and ${\alpha }_h\ge {\alpha }_l$ (representing a ‘good’ market), with associated probabilities $\lambda$ and $1-\lambda$, respectively. Label the ${\alpha }_h$ and ${\alpha }_l$ as a good and bad market, respectively. For simplicity, I assume that $\beta >0$ is a known parameter for both players.

This representation of uncertainty effectively captures the fundamental concept of market unpredictability [41] and is commonly applied in economics, marketing, and operations literature e.g., [17,42,43]. Let $\overline{\alpha }:=(1-\lambda ){\alpha }_h+\lambda {\alpha }_l$ denote the expected market potential. I assume that $\overline{\alpha }>\beta c$ to avoid trivial cases (see the proof of Lemma 1 for details).

3.2. Information Structure and Sequence of Moves

The analysis begins by outlining the information structure of the model. Initially, both the platform and the seller face uncertainty regarding market potential. Firms frequently commit to marketing actions in industries such as books, recorded music, and seasonal goods before market uncertainty is resolved [44]. Following standard industry practices, I assume that the platform and seller must set the commission rate and order quantity, respectively, prior to the revelation of market potential.

Notably, sellers are generally better positioned to predict market potential due to their expertise in product attributes and understanding of consumer perceptions [31]. To capture the information asymmetry between the platform and the seller regarding market potential, this work follows [43] and assumes that the seller learns the actual market potential after ordering the product but before setting the retail price.

Given these desirable properties, the sequence of moves resembles a Stackelberg model (see Figure 2 for the sequence of moves), highlighting the role of the e-commerce platform as a key player affecting sellers’ pricing and inventory strategies: (1) The platform, acting as the Stackelberg leader, sets the commission rate $\eta$. (2) Based on the platform’s commission policy, the seller decides on the order quantity q. (3) Once the stock is available, the seller sets the retail price p for the end consumers. The actual state of market potential is revealed to the seller between the second and third stages.

3.3. Equilibrium Analysis

By doing so, the model in this paper captures key aspects of online retail supply chains, such as platform influence and demand-side dynamics. Next, I characterize the equilibrium outcomes of this theoretical game model. In this paper, lemmas are presented as intermediate results that serve to establish the validity of the key statements, which are formalized as propositions. Propositions represent the main findings of the study, offering theoretical insights into the dynamics of the model. Once retail demand is realized, if the realized demand is less than or equal to the seller’s stocking level ($d\le q$), all demand will be satisfied, and the leftover inventory $d-q$ is salvaged at a value normalized to zero without loss of generality. Conversely, if the realized demand exceeds the seller’s stocking level ($d>q$), the seller meets q units of demand, resulting in lost sales for the unsatisfied demand.

In the third stage, the cost of ordering is sunk. Thus, for the specification of market potential, the seller sets the retail price p to maximize her revenue

$$\left\{\begin{array}{cc}\left(1-\eta \right)p\left(\alpha -\beta p\right)\hfill & \mathrm{if} \alpha -\beta p<q\hfill \\ \left(1-\eta \right)pq\hfill & \mathrm{if} \alpha -\beta p\ge q.\hfill \end{array}\right.$$

In fact, the optimal retail price $p^{*}=\alpha /\left(2\beta \right)$ if $q>\alpha /2$; otherwise, the optimal price $p^{*}=(\alpha -q)/\beta$. Given $q^{*}$, the seller’s optimal profit is

$$\left\{\begin{array}{cc}\left(1-\eta \right){\displaystyle \frac{{\alpha }^2}{4\beta }}-cq\hfill & \mathrm{if} q>\alpha /2\hfill \\ (1-\eta ){\displaystyle \frac{(\alpha -q)q}{\beta }}-cq\hfill & \mathrm{if} q\le \alpha /2.\hfill \end{array}\right.$$

In the second stage, the market potential is uncertain with two possibilities—$\alpha$ is either high or low. Let $r_1\left(q\right):=[\lambda {\alpha }_l^2+4(1-\lambda )({\alpha }_h-q)q]/\left(4\beta \right)$ and $r_2\left(q\right):=[\lambda ({\alpha }_l-q)q+(1-\lambda )({\alpha }_h-q)q]/\beta$. Thus, the seller’s problem in the second stage is equivalent to setting order quantity q to maximize her expected profit beyond the commission rate $\eta$:

$$\left\{\begin{array}{cc}\left(1-\eta \right){\displaystyle \frac{\lambda {\alpha }_l^2+\left(1-\lambda \right){\alpha }_h^2}{4\beta }}-cq\hfill & \mathrm{if} q>{\alpha }_h/2\hfill \\ \left(1-\eta \right)r_1\left(q\right)-cq\hfill & \mathrm{if} {\alpha }_l/2<q\le {\alpha }_h/2\hfill \\ \left(1-\eta \right)r_2\left(q\right)-cq\hfill & \mathrm{if} q\le {\alpha }_l/2.\hfill \end{array}\right.$$

(2)

Lemma 1 characterizes the seller’s optimal ordering quantity $q^{*}$ as a function of $\eta$.

Lemma 1. 

Define $q_1^{*}\left(\eta \right):={\displaystyle \frac{{\alpha }_h\left(1-\eta \right)\left(1-\lambda \right)-c\beta }{2\left(1-\eta \right)\left(1-\lambda \right)}}$ and $q_2^{*}\left(\eta \right):={\displaystyle \frac{\overline{\alpha }\left(1-\eta \right)-c\beta }{2\left(1-\eta \right)}}$. For a given η,

(a) 

The seller’s optimal order quantity

$$q^{*}\left(\eta \right)=\left\{\begin{array}{cc}{q}_1^{*}\left(\eta \right)\hfill & if \eta <z\left(\lambda \right)\hfill \\ q_2^{*}\left(\eta \right)\hfill & if z\left(\lambda \right)\le \eta <1-\beta c/\overline{\alpha }\hfill \\ 0\hfill & if\eta \ge 1-\beta c/\overline{\alpha },\hfill \end{array}\right.$$

(3)

where $z\left(x\right):=1-{\displaystyle \frac{\beta c}{({\alpha }_h-{\alpha }_l)(1-x)}}$.

(b) 

Consequently,$q^{*}\left(\eta \right)$ is decreasing in η, i.e., $\partial q^{*}\left(\eta \right)/\partial \eta \le 0$.

