Characterizing the Retailer's Preference for Demand–Control and Demand–Promotion Efforts

Abstract

This article investigates a setting where the retailer invests costly effort to shape demand, such as reducing demand fluctuation through demand–control effort or increasing demand levels with demand–promotion effort. We explore the preference bias between these two effort types and examine the impact of such a bias on profit performance. The experimental findings reveal that actual effort investments exhibit a significant preference bias for promotion effort in the low-profit condition, while no significant preference bias is observed in the high-profit condition. This behavioral pattern can be captured by a reference-dependent behavioral model incorporating the retailer's optimism level in the reference point. Additional analyses, such as robustness experiments and model extensions, provide further support for the promotion-effort bias in the low-profit condition. Our analysis presents a comprehensive understanding of demand-shaping effort preferences and extends the application of the reference-dependence framework. It provides insights for managers in identifying potential biases and mitigating profit loss in demand-shaping activities.

1. Introduction

The escalating importance of resource allocation for scaling up a retailer's business has been widely recognized in the industry and the academic literature. This issue entails the optimal distribution of a firm's budget across various activities that influence market conditions and customer behavior (Berry, 2014; Kung and Chen, 2012). Demand–control and demand–promotion efforts are two examples of such activities competing for a firm's limited resources. Demand–control efforts focus on activities that regulate demand fluctuations by enhancing demand forecasting accuracy, thus reducing supply–demand mismatch costs. In contrast, demand–promotion efforts involve practices that bolster demand levels through advertising and increased sales. In the face of rapidly evolving customer behavior and emerging competitors, resource-intensive retailers must promptly scale up their businesses to remain competitive.

Erroneous choices between the two effort types can lead to substantial profit losses. For instance, Walmart's expansion in Germany (2006) and Lidl's expansion in the United States (2018) were unsuccessful, largely due to insufficient investments in understanding local customer preferences (Pearson, 2018). Carrefour's failure in Singapore and Malaysia can be partly attributed to its reluctance to promote its brand (Marketing Interactive, 2012). Managers often resort to simple heuristics and rules to allocate resources, such as the “percentage-of-sales” rule for advertising budgets (Lilien et al., 1992) or constraining sales-force costs based on total sales percentage (Sinha and Zoltners, 2001). While practitioners and academics acknowledge the significance of each effort type for company performance (De Luca and Atuahene-Gima, 2007; Song and Parry, 1992), particularly with advancements in digital technologies that offer new operational possibilities (Mithas et al., 2012), the retailer's subjective preference for a specific effort type remains unclear.

To the best of our knowledge, only a few behavioral studies have simultaneously examined these two demand-shaping efforts, which are resource-intensive and vie for a firm's limited resources. Most studies have been theoretical, focusing on models that elicit a particular effort type primarily by configuring contract terms or transfer payments (Kurtuluş et al., 2012; Taylor, 2002; Taylor and Xiao, 2009). As such, through a behavioral lens, we aim to answer the following research questions: (a) Is there a preference bias for a specific demand-shaping effort, leading to overinvestment in that effort type relative to normative predictions? (b) Is this preference bias symmetrical between high- and low-profit margin products? (c) Which behavioral theory can account for this preference bias?

We address these questions in a newsvendor setting, where the retailer invests costly effort to shape demand before determining order quantity. The retailer chooses between control and promotion efforts, considering the influence of each effort type on demand—specifically, the degree to which demand uncertainty is reduced or demand level is increased. We develop a behavioral model based on reference dependence to predict the retailer's actual effort investments. This framework has been extensively studied in the context of newsvendor ordering decisions. Our proposed model predicts a preference bias for promotion effort over control effort, particularly for low-profit products. We also show that this bias decreases as the decision maker's level of optimism increases. Furthermore, our analysis predicts that the strength of the bias toward promotion effort is greater in the low-profit condition compared to its symmetric high-profit condition.

We conducted a laboratory experiment to empirically test our predictions. The experimental data revealed that (a) actual effort investments exhibit a significant preference bias for promotion effort in the low-profit condition, while no significant preference bias is observed in the high-profit condition and (b) a reference-dependent behavioral model incorporating optimism levels can capture such behavioral patterns. We included additional analyses, such as robustness experiments and model extensions, to provide more support for our findings.

Our study makes two significant contributions. First, to the best of our knowledge, this is the first study to analytically and experimentally demonstrate that retailers perceive the two demand-shaping effort types differently. Actual effort investments exhibit a preference bias for increasing the demand mean (promotion effort) rather than reducing the demand uncertainty (control effort) in the low-profit condition. In contrast, the preference bias can point in either direction in the high-profit condition. Such a bias depends on the decision maker's optimism level. These findings suggest that managers should reassess their optimism level when making effort decisions to properly evaluate each effort's impact.

Second, we use the reference-dependence framework proposed by Kirshner and Ovchinnikov (2018) to predict a preference bias in investing demand-shaping efforts. We validate the proposed behavioral theory using a laboratory experiment and derive managerial insights. Accordingly, we show that such a framework can be extended to predict behavioral biases in another operational setting, the retailer's costly demand-shaping effort. These findings contribute to the literature investigating a unified behavioral theory that can capture various retailer decisions. Our study represents a step toward finding robust, broadly applicable behavioral models.

This article is organized as follows. Section 2 presents a review of the related literature. Section 3 describes the setting of the demand-shaping effort investment and analyzes the normative and behavioral models. Section 4 details the experimental configuration and develops testable hypotheses. Section 5 analyzes the experimental data, provides tests for the hypotheses, and evaluates the behavioral model. Section 6 shows the robustness experiments and model extensions. Finally, Section 7 summarizes our results and discusses alternative potential explanatory factors, managerial and theoretical implications, and future research directions.

2. Literature Review

This study examines retailers’ preferences between two types of demand-shaping efforts. Retailers can influence demand by investing either control effort, which reduces demand uncertainty, or promotion effort, which increases demand. Theoretical studies on individual types of effort have been extensively developed, while few studies have simultaneously investigated both types of effort. One research stream focuses on the benefits of demand-shaping efforts for the supply chain. Chen and Xiao (2012) and Taylor and Xiao (2010) examine the influence of forecasting capabilities on supply chain performance among competing channel members. Tsay and Agrawal (2004) explore the potential advantages of including a direct sales channel in the presence of sales efforts. Kurtuluş et al. (2014) investigate the impact of category management on sales effort investment. Li (2023) examines how reducing demand uncertainty leads to increased profits for a newsvendor in a competitive environment.

Another research stream investigates how the interaction between supply chain contracts and the properties of demand-shaping efforts can induce the desired amount of control effort (Amornpetchkul et al., 2015; Marschak et al., 2015; Shin and Tunca, 2010; Taylor and Xiao, 2009) or promotion effort (Cachon and Lariviere, 2005; Krishnan et al., 2004; Mukhopadhyay et al., 2009; Taylor, 2002). Kurtuluş et al. (2012) observe higher control efforts under non-coordinating contracts. Taylor and Xiao (2009) demonstrate that the buyback contract outperforms the rebate contract in inducing control effort. Kouvelis and Shi (2020) reveal that the rebate contract yields higher promotion effort than the buyback contract when the retailer compensates the agent who invests the effort, whereas a wholesale price contract results in greater promotion effort than the two coordinating contracts when the manufacturer compensates the agent.

Notably, the literature typically examines control and promotion efforts independently, with little discussion on the preference for one type of effort over the other. One of our research contributions lies in exploring the preference bias between control and promotion efforts.

In contrast to theoretical studies, research in behavioral operations has developed a growing interest in understanding retailer decisions based on empirical data and refining standard theories. One robust finding is that these decisions usually deviate from predictions made by normative theory because of behavioral factors. One typical retailer decision is information sharing with upstream echelons, which can be influenced by lying aversion (Özer et al., 2011), gender (Ma et al., 2020), and communication methods (Johnsen et al., 2020). Another type of decision is bargaining over contract terms, which has been observed to be impacted by response time (Chen et al., 2023), loss aversion, reference dependence (Davis, 2015), and equality preference (Cui and Mallucci, 2016; Katok and Pavlov, 2013; Zhao et al., 2019). The most widely discussed topic is the ordering decisions of a newsvendor. Actual orders exhibit the “pull-to-center” effect, that is, above optimal levels in the low-profit condition and below optimal levels in the high-profit condition (Schweitzer and Cachon, 2000). This systematic regularity has been explained by ex-post inventory error regret (Kremer et al., 2014), anchoring and insufficient adjustment (Schweitzer and Cachon, 2000), and overconfidence (Ren et al., 2017; Ren and Croson, 2013). Becker-Peth and Thonemann (2018) comprehensively review behavioral articles in this setting. Our article contributes to the literature by studying another operational decision, the retailer's costly demand-shaping effort.

We conducted our study within the context of the newsvendor model and employed the reference-dependence framework to develop a behavioral model. Numerous theoretical studies and empirical evidence have substantiated this framework (Ho et al., 2010; Kőszegi and Rabin, 2006; Long and Nasiry, 2015). By incorporating selected reference points, the framework can accommodate both aggregate-level performance and individual-level heterogeneity, which other models, such as ex-post inventory regret or overconfidence, fail to address (Jammernegg et al., 2022; Uppari and Hasija, 2019).

Reference-dependent preference theory posits that decision-makers evaluate gains and losses relative to a reference point (Kahneman and Tversky, 1979). Ho et al. (2010) use realized demands as reference points and incorporate psychological costs related to inventory errors into the retailer's utility. This model has also been applied in other operational contexts (Davis, 2015; Zhao et al., 2020), offering additional empirical support for its theoretical foundations. Long and Nasiry (2015) propose another model, illustrating that the observed ordering tendencies of newsvendors can be explained using a reference point defined as a weighted average between the best and worst payoffs. This model has been utilized by Uppari and Hasija (2019) in a newsvendor context. In a subsequent study, Kirshner and Ovchinnikov (2018) show that the two aforementioned behavioral models are mathematically equivalent under loss and risk neutrality assumptions.

Similar to ordering decisions, investing in demand-shaping efforts involves dealing with uncertainty and weighing tradeoffs between gains and losses. Consequently, we base our behavioral model on the same framework as Kirshner and Ovchinnikov (2018), also assuming loss neutrality and risk neutrality. Such a formulation allows us to use the equivalency between the two previous reference-dependence models for predicting effort decisions. By employing this parsimonious model to explore effort decisions, we contribute empirical support to the robustness of Kirshner and Ovchinnikov's framework in predicting behavioral patterns, extending beyond newsvendor ordering decisions.