Lemma 1 indicates that when the commission rate is high ($\eta \ge 1-\beta c/\overline{\alpha }$), it is not optimal for the seller to place an order. For the seller to find it profitable to order, the platform must set a relatively low commission rate ($\eta <1-\beta c/\overline{\alpha }$), allowing the marginal revenue of ordering at zero stocking level ($(1-\eta )\overline{\alpha }/\beta$) to exceed the unit ordering cost (c). In this scenario, ordering becomes beneficial for the seller.

Furthermore, if the commission rate is sufficiently low ($\eta <z\left(\lambda \right)$), the seller’s order quantity matches the realized demand when the market potential is high, i.e., $q_1^{*}\left(\eta \right)={\alpha }_h-\beta p^{*}$. In this case, the order quantity will exceed demand in the low market potential state, which I refer to as the overstocking scenario. Conversely, if the commission rate is moderate ($z\left(\lambda \right)\le \eta <1-\beta c/\overline{\alpha }$), understocking may occur. Here, the seller’s order quantity is less than the realized demand in the high market potential state but equal to that in the low market potential state, i.e., $q_2^{*}\left(\eta \right)={\alpha }_l-\beta p^{*}$.

Lastly, Lemma 1 implies that the optimal stocking level $q^{*}$ decreases as the commission rate $\eta$ increases. Intuitively, the seller is more motivated to improve their stocking level when they retain a larger share of the revenues.

Note that when $\eta <z\left(\lambda \right)$, ${\alpha }_l/2<q_1^{*}\left(\eta \right)<{\alpha }_h/2$, whereas, when $z\left(\lambda \right)\le \eta <1-\beta c/\overline{\alpha }$, $q_2^{*}\left(\eta \right)\le {\alpha }_l/2$. Thus, from (2), the seller’s equilibrium profit is

$${\pi }_S\left(\eta \right):=\left\{\begin{array}{cc}\left(1-\eta \right)r_1\left(q_1^{*}\right)-c·q_1^{*}\hfill & \mathrm{if} \eta <z\left(\lambda \right)\hfill \\ \left(1-\eta \right)r_2\left(q_2^{*}\right)-c·q_2^{*}\hfill & \mathrm{if} z\left(\lambda \right)\le \eta <1-\beta c/\overline{\alpha }\hfill \\ 0\hfill & \mathrm{if} \eta \ge 1-\beta c/\overline{\alpha }.\hfill \end{array}\right.$$

(4)

Recall that the platform charges a fixed proportion $\eta$ of the seller’s revenue. Therefore, the platform’s problem in the first stage is equivalent to setting the commission rate $\eta$ in order to maximize his expected profit

$$\underset{\eta }{max}{\pi }_P\left(\eta \right):=\left\{\begin{array}{cc}\eta ·r_1\left(q_1^{*}\right)\hfill & \mathrm{if} \eta <z\left(\lambda \right)\hfill \\ \eta ·r_2\left(q_2^{*}\right)\hfill & \mathrm{if} z\left(\lambda \right)\le \eta <1-\beta c/\overline{\alpha }\hfill \\ 0\hfill & \mathrm{if} \eta \ge 1-\beta c/\overline{\alpha }.\hfill \end{array}\right.$$

(5)

Let ${\eta }_1^{*}$ denote the solution to

$${\displaystyle \frac{c^2{\beta }^2\left(1+\eta \right)}{{\left(1-\eta \right)}^3}}={\alpha }_h^2{\left(1-\lambda \right)}^2+{\alpha }_l^2\lambda \left(1-\lambda \right),$$

(6)

and ${\eta }_2^{*}$ denote the solution to

$${\displaystyle \frac{c^2{\beta }^2(1+\eta )}{{(1-\eta )}^3}}={\overline{\alpha }}^2.$$

(7)

Lemma 2. 

Let $f\left(x\right):=x^2/{(1-x)}^3$. Then, there exists a threshold m such that $f\left({\eta }_1^{*}\right)/f\left({\eta }_2^{*}\right)=1-\lambda$ when $\beta c=m$. Thus, the solution of (5) is ${\eta }^{*}={\eta }_1^{*}$ if $c<m/\beta$ and ${\eta }^{*}={\eta }_2^{*}$ if $c>m/\beta$. Moreover, ${\eta }_1^{*}<{\eta }_2^{*}$.

Lemma 2 suggests that when the ordering cost is low ($c<m/\beta$), the platform’s equilibrium choice leads to an overstocking equilibrium, as ${\eta }_1^{*}<z\left(\lambda \right)$. Conversely, if the ordering cost is high ($c>m/\beta$), the platform opts for a relatively high commission rate (${\eta }_2^{*}>z\left(\lambda \right)$), resulting in an understocking equilibrium.

4. Analysis and Results

4.1. Optimistic Seller

This paper first analyzes a scenario where the online platform remains unbiased while the seller is the only optimistic party. Optimism bias refers to the tendency of individuals to overestimate the likelihood of positive outcomes while underestimating the probability of negative ones. Under this assumption, the optimistic seller perceives the market potential as $\phi$ rather than $\alpha$, where $\phi$ can take one of two values, ${\alpha }_l$ and ${\alpha }_h$, with probabilities $\kappa ·\lambda$ and $1-\kappa ·\lambda$, respectively, for $\kappa \in [0,1]$.

The parameter $\kappa \in [0,1]$ represents the level of optimism bias. When $\kappa <1$, the seller exhibits optimism bias by underestimating the likelihood of experiencing a low market state as $\kappa \lambda <\lambda$. The smaller the value of $\kappa$, the greater the seller’s bias. At the extreme of $\kappa =0$, the seller demonstrates infinite optimism, acting as though the market potential is constantly in its high state, ${\alpha }_h$. Conversely, $\kappa =1$ indicates an unbiased seller. This approach to modeling optimism bias, by reflecting an underestimation of negative event risks, has been widely applied to gain insights. For instance, ref. [45] model optimism as a cognitive bias where an agent underestimates the likelihood of encountering unfavorable economic conditions. Similarly, ref. [46] assume that an optimistic manager underestimates the probability of facing low-valuation consumers.