3. Model Analysis

The retailer's demand-shaping effort investment model is formulated within a newsvendor setting. The sequence of events is as follows. At the beginning of the selling season, the retailer initially decides whether to invest in control or promotion efforts. In practice, a company can invest in both efforts simultaneously while still needing to consider allocating resources between the two efforts (Erickson and Jacobson, 1992; Peterson and Jeong, 2010; Vinod and Rao, 2000). Without loss of generality, we assume that only one type of effort can be invested at a time. Both efforts have an associated cost, E. The underlying demand distribution is updated based on the chosen effort. Subsequently, the retailer selects an order quantity, considering a unit wholesale price ( $w$ ) and a unit selling price ( $p$ ). Unmet demand and leftover units are not carried over to the next selling season.

The original market demand distribution (prior to effort investment) is denoted by $D=\mu +\mathrm{\Psi }$ , where $\mu$ is a positive constant representing the mean demand, and $\mathrm{\Psi }$ is a zero-mean uniform random variable with support on $[-\psi ,\psi ]$ , ensuring that $-\psi +\mu \ge 0$ . We assume that demand follows a uniform distribution. The extension of our model to accommodate a general distribution can be found in Section 6.2.2. To facilitate the theoretical analysis, we set $\mu =\psi =B/2$ , making zero the minimum demand and B the maximum demand. The decision variable for demand-shaping effort is denoted by $I_i$ (binary variable), where $i\in C,P$ ( $C$ represents control effort and P represents promotion effort). We assume binary effort investments rather than effort amounts because we focus on how the retailer perceives different effort types. The binary-effort approach is also commonly used in the literature (Li et al., 2021; Schöttner, 2017). This simplified setting enables clear comparisons and precise theoretical predictions.

If the retailer invests in control effort ( $I_C=1$ ), the demand uncertainty (i.e., the spread of $\mathrm{\Psi }$ ) is reduced by the control-effort factor $\upsilon >0$ . The transformed demand is denoted by $D_C=\mu +{\mathrm{\Psi }}_C$ , where ${\mathrm{\Psi }}_C\sim U(-\psi +\upsilon ,\psi -\upsilon ).$ Consequently, control effort results in $D_C\sim U(\upsilon ,B-\upsilon ),$ while the mean demand remains at $\mu$ . ¹ If the retailer invests in promotion effort ( $I_P=1$ ), the mean demand increases by the promotion-effort factor $\xi >0$ . The transformed demand is denoted by $D_P=\mu +\xi +\mathrm{\Psi }$ . As a result, promotion effort leads to $D_P\sim U(\xi ,B+\xi )$ , while the demand uncertainty remains unchanged as Ψ. These definitions of how effort impacts demand align with those in the literature (Güler, 2019; Khouja and Robbins, 2003; Mukhopadhyay et al., 2009). In the following sections, we develop and analyze the normative and behavioral models based on this setting. Readers are referred to Sections A and B of the Supplemental Material for detailed derivations and proofs.

3.1 Normative Model

We calculate the retailer's first-best expected profit ${\mathrm{\Pi }}_i$ under effort i following the newsvendor setting. Given the unit wholesale and selling prices w and p, the expressions for the expected profit are as follows, where $\mathrm{\Pi }$ without subscripts denotes the case under a no-effort situation:

$$\begin{array}{rl}{\mathrm{\Pi }}_C& =(p-w)Q_C-\frac{{(Q_C-\upsilon )}^2}{2(B-2\upsilon )}p-E,\end{array}$$

(1)

$$\begin{array}{rl}{\mathrm{\Pi }}_P& =(p-w)Q_P-\frac{{(Q_P-\xi )}^2}{2B}p-E,\end{array}$$

(2)

$$\begin{array}{rl}\mathrm{\Pi }& =(p-w)Q-\frac{Q^2}{2B}p.\end{array}$$

(3)

$Q_i$ is the optimal order quantity under effort i, and it is calculated by

$$\begin{array}{rl}{Q}_C& =\upsilon +(B-2\upsilon )\cdot CR,\end{array}$$

(4)

$$\begin{array}{rl}{Q}_P& =\xi +B\cdot CR,\end{array}$$

(5)

$$\begin{array}{rl}Q& =B\cdot CR,\end{array}$$

(6)

where $CR=c_u/(c_u+c_o)$ is the critical ratio with stockout cost $c_u=p-w$ and leftover cost $c_o=w$ . $CR>0.5$ represents the high-profit condition (hereinafter: $hp$ ), and $CR<0.5$ is the low-profit condition (hereinafter: $lp$ ). Finally, we assume that both efforts outperform no effort; that is, ${\mathrm{\Pi }}_C>\mathrm{\Pi }$ and ${\mathrm{\Pi }}_P>\mathrm{\Pi }$ (equivalently, $E\le min\xi (p-w),w\upsilon (p-w)/p$ ).

We compare the retailer's profit under each investment type to better understand the tradeoffs between the two effort investments. The retailer's relative profit from investing promotion effort over control effort is expressed as $\mathrm{\Delta }{\mathrm{\Pi }}_P={\mathrm{\Pi }}_P-{\mathrm{\Pi }}_C=CR\cdot (\xi p-w\upsilon )$ . Accordingly, the sign of $\mathrm{\Delta }{\mathrm{\Pi }}_P$ is dependent on $\xi p-w\upsilon$ and determines the retailer's preference under the normative model. The larger the magnitude of $\mathrm{\Delta }{\mathrm{\Pi }}_P$ is, the larger the retailer's willingness to invest promotion effort. Proposition 1 establishes that a threshold of the promotion-effort factor, denoted by $\overline{\xi }$ , exists that separates the retailer's preference space into regions and determines the optimal effort.

Proposition 1.

There is a threshold of the promotion-effort factor, $\overline{\xi }=\upsilon \cdot (1-CR)$ , such that the retailer

1. (a)

invests control effort ( $I_C=1$ ) when $\xi <\overline{\xi }$ ; or

(b)

invests promotion effort ( $I_P=1$ ) when $\xi >\overline{\xi }$ .

Otherwise, the retailer is indifferent between control and promotion efforts. Additionally, $\overline{\xi }$ decreases in $CR$ .

Proposition 1 indicates that $\mathrm{\Delta }{\mathrm{\Pi }}_P$ is monotonic in $\xi$ . Figure 1 shows the areas of the retailer's optimal decisions with respect to combinations $(\upsilon ,\xi )$ , under the $lp$ ( $C{R}_{lp}=0.25<0.5$ ) and its symmetric $hp$ ( $C{R}_{hp}=1-C{R}_{lp}=0.75>0.5$ ) conditions. As CR decreases, the slope of the threshold $\overline{\xi }$ increases. Hence, Figure 1 shows that in the $lp$ condition, the area of the $(\upsilon ,\xi )$ -combination where promotion effort is optimal is smaller than that in the $hp$ condition.

Figure 1.

Optimal efforts under combinations of $(\upsilon ,\xi )$ in low- and high-profit conditions ( $\mu =50$ , $\psi =50$ , and $E=20$ ).

3.2 Behavioral Model

Individuals have been observed to make decisions based on outcomes relative to a reference point rather than absolute outcomes (Kahneman and Tversky, 1979). Such a reference-dependence framework is well-developed and is primarily used in behavioral operations to predict newsvendor ordering decisions (Ho et al., 2010; Jammernegg et al., 2022; Long and Nasiry, 2015; Uppari and Hasija, 2019). We apply this framework to predict behavioral biases in another fundamental operations management problem, namely, selecting demand-shaping efforts.

Our behavioral model is based on the reference-dependence framework by Kirshner and Ovchinnikov (2018). These authors demonstrate that the models of Long and Nasiry (2015) and Ho et al. (2010) result in identical ordering quantity predictions (i.e., mathematical equivalence) under specific reparameterizations. In this article, we use this framework to formulate our behavioral model because of its parsimony and significant theoretical findings. In Section 4.2, we add bounded rationality theory to develop research hypotheses considering the random errors existing in human decisions.

Let $\pi (x,Q)=pmin(x,Q)-wQ$ be the payoff for demand x and order quantity Q. The reference point under Q is denoted by $\mathrm{\Omega }(Q)$ . $\mathrm{\Omega }(Q)$ is a convex combination of the maximum and minimum payoffs, that is, $\mathrm{\Omega }(Q)=\beta \cdot \pi (Q,Q)+(1-\beta )\cdot \pi (\underset{\_}{D},Q)$ , where $\underset{\_}{D}$ denotes the minimum demand, and $\beta$ ( $0\le \beta \le 1$ ) is defined as the optimism parameter. $\mathrm{\Omega }(Q)$ is prospect-dependent because it is a function of the prospect Q and independent of the stochastic element x (Uppari and Hasija, 2019). We use the convex combination of the minimum and maximum payoff values as the reference point because it can effectively capture a wide range of salient outcomes relevant to our context. Depending on the values of $\beta$ , reference points can transition from the minimum to the maximum payoff. Other commonly used reference points, such as “MaxMin” (the maximum realized payoff evaluated at the minimum demand $\underset{\_}{D}$ , $(p-w)\underset{\_}{D}$ ) and “Status Quo” (zero payoff), lack this flexibility. Furthermore, integrating such a reference point into a mental model is more straightforward than the stochastic reference point suggested by Uppari and Hasija (2019). In Section 7.1, we address these alternative reference points, as well as other explanatory factors, further supporting the appropriateness of our model to the operational context.

Accordingly, the reference effect function $r(x,Q)$ is defined as

$$\begin{array}{rl}r(x,Q)& =\eta \cdot (\pi (x,Q)-\mathrm{\Omega }(Q))\\ & =\eta \cdot p\cdot (min(x,Q)-\beta Q-(1-\beta )\underset{\_}{D}),\end{array}$$

(7)

where $\eta (>0)$ is the reference effect strength, capturing the sensitivity to deviations from the reference point. Thus, the retailer measures control- and promotion-effort performance relative to a prospect-based reference point—the convex combination of the maximum and minimum realized profits associated with a particular effort type. The retailer's utility is then calculated by $u(x,Q)=\pi (x,Q)+r(x,Q)$ , and the corresponding expected utility is given by

$$\begin{array}{r}U(Q)=\mathrm{\Pi }(Q)+R(Q),\end{array}$$

(8)

where the first term $\mathrm{\Pi }(Q)$ is the same as the normative model, reflecting the expected profits (the rational part of the utility), and the second term $R(Q)$ denotes the gain-loss utility (the psychological part of the utility). The model nests the normative model when $\eta =0$ .