When the platform is unbiased, his commission rate policy follows standard theory. Specifically, the platform’s commission rate decision aligns with Lemma 2: if the ordering cost is relatively low $(c<m/\beta )$, then the commission rate is ${\eta }_1^{*}$, whereas if the ordering cost is relatively high $(c>m/\beta )$, then the equilibrium commission rate is ${\eta }_2^{*}$. With an optimism parameter $\kappa$, the biased seller behaves as though her profit is given by expression (2), but with $\kappa \lambda$ replacing $\lambda$. Let $\overline{\phi }:=(1-\kappa \lambda ){\alpha }_h+\kappa \lambda {\alpha }_l$. For a given $\eta$, form Lemma 1, the biased seller places an order of

$$\left\{\begin{array}{cc}{\widehat{q}}_1(\kappa ,\eta )\hfill & \mathrm{if} \eta <z\left(\kappa \lambda \right)\hfill \\ {\widehat{q}}_2(\kappa ,\eta )\hfill & \mathrm{if} z\left(\kappa \lambda \right)\le \eta <1-\beta c/\overline{\phi }\hfill \\ 0\hfill & \mathrm{if} \eta \ge 1-\beta c/\overline{\phi },\hfill \end{array}\right.$$

where ${\widehat{q}}_1(\kappa ,\eta ):={\displaystyle \frac{{\alpha }_h\left(1-\eta \right)\left(1-\kappa \lambda \right)-c\beta }{2\left(1-\eta \right)\left(1-\kappa \lambda \right)}}$ and ${\widehat{q}}_2(\kappa ,\eta ):={\displaystyle \frac{\overline{\phi }\left(1-\eta \right)-c\beta }{2\left(1-\eta \right)}}$.

Lemma 3. 

For a given optimism parameter κ,

(a) 

When $c<m/\beta$, the seller’s equilibrium order quantity $\widehat{q}\left(\kappa \right)={\widehat{q}}_1(\kappa ,{\eta }_1^{*})$, whereas when $c>m/\beta$,

$$\widehat{q}\left(\kappa \right)=\left\{\begin{array}{cc}{\widehat{q}}_1(\kappa ,{\eta }_2^{*})\hfill & if \kappa <z\left({\eta }_2^{*}\right)/\lambda \hfill \\ {\widehat{q}}_2(\kappa ,{\eta }_2^{*})\hfill & if \kappa \ge z\left({\eta }_2^{*}\right)/\lambda .\hfill \end{array}\right.$$

(b) 

Consequently, $\widehat{q}\left(\kappa \right)$ decreases in κ.

Lemma 3 indicates that the distortion caused by optimism results in the equilibrium order quantity decreasing as the optimism parameter $\kappa$ increases. In other words, the more optimistic a seller is about market potential, the more they order. This occurs because the seller tends to purchase a higher quantity to meet the overestimated market demand. As a result, the stock level ordered by an optimistic seller is consistently higher than that ordered by an unbiased seller.

When $c<m/\beta$, I have ${\alpha }_l/2<\widehat{q}\left(\kappa \right)={\widehat{q}}_1(\kappa ,{\eta }_1^{*})<{\alpha }_h/2$ (see the proof of Lemma 3). From (2), the ensuing seller’s equilibrium profit is ${\widehat{\pi }}_S\left(\kappa \right)=(1-{\eta }_1^{*})·r_1\left({\widehat{q}}_1(\kappa ,{\eta }_1^{*})\right)-c·{\widehat{q}}_1(\kappa ,{\eta }_1^{*})$, while the platform’s equilibrium profit is ${\widehat{\pi }}_P\left(\kappa \right)={\eta }_1^{*}·r_1\left({\widehat{q}}_1(\kappa ,{\eta }_1^{*})\right)$. When $c>m/\beta$, similarly, it can be verified that

$$\left\{\begin{array}{cc}{\alpha }_l/2<\widehat{q}\left(\kappa \right)={\widehat{q}}_1(\kappa ,{\eta }_2^{*})<{\alpha }_h/2\hfill & \mathrm{if} \kappa <z\left({\eta }_2^{*}\right)/\lambda \hfill \\ \widehat{q}\left(\kappa \right)={\widehat{q}}_2(\kappa ,{\eta }_2^{*})\le {\alpha }_l/2\hfill & \mathrm{if} \kappa \ge z\left({\eta }_2^{*}\right)/\lambda .\hfill \end{array}\right.$$

Thus, from (2), the ensuing seller’s equilibrium profit is

$${\widehat{\pi }}_S\left(\kappa \right)=\left\{\begin{array}{cc}(1-{\eta }_2^{*})·r_1\left({\widehat{q}}_1(\kappa ,{\eta }_2^{*})\right)-c·{\widehat{q}}_1(\kappa ,{\eta }_2^{*})\hfill & \mathrm{if} \kappa <z\left({\eta }_2^{*}\right)/\lambda \hfill \\ (1-{\eta }_2^{*})·r_2\left({\widehat{q}}_2(\kappa ,{\eta }_2^{*})\right)-c·{\widehat{q}}_2(\kappa ,{\eta }_2^{*})\hfill & \mathrm{if} \kappa \ge z\left({\eta }_2^{*}\right)/\lambda ,\hfill \end{array}\right.$$

while the platform’s equilibrium profit is

$${\widehat{\pi }}_P\left(\kappa \right)=\left\{\begin{array}{cc}{\eta }_2^{*}·r_1\left({\widehat{q}}_1(\kappa ,{\eta }_2^{*})\right)\hfill & \mathrm{if}\kappa <z\left({\eta }_2^{*}\right)/\lambda \hfill \\ {\eta }_2^{*}·r_2\left({\widehat{q}}_2(\kappa ,{\eta }_2^{*})\right)\hfill & \mathrm{if}\kappa \ge z\left({\eta }_2^{*}\right)/\lambda .\hfill \end{array}\right.$$

I next investigate the impact of optimism on the platform’s profit ${\widehat{\pi }}_P\left(\kappa \right)$, and the seller’s expected profit ${\widehat{\pi }}_S\left(\kappa \right)$.

Proposition 1. 

When the platform is unbiased while the seller is biased, described by optimism parameter κ, the seller’s profit ${\widehat{\pi }}_S\left(\kappa \right)$ is increasing in κ. However, the platform’s profit ${\widehat{\pi }}_P\left(\kappa \right)$ is decreasing in κ.

Proposition 1 reveals an intriguing dynamic: while seller optimism invariably leads to self-harm, it serves as an advantage for the platform. To illustrate this, I plot how ${\widehat{\pi }}_S\left(\kappa \right)$ and ${\widehat{\pi }}_P\left(\kappa \right)$ change with respect to the parameter $\kappa$ in Figure 3. Let ${\alpha }_h=20$, ${\alpha }_l=10$, $\lambda =0.4$, $\beta =1$, and $c=5$, which ensure that $\overline{\alpha }>\beta c$. As per this figure, the seller’s profit increases with the parameter $\kappa$, while the platform’s profit decreases with the parameter $\kappa$. In other words, as the seller becomes more biased ($\kappa$ decreases), the seller is worse off while the platform’s performance improves. This can be explained as follows. The optimistically biased seller consistently over-orders relative to an unbiased seller, leading to sub-optimal decision-making that ultimately harms the seller’s profitability. However, higher levels of seller optimism drive up stocking levels, allowing the platform to generate increased revenue from product sales and ultimately benefiting from the seller’s optimism.

In essence, seller optimism creates a win-lose situation: while it benefits the platform, this gain comes at the seller’s expense. Given these dynamics, I focus on the system’s profit ${\widehat{\pi }}_E\left(\kappa \right):={\widehat{\pi }}_S\left(\kappa \right)+{\widehat{\pi }}_P\left(\kappa \right)$ to assess the effect of seller optimism on the overall supply chain.

Proposition 2. 

When the platform is unbiased while the seller is described by optimism parameter κ, there exists a threshold $\widehat{\kappa }$, $0\le \widehat{\kappa }<1$, such that ${\widehat{\pi }}_E$ decreases with κ over a region of $\kappa \in [\widehat{\kappa },1]$.

Proposition 2 suggests that the entire ORSC can potentially benefit from seller optimism. For illustration, Figure 3 indicates that the whole performance of ORSC improves as the parameter $\kappa$ decreases. Specifically, seller optimism can bring more advantages to the platform than the harm it imposes on the seller. To explore the intuition behind Proposition 2, I consider a hypothetical scenario where an unbiased central planner maximizes the combined expected profits of the platform and seller to determine the first-best order quantity. The expected profit for this centralized system, given an order quantity q, is represented as expression (2) with $\eta =0$. Therefore, the first-best order quantity is $q^{*}\left(0\right)$, as characterized by Lemma 1. Specifically,

$$q^{*}\left(0\right)=\left\{\begin{array}{cc}{q}_1^{*}\left(0\right)\hfill & \mathrm{if} 0<z\left(\lambda \right)\hfill \\ q_2^{*}\left(0\right)\hfill & \mathrm{if} 0\ge z\left(\lambda \right).\hfill \end{array}\right.$$

(8)

Intuitively, the overall performance of ORSC hinges on the relationship between the equilibrium order quantity $\widehat{q}\left(\kappa \right)$ and the system-wide first-best stock level $q^{*}\left(0\right)$. The closer $\widehat{q}\left(\kappa \right)$ is to $q^{*}\left(0\right)$, the better is the equilibrium performance of ORSC. According to Lemma 1b, if the channel were unbiased ($\kappa =1$), $q^{*}\left(0\right)$ would always exceed $\widehat{q}(\kappa =1)$, meaning that the decentralized, unbiased channel under-orders relative to the first-best benchmark. This under-ordering occurs because, in a decentralized setting, the seller only captures a portion of the product revenue, reducing their incentive to order optimally.

This observation suggests that decentralization within the e-commerce supply chain introduces distortions relative to the quantity a central planner would ideally set. However, as $\widehat{q}\left(\kappa \right)$ decreases with $\kappa$, the order quantity increases as the channel becomes more optimistic (as shown in Lemma 3). Consequently, optimism can counterbalance the distortion caused by decentralization, thereby enhancing the performance of the channel as a whole.

Proposition 3. 

If $({\alpha }_h-{\alpha }_l){(1-\lambda )}^2/\beta <c<m/\beta$, then there exists ${\kappa }^{*}\in (0,1)$ such that the seller with optimism level $\kappa ={\kappa }^{*}$ orders the same quantity that an (unbiased) central planner orders, i.e., $\widehat{q}\left({\kappa }^{*}\right)=q^{*}\left(0\right)$.

Proposition 3 further suggests that the order quantity in a biased supply chain can match that of an unbiased, centralized supply chain. In other words, seller optimism serves as a bias that can counteract the under-ordering effect typically caused by decentralization, enabling the biased channel to achieve an order quantity equivalent to its centralized benchmark.

Together, Propositions 2 and 3 demonstrate that seller optimism can benefit the entire channel and may even facilitate system-wide coordination. However, it is essential to note that seller optimism consistently results in self-harm. Next, I examine the case of platform optimism, showing that while platform optimism is always detrimental to the channel as a whole, it does not necessarily lead to self-harm for the platform itself.

4.2. Optimistic Platform

In parallel with Section 4.1, I now examine the case where the platform is biased while the seller remains unbiased, allowing us to isolate the effects of platform optimism. When the platform’s optimism level is $\kappa$, he expects the seller’s ordering decision for a given $\eta$ to follow expression (3), with $\kappa \lambda$ substituting for $\lambda$. Define ${\widehat{r}}_1\left(q\right):=[\kappa \lambda {\alpha }_l^2+4(1-\kappa \lambda )({\alpha }_h-q)q]/\left(4\beta \right)$ and ${\widehat{r}}_2\left(q\right):=[\kappa \lambda ({\alpha }_l-q)q+(1-\kappa \lambda )({\alpha }_h-q)q]/\beta$. Thus, from (5), the biased platform behaves as if he were enjoying a profit of

$$\left\{\begin{array}{cc}\eta ·{\widehat{r}}_1\left({\widehat{q}}_1\right)\hfill & if\eta <z\left(\kappa \lambda \right)\hfill \\ \eta ·{\widehat{r}}_2\left({\widehat{q}}_2\right)\hfill & ifz\left(\kappa \lambda \right)\le \eta <1-\beta c/\overline{\phi }\hfill \\ 0\hfill & if\eta \ge 1-\beta c/\overline{\phi }.\hfill \end{array}\right.$$

Let ${\widehat{\eta }}_1\left(\kappa \right)$ denote the solution to (6), except with $\kappa \lambda$ in place of $\lambda$, while let ${\widehat{\eta }}_2\left(\kappa \right)$ denote the solution to (7), except with $\kappa \lambda$ in place of $\lambda$.

Proposition 4. 