To better understand how the framework influences the retailer's behavior, we examine the attributes of expected reference effect function $R(Q)$ . First, $R(Q)$ under promotion effort is constant in the promotion-effort factor $\xi$ (i.e., $R_P(Q_P)=\eta p\cdot CR\cdot \frac{B}{2}\cdot (2(1-\beta )-CR)$ ). Thus, changing the demand level does not influence the reference-dependence utility. Second, $R(Q)$ is a function of $\upsilon$ under control effort (i.e., $R_C(Q_C)=\eta p\cdot CR\cdot \left(\frac{B-2\upsilon }{2}\right)\cdot (2(1-\beta )-CR)$ ), implying that the absolute value of the reference-dependence utility decreases with $\upsilon$ . Third, it can be easily proven that $|R_P(Q_P)|>|R_C(Q_C)|$ . This result shows that the reference-dependence utility holds less weight for control effort than promotion effort. Accordingly, the influence of the gain-loss utility on the effort investments is highly dependent on the signs of $R_P(Q_P)$ and $R_C(Q_C)$ .

Since we focus on demand-shaping effort investments, we assume that the order quantity remains unaffected by the reference-dependent framework. Instead, the order quantity is set to optimize the expected profit given the chosen effort. This assumption is reasonable, as operational decisions are typically decentralized and handled by different departments within a company (Papier and Thonemann, 2021; Scheele et al., 2018). In our context, the sales or marketing department is responsible for effort investments, while the operations department manages inventory decisions. It is common for one department to presume that the other is making the best decisions. Also, technological advancements enable automatic inventory replenishment (Zhang et al., 2016), aligning with the optimal ordering assumption. In Section 6.2.1, we show that our findings remain consistent without this assumption.

Now, we analyze the influence of the reference-dependent framework on effort choice. We use $U_P$ and $U_C$ to denote the expected utilities for investing promotion and control effort, respectively. Then, the relative net utility from investing promotion effort over control effort is given by

$$\begin{array}{r}\mathrm{\Delta }{U}_P=U_P-U_C=\mathrm{\Delta }{\mathrm{\Pi }}_P+\overline{U},\end{array}$$

(9)

where $\overline{U}=\eta p[CR\cdot \upsilon (2(1-\beta )-CR)]$ is the additional utility induced by reference dependence to the retailer's relative profit $\mathrm{\Delta }{\mathrm{\Pi }}_P$ . We define the magnitude of $\overline{U}$ as the strength of the promotion-effort preference bias. Hence, $\overline{U}>0$ implies a preference bias for promotion effort. $\overline{U}<0$ implies a preference bias for control effort. Otherwise, we have $\mathrm{\Delta }{U}_P=\mathrm{\Delta }{\mathrm{\Pi }}_P$ , and the prediction of the behavioral model is the same as that of the normative model.

Setting $\mathrm{\Delta }{U}_P=0$ , we obtain the threshold ${\overline{\xi }}^{Beh}$ for the behavioral model: ${\overline{\xi }}^{Beh}=\upsilon [1-CR+\eta (CR-2(1-\beta ))]$ . This threshold has a similar structure to the normative model's threshold $\overline{\xi }$ . Consequently, ${\overline{\xi }}^{Beh}<\overline{\xi }$ leads to effort investments biased toward promotion effort, while ${\overline{\xi }}^{Beh}>\overline{\xi }$ results in investments biased toward control effort. Given $\eta >0$ , Proposition 2 summarizes the preference bias for efforts under different values of $\beta$ .

Proposition 2.

If $\beta <1-\frac{CR}{2}$ , then $\overline{U}>0$ and ${\overline{\xi }}^{Beh}<\overline{\xi }$ , implying a preference bias for promotion effort. If $\beta >1-\frac{CR}{2}$ , then $\overline{U}<0$ and ${\overline{\xi }}^{Beh}>\overline{\xi }$ , implying a preference bias for control effort. Otherwise, if $\beta =1-\frac{CR}{2}$ , then $\overline{U}=0$ and ${\overline{\xi }}^{Beh}=\overline{\xi }$ , implying no preference bias.

We provide the following explanations for Proposition 2 and the mechanisms leading to the preference bias. It is important to note that the reference effect resulting from promotion effort remains unchanged with respect to $\xi$ since the demand distribution remains constant. Conversely, the reference effect under control effort varies with $\upsilon$ . Then, the following explanations are based on this distinct correlation of the reference effect with $\upsilon$ .

When $\beta$ is small ( $\beta <1-\frac{CR}{2}$ ), the retailer is pessimistic and places significant weight on the worst-case outcome (i.e., the realized demand equals the lower bound), resulting in lower reference payoffs. Thus, the retailer perceives more relative gains than losses, as additional positive utility arises when profits exceed the reference point ( $R_P(Q_P)>0$ and $R_C(Q_C)>0$ ). In this case, mitigating demand uncertainty results in decreases across both the gain and loss domains. Since, with $\beta <1-\frac{CR}{2}$ , the reference point consistently falls below the mean demand, the reduction in the gain domain outpaces that in the loss domain. Therefore, the retailer prefers to maintain the amount of demand uncertainty instead of reducing it to benefit from the instances of relative gains. Accordingly, the effort investments exhibit a preference bias for promotion effort. Also, we observe that $\overline{U}$ increases in the control-effort factor $\upsilon$ when $\beta <1-\frac{CR}{2}$ , indicating that the preference bias for promotion effort becomes stronger when the control effort leads to an increased reduction in demand uncertainty.

On the other hand, when $\beta$ is large ( $\beta >1-\frac{CR}{2}$ ), the retailer is optimistic and places significant weight on the best-case outcome (i.e., demand equals the order quantity, $x=Q$ ), resulting in higher reference payoffs and the retailer perceiving more relative losses than gains ( $R_P(Q_P)<0$ and $R_C(Q_C)<0)$ . In this case, reducing demand uncertainty leads to more reduction in the loss domain than the gain domain. Hence, the retailer prefers to reduce the demand uncertainty to mitigate the relative loss. Consequently, the effort investments exhibit a preference bias for control effort. In addition, we observe that $\overline{U}$ decreases in $\upsilon$ when $\beta >1-\frac{CR}{2}$ , indicating that the preference bias for control effort becomes stronger when the demand uncertainty is further reduced.

From the above discussion, it can be observed that the value of $\beta$ is key for predicting the preference bias under the behavioral model. Based on existing studies (Ho et al., 2010; Kirshner and Ovchinnikov, 2018), we can derive a reasonable assumption for the interval of $\beta$ . Kirshner and Ovchinnikov (2018) demonstrate that $\beta$ can be equivalently interpreted by the retailer's mental accounting of inventory errors defined in Ho et al. (2010). They show that $\beta =\frac{{\delta }_o}{{\delta }_o+{\delta }_u}$ , where ${\delta }_o$ is the mental cost induced by inventory surplus, and ${\delta }_u$ is the mental cost induced by inventory shortage. Such equivalence between optimism parameter $\beta$ and mental costs $\frac{{\delta }_o}{{\delta }_o+{\delta }_u}$ should be independent of the decision context because the relative utility induced by inventory difference exists objectively in any newsvendor-like setting. Hence, we utilize this equivalence to derive the assumption for the values of $\beta$ .

Following the results of Ho et al. (2010), $\frac{{\delta }_u}{{\delta }_o+{\delta }_u}$ ( $=1-\beta$ ) is always smaller than $CR$ in the $hp$ condition ( $C{R}_{hp}$ ) and larger than $CR$ in the $lp$ condition ( $C{R}_{lp}$ ). This finding hints at the relationship $1-C{R}_{hp}<\beta <1-C{R}_{lp}$ , which becomes particularly apparent when $C{R}_{hp}$ and $C{R}_{lp}$ significantly deviate from $\frac{1}{2}$ . Thus, in the following discussions, we base all propositions on this assumption. In the experiment outlined in Section 4, we set $C{R}_{hp}=0.75$ and $C{R}_{lp}=0.25$ , and empirically verify this assumption $1-C{R}_{hp}<\beta <1-C{R}_{lp}$ using experimental data.

Proposition 2 proposes the conditions for the effort preferences. In the $lp$ condition, the interval assumption $\beta <1-C{R}_{lp}$ consistently yields $\beta <1-\frac{C{R}_{lp}}{2}$ . This results in ${\overline{U}}_{lp}>0$ , implying that the promotion-effort bias persistently emerges in the $lp$ condition. However, in the $hp$ condition, the interval assumption $1-C{R}_{hp}<\beta <1-C{R}_{lp}$ cannot uniquely predict whether $\beta$ is higher or lower than the threshold $1-\frac{C{R}_{hp}}{2}$ . Hence, we first propose Proposition 3 for the effort preference in the $lp$ condition:

Proposition 3.

In the low-profit condition, effort investments exhibit a preference bias for promotion effort, that is, ${\overline{U}}_{lp}>0$ , because $\beta <1-C{R}_{lp}<1-\frac{C{R}_{lp}}{2}$ .

The consistent preference bias shown in the $lp$ condition can be easily understood because the order quantity does not exceed the mean demand, so the reference payoff is always lower than the profit from selling the mean demand. As a result, the retailer experiences more relative gains than losses across all demand instances. The retailer prefers to maintain the demand spread to benefit from the instances of relative gains. Consequently, effort investments exhibit a preference bias for promotion effort.

For the $hp$ condition, order quantity $Q_i$ always exceeds the mean demand. However, the reference payoff can be either above or below the profit from selling the mean demand, depending on the value of $\beta$ . Note that the threshold of $\beta$ in the $hp$ condition is $1-\frac{C{R}_{hp}}{2}$ . When $\beta$ is lower than the threshold, the reference payoff, $p(\beta Q+(1-\beta )\underset{\_}{D})-wQ=p(\beta CR(\overline{D}-\underset{\_}{D})+\underset{\_}{D})-wQ$ (where $\overline{D}$ is the maximum demand), is below that from selling the mean demand, $p\left(\frac{1}{2}(\overline{D}-\underset{\_}{D})+\underset{\_}{D}\right)-wQ$ . As explained above, maintaining the demand spread is favored, so the promotion effort is preferred. When $\beta$ is larger than the threshold, the reference payoff is higher than that from selling the mean demand. Hence, the preference bias is toward the control effort. Consequently, there is no unique prediction for the preference bias in the $hp$ condition.

In addition, we compare the strength of the preference bias (measured by $\overline{U}$ ) between the $lp$ condition and its symmetric $hp$ condition ( $C{R}_{hp}+C{R}_{lp}=1$ ). It is evident that the promotion-effort bias is stronger in the $lp$ condition than in the $hp$ condition when $\beta >1-\frac{C{R}_{hp}}{2}$ . This relationship continues to hold as long as $\beta >\frac{1}{2}$ , where $\frac{1}{2}$ is the lower bound of $1-\frac{C{R}_{hp}}{2}$ . On the other hand, when $\beta <\frac{1}{2}$ , the promotion-effort bias is present in both profit conditions, and it becomes stronger in the $hp$ condition. Propositions 4a and 4b summarize these results:

Proposition 4a.