For a given optimism parameter κ,

(a) 

There exists a threshold $\widehat{m}$ such that $f\left({\widehat{\eta }}_1\right)/f\left({\widehat{\eta }}_2\right)=1-\kappa \lambda$ when $\beta c=\widehat{m}$. Thus, the biased platform’s commission rate $\widehat{\eta }\left(\kappa \right)={\widehat{\eta }}_1\left(\kappa \right)$ if $c<\widehat{m}/\beta$ and $\widehat{\eta }\left(\kappa \right)={\widehat{\eta }}_2\left(\kappa \right)$ if $c>\widehat{m}/\beta$.

(b) 

Consequently, $\widehat{\eta }\left(\kappa \right)$ is decreasing in κ—that is, $\partial {\widehat{\eta }}_1\left(\kappa \right)/\partial \kappa <0$ and $\partial {\widehat{\eta }}_2\left(\kappa \right)/\partial \kappa <0$.

Similar to Lemma 2, Proposition 4 indicates that a biased platform sets a low commission rate when the ordering cost is low and a high commission rate when the ordering cost is high. Moreover, Proposition 4 shows that optimism bias distorts the equilibrium commission rate, which decreases with optimism parameter $\kappa$. Intuitively, as the platform becomes more optimistic, it anticipates that the seller will order more products at a given $\eta$ and thus sets a higher commission rate to capitalize on the additional revenue from increased stocking.

Given the biased commission rate $\widehat{\eta }\left(\kappa \right)$, the unbiased seller’s ordering decision accordingly follows Lemma 1. Consequently, the ensuing seller’s expected profit is ${\tilde{\pi }}_S\left(\kappa \right):={\pi }_S\left(\widehat{\eta }\right)$, as defined in (4), while the platform’s equilibrium profit is ${\tilde{\pi }}_P\left(\kappa \right):={\pi }_P\left(\widehat{\eta }\right)$, as defined in (5). Let ${\tilde{\pi }}_E\left(\kappa \right):={\tilde{\pi }}_S\left(\kappa \right)+{\tilde{\pi }}_P\left(\kappa \right)$ be the supply chain’s total profit.

Proposition 5. 

When the seller is unbiased while the platform is biased with optimism parameter κ, neither the seller nor the whole system benefits from platform optimism—that is, $\partial {\tilde{\pi }}_S\left(\kappa \right)/\partial \kappa >0$ and $\partial {\tilde{\pi }}_E\left(\kappa \right)/\partial \kappa >0$.

In contrast to the finding that seller optimism can benefit the supply chain as a whole (Proposition 2), Proposition 5 indicates that platform optimism consistently harms the entire channel. To illustrate these results, I plot the performance of the retailer, the supplier, and the system in Figure 4. To ensure that $\overline{\alpha }>\beta c$, I also assume that ${\alpha }_h=20$, ${\alpha }_l=10$, $\lambda =0.4$, $\beta =1$, and $c=5$. As per this figure, the seller’s and system’s profits increase with the parameter $\kappa$. In other words, as the platform becomes more biased ($\kappa$ decreases), both the seller and the whole system are worse off. The reasoning behind this is that the seller typically under-orders relative to the optimal stocking level, and platform optimism drives up the commission rate, pushing the stocking level even further from this first-best benchmark, thereby negatively impacting the entire system. Additionally, Proposition 5 underscores that platform optimism harms the unbiased seller, as a more optimistic platform claims a larger share of sales revenue.

Although Figure 4 shows that the platform’s profit decreases as the platform becomes more biased, Proposition 6 indicates that this may not be true under some conditions. In other words, the platform can benefit from optimism bias in the same way as in the optimistic seller scenario.

Proposition 6. 

When the seller is unbiased while the platform is biased with optimism parameter κ, there exist thresholds $t_1$ and $t_2$ such that

(a) 

The platform’s profit ${\tilde{\pi }}_P\left(\kappa \right)$ always increases in κ either when $c\le t_1$ or when $c\ge {\displaystyle \frac{2{({\alpha }_h-{\alpha }_l)}^2}{\beta {\alpha }_h}}$.

(b) 

The platform’s profit ${\tilde{\pi }}_P\left(\kappa \right)$ can decrease in κ when $t_1<c<t_2$ and ${\alpha }_h/{\alpha }_l<5/2$.

In contrast to seller optimism (Proposition 1), Proposition 6 shows that platform optimism does not necessarily lead to self-harm. Specifically, when the ordering cost is relatively moderate $(t_1<c<t_2)$, the platform can benefit from his own bias. This is noteworthy because an unbiased platform can accurately anticipate the seller’s response to his commission rate and set an optimal rate accordingly. However, as the platform’s optimism bias increases, his chosen commission rate $\widehat{\eta }\left(\kappa \right)$ deviates further from the unbiased commission rate $\widehat{\eta }\left(0\right)$. Figure 5 illustrates how $\widehat{\eta }\left(\kappa \right)$ varies as $\kappa$ decreases, along with the corresponding changes in the platform’s profit based on its commission rate. The figure depicts three scenarios with differing equilibrium profits. For a relatively low (or high) ordering cost, i.e., $c\le t_1$ (or $c\ge {\displaystyle \frac{2{({\alpha }_h-{\alpha }_l)}^2}{\beta {\alpha }_h}}$), the platform’s equilibrium commission rate is ${\widehat{\eta }}_1$ (or ${\widehat{\eta }}_2$), respectively. As the platform’s optimism bias increases, it sets a higher commission rate, which can harm its own performance. However, when the ordering cost is moderate ($t_1<c<t_2$), an increase in the commission rate due to platform bias can substantially improve the platform’s profit, particularly when ${\widehat{\eta }}_1\in \left[z\left(\lambda \right),{\eta }_2^{*}\right]$.

4.3. Optimistic Platform and Optimistic Seller

The analysis of Section 4.1 and Section 4.2 demonstrated that while seller optimism can benefit the entire channel, platform optimism is consistently detrimental to the system. To synthesize the findings from Section 3 and Section 4, I now examine the combined effect of seller and platform optimism by considering a scenario where both the platform and seller are optimistic simultaneously. In this section, I assume that both parties’ optimism levels are identical. This assumption removes any asymmetry in optimism, allowing for a more explicit analysis of the aggregate impact of combined seller and platform optimism.