When $\beta >\frac{1}{2}$ , the strength of the promotion-effort bias is always stronger in the low-profit condition than in its symmetric high-profit condition, that is, ${\overline{U}}_{lp}>{\overline{U}}_{hp}$ .

Proposition 4b.

When $\beta <\frac{1}{2}$ , the strength of the promotion-effort bias is stronger in the high-profit condition than in its symmetric low-profit condition, that is, $0<{\overline{U}}_{lp}<{\overline{U}}_{hp}$ .

Propositions 4a and 4b can be explained as follows. When $\beta >\frac{1}{2}$ , effort investments under the $lp$ condition are exclusively biased toward promotion effort. However, effort investments under the $hp$ condition are predominately biased toward control effort or only slightly favor promotion effort. Thus, under this scenario, the strength of promotion-effort bias is stronger in the $lp$ than in its symmetric $hp$ condition (Proposition 4a). When $\beta <\frac{1}{2}$ , effort investments in both $hp$ and $lp$ conditions exhibit a preference bias for promotion effort ( $\beta <1-\frac{C{R}_{hp}}{2},\beta <1-\frac{C{R}_{lp}}{2}$ ) because reference profits in both conditions are lower than the profit from selling the mean demand. Considering the higher profit margin in the $hp$ condition compared to the $lp$ condition, the retailer experiences greater gain utility in the former when the payoff exceeds the reference payoff. Hence, under this scenario, the strength of promotion-effort bias is stronger in the $hp$ condition than in the $lp$ condition (Proposition 4b).

In summary, we predict a decision pattern different from the normative model when reference dependence is considered during the retailer's decision process. We expect a significant promotion-effort preference bias in the $lp$ condition and an asymmetric strength of the preference biases between the $lp$ and $hp$ conditions. In the following sections (Sections 4 and 5), we conduct a laboratory experiment to empirically test these predictions.

4. Experimental Design and Hypotheses

We conducted the experiment for both the $lp$ and $hp$ conditions. The initial demand (before effort investment) follows a uniform distribution $U(0,100)$ , and subjects choose between control and promotion efforts. The subjects were informed in advance that the order quantities after each effort decision would be automatically determined to maximize expected profits. Screenshots of the experiment interfaces are found in Section C of the Supplemental Material (see Figure C1–C7).

4.1 Experimental Design

Each subject participated in only one profit condition and made demand-shaping effort decisions for nine independent tasks. Following the normative model, (a) three tasks had $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ ; (b) three tasks had $\mathrm{\Delta }{\mathrm{\Pi }}_P=0$ ; and (c) three tasks had $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ . The parameters for these tasks are summarized in Table 1 and illustrated in Figure 2. To facilitate better comparisons, we assigned the same value sets of control-effort factor $\upsilon$ and $|\mathrm{\Delta }{\mathrm{\Pi }}_P|$ between the two profit conditions as well as between scenarios $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ and $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ : $\upsilon \in 16,28,40$ and $|\mathrm{\Delta }{\mathrm{\Pi }}_P|\in 0,27,54$ .

Figure 2.

Experimental predictions based on effort factor $(\upsilon ,\xi )$ combinations.

Table 1.

Summary of experimental parameters.

$lp$ condition $(p=12,w=9,CR=0.25,E=20)$			Predictions	$hp$ condition $(p=12,w=3,CR=0.75,E=20)$
$(\upsilon ,\xi )$	${\mathrm{\Pi }}_C$	${\mathrm{\Pi }}_P$	$\mathrm{\Delta }{\mathrm{\Pi }}_P$	$(\upsilon ,\xi )$	${\mathrm{\Pi }}_C$	${\mathrm{\Pi }}_P$
(16,21)	53.5	80.5	27	(16,7)	353.5	380.5
(28,30)	80.5	107.5	27	(28,10)	380.5	407.5
(16,30)	53.5	107.5	54	(16,10)	353.5	407.5
(16,12)	53.5	53.5	0	(16,4)	353.5	353.5
(28,21)	80.5	80.5	0	(28,7)	380.5	380.5
(40,30)	107.5	107.5	0	(40,10)	407.5	407.5
(28,12)	80.5	53.5	−27	(28,4)	380.5	353.5
(40,21)	107.5	80.5	−27	(40,7)	407.5	380.5
(40,12)	107.5	53.5	−54	(40,4)	407.5	353.5

The experimental design was motivated by the following factors. First, the experiment task directly replicated the costly demand-shaping effort setting modeled in Section 3.1. It aimed to directly elicit subjects’ preferences between control and promotion efforts. Subjects chose between the two demand-shaping efforts based solely on each one's direct impact on the original demand distribution and realized profits. Second, as shown in Figure 2, we used three sets of $(\upsilon ,\xi )$ -combinations for each $\mathrm{\Delta }{\mathrm{\Pi }}_P$ -scenario, spanning an equidistant grid over the feasible region of control- and promotion-effort factor values to rigorously observe subjects’ effort investment behavior. Limited decision-making periods have been employed in other behavioral operations articles (Becker-Peth et al., 2018; Gavirneni and Isen, 2010; Zhang et al., 2016) and have successfully captured subjects’ true preferences. The simplicity of the design facilitated the understanding of the operational setting and experimental tasks for subjects.