With a typical optimism level $\alpha$, the biased platform sets the commission rate according to Proposition 4a. It is easy to verify that ${\widehat{\eta }}_1\left(\kappa \right)<z\left(\kappa \lambda \right)$ and $z\left(\kappa \lambda \right)\le {\widehat{\eta }}_2\left(\kappa \right)<1-\beta c/\overline{\phi }$. Thus, the equilibrium order quantity of the biased seller is

$$\breve{q}\left(\kappa \right)=\left\{\begin{array}{cc}{\widehat{q}}_1(\kappa ,{\widehat{\eta }}_1)\hfill & \mathrm{if} c<\widehat{m}/\beta \hfill \\ {\widehat{q}}_2(\kappa ,{\widehat{\eta }}_2)\hfill & \mathrm{if} c>\widehat{m}/\beta .\hfill \end{array}\right.$$

Lemma 4. 

$\breve{q}\left(\kappa \right)$ is decreasing in κ.

Lemma 4 extends the findings from the seller-only optimism case (Lemma 3b), showing that the equilibrium order quantity increases as both the seller and platform become more biased. Intuitively, seller and platform optimism exert opposing influences on the equilibrium order quantity. Platform optimism has a negative effect, as a biased platform sets a higher commission rate, prompting the seller to reduce inventory levels. Conversely, seller optimism has a positive impact; a more substantial optimism bias encourages the seller to order more, aligning with Lemma 3b. When both seller and platform optimism are present, the (positive) effect of seller optimism is substantial enough to outweigh the (negative) impact of platform optimism, ultimately leading the biased seller to increase their order quantity.

Given $\widehat{\eta }\left(\kappa \right)$ and $\breve{q}\left(\kappa \right)$, the ensuing seller’s expected profit is

$${\breve{\pi }}_S\left(\kappa \right)=\left\{\begin{array}{cc}(1-{\widehat{\eta }}_1)·r_1\left({\widehat{q}}_1(\kappa ,{\widehat{\eta }}_1)\right)-c·{\widehat{q}}_1(\kappa ,{\widehat{\eta }}_1)\hfill & \mathrm{if} c<\widehat{m}/\beta \hfill \\ (1-{\widehat{\eta }}_2)·r_2\left({\widehat{q}}_2(\kappa ,{\widehat{\eta }}_2)\right)-c·{\widehat{q}}_2(\kappa ,{\widehat{\eta }}_2)\hfill & \mathrm{if} c>\widehat{m}/\beta .\hfill \end{array}\right.$$

Accordingly, the platform’s equilibrium profit is

$${\breve{\pi }}_P\left(\kappa \right)=\left\{\begin{array}{cc}{\widehat{\eta }}_1·r_1\left({\widehat{q}}_1(\kappa ,{\widehat{\eta }}_1)\right)\hfill & \mathrm{if} c<\widehat{m}/\beta \hfill \\ {\widehat{\eta }}_2·r_2\left({\widehat{q}}_2(\kappa ,{\widehat{\eta }}_2)\right)\hfill & \mathrm{if} c>\widehat{m}/\beta .\hfill \end{array}\right.$$

Proposition 7. 

When both the platform and seller are biased as described by optimism parameter κ, we have ${\left.{\displaystyle \frac{\partial {\breve{\pi }}_S}{\partial \kappa }}\right|}_{\kappa =1}>0$ and ${\left.{\displaystyle \frac{\partial {\breve{\pi }}_P}{\partial \kappa }}\right|}_{\kappa =1}<0$.

Recall that both seller and platform optimism are consistently harmful to the seller (as shown in Proposition 1 and Proposition 5). In the scenario where both the seller and platform are optimistic, the analysis shows that optimism within the e-commerce supply chain can further disadvantage the seller, as indicated by ${\left.{\displaystyle \frac{\partial {\breve{\pi }}_S}{\partial \kappa }}\right|}_{\kappa =1}>0$. On the other hand, Proposition 1 and Proposition 6b demonstrate that seller and platform optimism can each potentially benefit the platform individually. In this light, Proposition 7 further shows that the combined effect of seller and platform optimism can also benefit the platform, with ${\left.{\displaystyle \frac{\partial {\breve{\pi }}_P}{\partial \kappa }}\right|}_{\kappa =1}<0$.

I have shown that seller and platform optimism exert opposite effects on the overall supply chain performance. Specifically, while the entire system can benefit from seller optimism (Proposition 2), it consistently suffers from platform optimism (Proposition 5). Given these insights, I now examine the combined effect of seller and platform optimism on the system’s performance. Define ${\breve{\pi }}_E\left(\kappa \right):={\breve{\pi }}_S\left(\kappa \right)+{\breve{\pi }}_P\left(\kappa \right)$ as the whole system’s expected profit.

Proposition 8. 

For a given optimism parameter κ, $\partial {\breve{\pi }}_E\left(\kappa \right)/\partial \kappa <0$ holds in the following conditions: (a) When $\beta c<\widehat{m}$ while ${\widehat{\eta }}_1>{\displaystyle \frac{\lambda -\kappa \lambda }{1-\kappa \lambda }}$. (b) When $\beta c>\widehat{m}$ while ${\widehat{\eta }}_2>{\displaystyle \frac{\overline{\phi }-\overline{\alpha }}{\overline{\phi }-\overline{\alpha }+\beta c}}$.

Proposition 8 suggests that channel profit can increase as optimism bias intensifies, thereby enhancing the supply chain’s overall performance. The benefits that seller optimism brings to the system outweigh the negative effects of platform optimism. Intuitively, optimism bias in both the seller and the platform drives a higher stocking level. As a result, optimism throughout the supply chain can offset the distortions introduced by decentralization, ultimately benefiting the channel as a whole.

5. Robustness Check: The Case of AI-Driven Platform

Today, e-commerce platforms increasingly leverage artificial intelligence (AI) to automate and optimize operations, especially in monitoring seller decisions and performance. For instance, Amazon uses AI-driven algorithms to track and adjust prices in real time based on factors like demand fluctuations, competitor pricing, and inventory levels. Amazon’s system continuously analyzes pricing patterns and may flag sellers who are offering significantly higher or lower prices compared to the market, monitoring sellers’ performance. With these AI capabilities, the platform becomes not only unbiased but also fully aware of the seller’s optimism bias. To evaluate the robustness of the main findings, this section extends the analysis to consider the case of an AI-driven platform.

In this scenario, the AI-driven platform can anticipate the seller’s stocking behavior and incorporate it into its decision-making process. Thus, the platform’s objective is to select the commission rate $\eta$ that maximizes his expected profit, taking into account the seller’s equilibrium order quantities $\widehat{q}\left(\kappa \right)$,

$$\left\{\begin{array}{cc}\eta ·r_1\left({\widehat{q}}_1\right)\hfill & \mathrm{if} \eta <z\left(\kappa \lambda \right)\hfill \\ \eta ·r_2\left({\widehat{q}}_2\right)\hfill & \mathrm{if} z\left(\kappa \lambda \right)\le \eta <1-\beta c/\overline{\phi }\hfill \\ 0\hfill & \mathrm{if} \eta \ge 1-\beta c/\overline{\phi }.\hfill \end{array}\right.$$

(9)

Equation (9) implies that if the commission rate $\eta$ is set too high, the seller would not order the product, leading to the platform not obtaining any revenue. In other words, the AI-driven platform has to choose the optimal commission rate within a lower range. Then, Lemma 5 below provides the solution of (9).

Lemma 5. 

With AI-driven platform and biased seller described by parameter κ, there exist two thresholds $\underline{M}\left(\kappa \right)$ and $\overline{M}\left(\kappa \right)$ such that:

(a) 

If $\beta c\le \underline{M}\left(\kappa \right)$, then the equilibrium commission rate ${\widehat{\eta }}^o\left(\kappa \right)={\widehat{\eta }}_1^o\left(\kappa \right)$, where ${\widehat{\eta }}_1^o\left(\kappa \right)$ is characterised by

$${\displaystyle \frac{c^2{\beta }^2\left(1+\eta \right)}{{\left(1-\eta \right)}^3}}={\displaystyle \frac{{\left(1-\kappa \lambda \right)}^2\left[{\alpha }_h^2\left(1-\lambda \right)+{\alpha }_l^2\lambda \right]}{1-\lambda }}.$$

(10)

(b) 

If $\overline{M}\left(\kappa \right)\le \beta c$, then the equilibrium commission rate ${\widehat{\eta }}^o\left(\kappa \right)={\widehat{\eta }}_2^o\left(\kappa \right)$, where ${\widehat{\eta }}_2^o\left(\kappa \right)$ is characterised by

$$\begin{array}{cc}\hfill {\displaystyle \frac{c^2{\beta }^2\left(1+\eta \right)}{{\left(1-\eta \right)}^3}}& ={\alpha }_l^2\left(2-\kappa \right)\kappa {\lambda }^2+{\alpha }_h^2\left(1-2\lambda +\left(2-\kappa \right)\kappa {\lambda }^2\right)+2{\alpha }_h{\alpha }_l\lambda \left(1-2\kappa \lambda +{\kappa }^2\lambda \right)\hfill \\ \hfill & +{\displaystyle \frac{2({\alpha }_h-{\alpha }_l)c\beta \lambda \left(1-\kappa \right)}{{\left(1-\eta \right)}^2}}.\hfill \end{array}$$

(11)

(c) 

If $\underline{M}\left(\kappa \right)<\beta c<\overline{M}\left(\kappa \right)$, then ${\widehat{\eta }}^o\left(\kappa \right)={\widehat{\eta }}_1^o\left(\kappa \right)$ if ${\widehat{\eta }}_1^o·r_1\left({\widehat{q}}_1\left({\widehat{\eta }}_1^o\right)\right)>{\widehat{\eta }}_2^o·r_2\left({\widehat{q}}_2\left({\widehat{\eta }}_2^o\right)\right)$ and ${\widehat{\eta }}^o\left(\kappa \right)={\widehat{\eta }}_2^o\left(\kappa \right)$ otherwise.

(d) 

Moreover, ${\widehat{\eta }}_1^o$ and ${\widehat{\eta }}_2^o$ are decreasing in the parameter κ, i.e., $\partial {\widehat{\eta }}_1^o/\partial \kappa <0$ and $\partial {\widehat{\eta }}_2^o/\partial \kappa <0$.

Lemma 5 extends Proposition 2, which applies to an unbiased platform, by showing that the AI-driven platform also exhibits both overstocking and understocking equilibrium. The lemma further indicates that, in both the overstocking equilibrium (${\widehat{\eta }}^o={\widehat{\eta }}_1^o$) and the understocking equilibrium (${\widehat{\eta }}^o={\widehat{\eta }}_2^o$), the AI-driven platform would increase its commission rate as the seller’s optimism bias intensifies. This is because the AI-driven platform can anticipate the seller’s tendency to over-order due to optimism bias, leading to higher revenue for the platform. Consequently, the platform sets a higher commission rate to capture a portion of this additional revenue generated by the seller’s stocking behavior.

Lemma 6. 

In the overstocking equilibrium, i.e., ${\widehat{\eta }}^o={\widehat{\eta }}_1^o$, the seller’s equilibrium order quantity is ${\widehat{q}}_1(\kappa ,{\widehat{\eta }}_1^o)$. In the understocking equilibrium, i.e., ${\widehat{\eta }}^o={\widehat{\eta }}_2^o$, the seller’s equilibrium order quantity is ${\widehat{q}}_2(\kappa ,{\widehat{\eta }}_2^o)$. Moreover, $\partial {\widehat{q}}_1(\kappa ,{\widehat{\eta }}_1^o)/\partial \kappa <0$ and $\partial {\widehat{q}}_2(\kappa ,{\widehat{\eta }}_2^o)/\partial \kappa <0$.

Lemma 6 indicates that the seller’s stock level decreases with the parameter $\kappa$, suggesting that the seller increases her stock level as her optimism bias intensifies. Two factors drive this impact of $\kappa$ on the seller’s stock level. The first is a direct effect: optimism bias increases the seller’s stock level. The second is an indirect effect: as shown in Lemma 5, a higher degree of optimism bias prompts the platform to raise the commission rate, which, according to Lemma 3, results in a reduced stock level. Ultimately, the direct effect outweighs the indirect one, leading the biased seller to over-order compared to an unbiased seller, even when AI drives the platform.

In the overstock case, the ensuing seller’s profit is ${\mathring{\pi }}_S\left(\kappa \right)=(1-{\widehat{\eta }}_1^o)·r_1\left({\widehat{q}}_1(\kappa ,{\widehat{\eta }}_1^o)\right)-c·{\widehat{q}}_1(\kappa ,{\widehat{\eta }}_1^o)$, and accordingly, the platform’s expected profit is ${\mathring{\pi }}_P\left(\kappa \right)={\widehat{\eta }}_1^o·r_1\left({\widehat{s}}_1\left({\widehat{\eta }}_1^o\right)\right)$. In the understock case, the seller’s resulting expected profit is ${\mathring{\pi }}_S\left(\kappa \right)=(1-{\widehat{\eta }}_2^o)·r_2\left({\widehat{s}}_2\left({\widehat{\eta }}_2^o\right)\right)-c·{\widehat{s}}_2\left({\widehat{\eta }}_2^o\right)$, and accordingly, the platform’s expected profit is ${\mathring{\pi }}_P\left(\kappa \right)={\widehat{\eta }}_2^o·r_2\left({\widehat{s}}_2\left({\widehat{\eta }}_2^o\right)\right)$. Given ${\mathring{\pi }}_S\left(\kappa \right)$ and ${\mathring{\pi }}_P\left(\kappa \right)$, the total expected profit of ORSC is ${\mathring{\pi }}_E\left(\kappa \right):={\mathring{\pi }}_S\left(\kappa \right)+{\mathring{\pi }}_P\left(\kappa \right)$.

Proposition 9. 

With the AI-driven platform, seller optimism can be a positive force for the platform and the entire channel but can lead to self-harm.

Proposition 9 indicates that the main findings remain valid in the case of an AI-driven platform. Specifically, the platform’s and the system’s profit increase as seller optimism becomes more severe ($\kappa$ decreases), as shown in Figure 6. Furthermore, this figure demonstrates that the seller also suffers from her bias even if AI drives the platform. For insight, optimism bias leads the seller to make sub-optimal decisions by increasing the stocking level, ultimately harming her own profitability. Additionally, the AI-driven platform raises the commission rate in response to the seller’s optimism bias, which reduces the seller’s share of sales revenue and further decreases her profit. In summary, optimism bias negatively impacts the seller’s profit. However, the increase in both stocking level and commission rate benefits the platform, making seller optimism advantageous for the performance of the AI-driven platform. Furthermore, seller optimism helps to mitigate the ordering distortion caused by decentralization, thereby enhancing the performance of the entire channel.

6. Conclusion Remarks

This paper examines the effects and implications of optimism bias in an online retail supply chain, where a third-party seller distributes products through an e-commerce platform. In this context, optimism bias refers to a cognitive bias that leads decision-makers to overestimate the likelihood of high market potential and underestimate the probability of low market potential. Using a game-theoretic model, I characterize the decision strategies of the platform and the seller at equilibrium, showing that their expected profits can either increase or decrease with the level of optimism bias, depending on the firm’s specific bias and production costs.

6.1. Theoretical Contributions

This paper introduces optimism bias into the decision-making processes of both sellers and platforms to investigate its theoretical implications in online retail supply chains. Through this approach, this work makes three key contributions.

First, the results demonstrate that increasing optimism on the part of the third-party seller can enhance the performance of both the online platform and the overall online retail supply chain. Second, this work shows that, while seller optimism consistently results in self-harm, platform optimism can benefit the platform under certain production cost conditions. Third, the analysis reveals that the combined effect of seller and platform optimism on the online retail supply chain can be positive. Through these contributions, this work highlights the significance of optimism bias in e-tailing distribution channels, offering valuable insights for platform managers and third-party sellers.

6.2. Managerial Implications

Previous research suggests that optimism bias can lead to poor decision-making, sometimes with disastrous outcomes [11]. Various approaches have been proposed to reduce optimism bias e.g., [47]. In light of this theoretical examination of the effects of optimism bias in ORSCs, these results offer crucial managerial insights for firms adopting strategies in response to optimism bias. Since bias mitigation could potentially be implemented by the platform, the seller, or external forces, the results in this paper emphasize strategic considerations for such efforts:

• Targeting E-Commerce Platform Operators by External Forces. External forces, such as government agencies, can effectively enhance the overall performance of ORSCs by reducing optimism bias among platform operators, given their central role in decision-making and coordination. External forces could implement regulatory frameworks or provide tools that promote accurate demand forecasting and operational transparency among platform operators. For example, introducing incentives for platforms to use unbiased demand forecasting algorithms would reduce systemic inefficiencies.
• Seller-Led Mitigation Programs. If sellers address optimism bias within their organization, consistency in implementation is critical to avoid fragmented decision-making. The platform’s role should remain supervisory, ensuring sellers do not deviate from optimal coordination practices. Sellers could develop internal training programs or adopt behavioral analytics tools that detect and correct bias in decision-making. The platform, in turn, can monitor seller practices to ensure alignment with overall supply chain objectives.
• Platform-Led Mitigation Efforts. Platform-initiated optimism mitigation programs might not always lead to benefits, especially when production costs are moderate. However, sellers should support such efforts to foster alignment and efficiency. Platforms can focus on collaborative mitigation frameworks, such as joint forecasting initiatives with sellers, where bias correction mechanisms are shared but led by the platform.

The proposed strategies are directly derived from the study’s findings, which highlight the nuanced roles of platforms and sellers in addressing optimism bias. These strategies reflect how different entities—external forces, sellers, and platforms—can effectively mitigate bias depending on their specific context and position within the supply chain.

6.3. Further Research

Several directions for future research would be valuable to explore. First, the current study investigates explicitly the interactions between one third-party seller and one e-commerce platform to analyze how optimism as a cognitive bias affects their decision-making and the resulting equilibrium strategies. This focus allows us to delve deeply into the seller-platform dynamics and maintain analytical tractability. However, including other players, such as customers and regulatory bodies, would enrich the analysis by capturing additional layers of complexity and interdependencies in the supply chain. For example, customer behavior (e.g., preferences, price sensitivity, and quality) could provide insights into demand-side dynamics. At the same time, regulatory bodies could add constraints or incentives that shape seller and platform behavior. While this study focuses on seller-platform interaction, the framework developed here can be extended to incorporate these additional players in future research.

Second, the current study assumes that transaction prices are determined by market externalities, focusing on the interaction between third-party sellers and the e-commerce platform. In fact, factors such as government policies, market volatility, and the actions of other market participants also play a significant role in shaping transaction prices. For example, taxation, subsidies, or antitrust regulations could directly influence pricing strategies. Competitor strategies, such as price wars or product differentiation, could impact equilibrium prices. Extending the model to include these factors would allow for a more comprehensive analysis and broader applicability to real-world scenarios.

Third, this research is theoretical and does not incorporate empirical data from a specific time period or platform. This limits the ability to validate the findings against dynamic changes in market conditions or technological developments in e-commerce. Therefore, this study could be extended to experimental or empirical contexts. For instance, further research could adopt approaches from the literature [48] to measure decision biases and assess the impact of optimism on firm performance within e-commerce environments.