To mitigate potential confounding factors, we included the following configurations. First, 18 unique sequences of the nine tasks were generated by a Latin Square design and randomly assigned to subjects to eliminate the sequential effect (Bradley, 1958). Second, we provided decision support to reduce task complexity and control cognitive limitations in information processing. For example, the subjects were shown how the demand distribution was transformed based on the chosen effort in the main-task interface. A plot of 30 demand realizations before and after each demand shaping is shown for each effort. In addition, a plot of the realized profits associated with an effort investment across all possible demand realization was provided. Instead of explicitly showing the order quantities, we implemented these design configurations to avoid unintentional anchoring. Third, we randomized the locations of the effort-choice buttons to reduce the influence of perfunctory selection or consistent favoring of one side of the screen (Harrison and Rutström, 2009). Half of the subjects saw the control-effort button on the left and the promotion-effort button on the right, while the other half were presented with the opposite layout. Fourth, each task ended with a 10-s waiting screen, indicating the termination of a particular task and ensuring the understanding of a new combination $(\upsilon ,\xi )$ in the next task. Lastly, no feedback was provided before the end of all tasks to mitigate the influence of learning.

The experiment was programmed in z-Tree (Fischbacher, 2007) and implemented online using z-Tree Unleashed (Duch et al., 2020). Subjects were recruited on campus through the ORSEE recruitment platform (Greiner, 2004). We recruited 184 subjects ( $N_{hp}=97$ and $N_{lp}=87$ ), in line with the recommendations in Lonati et al. (2018) for running behavioral experiments. At the beginning of the session, subjects were asked to fully concentrate throughout the experiment and read the experiment instructions. They then took a comprehension quiz to ensure they understood the main tasks. They were allowed to participate in the experiment only after correctly answering all questions. Next, they entered the main experimental phase and made effort decisions. At the end of the ninth task, subjects received a summary of their performance, including effort decisions, corresponding order quantities, realized demands, and realized profits.

Following the main experimental phase, subjects participated in a loss-aversion elicitation task adapted from Bolton et al. (2023) and Gächter et al. (2022). This task was included to empirically analyze whether loss aversion influenced effort decisions. The task consisted of seven choices in which the winning prize was fixed and the losing prize varied (see Figure C7 in Section C of the Supplemental Material for details and a screenshot). Our behavioral model and the corresponding analyses are based on the model of Kirshner and Ovchinnikov (2018), which assumes loss neutrality. Thus, examining the influence of loss aversion helps validate our behavioral model.

At the end of the experiment session, the subjects viewed a summary of their earnings. They were paid based on their overall performance in the main and loss-aversion elicitation tasks. The exchange rates were 1 euro = 400 ECU (Experimental Currency Units) and 1 euro = 85 ECU for the $hp$ and $lp$ conditions, respectively. The different exchange rates guaranteed equivalent expected rewards under each profit condition. Each session lasted 60 min. The average earnings, including a 2-euro participation fee, were 12.48 euros and 12.63 euros for the $hp$ and $lp$ conditions, respectively.

4.2 Experimental Hypotheses

Following the perfectly rational theory assumed in the normative model, the retailer chooses promotion effort if $\mathrm{\Delta }{\mathrm{\Pi }}_P$ is positive and control effort otherwise. Nevertheless, behavioral studies show that decisions are mostly not rational. Following the assumption of bounded rationality (Su, 2008), the retailer chooses effort with a probability positively correlated with the corresponding profit. Denoting the probabilities of investing promotion and control efforts as $P{r}_P$ and $P{r}_C$ , respectively, they are calculated with the logit choice model $P{r}_P=\frac{e^{(\mathrm{\Delta }{\mathrm{\Pi }}_P)/\tau }}{1+e^{(\mathrm{\Delta }{\mathrm{\Pi }}_P)/\tau }}$ and $P{r}_C=1-P{r}_P$ , where $\tau$ is the bounded rationality parameter. The retailer may make completely random decisions, $\tau \to \mathrm{\infty }$ , or may select the profit-maximizing choice, $\tau \to 0$ . Accordingly, a subject should overall choose promotion and control efforts with equal probability across all nine tasks because of the symmetric configurations ( $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ , $\mathrm{\Delta }{\mathrm{\Pi }}_P=0$ , and $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ ). Hence, we have a baseline hypothesis for a subject's decisions as follows:

Hypothesis 1. Across all nine tasks, the value of $P{r}_P$ is equal to $P{r}_C$ , for both the high-profit condition and the low-profit condition, that is, $P{r}_P=P{r}_C$ .

Any deviation from this perfectly rational prediction indicates a preference for a specific effort. According to the behavioral model, the relative net utility from investing promotion effort over control effort is given by $\mathrm{\Delta }{U}_P=\mathrm{\Delta }{\mathrm{\Pi }}_P+\overline{U}$ , and the probabilities of investing promotion and control efforts are $P{r}_P=\frac{e^{(\mathrm{\Delta }{U}_P)/\tau }}{1+e^{(\mathrm{\Delta }{U}_P)/\tau }}$ and $P{r}_C=1-P{r}_P$ . The retailer's effort investments exhibit a preference bias for promotion effort if $\overline{U}>0$ ; otherwise, there is a preference bias for control effort. Proposition 2 suggests a preference bias for the promotion effort in the $lp$ condition. Then, we have the following hypothesis, contrasting with Hypothesis 1:

Hypothesis 2. Across all nine tasks, the value of $P{r}_P$ is higher than $P{r}_C$ in the low-profit condition, that is, $P{r}_{P,lp}>P{r}_{C,lp}$ .

In the $hp$ condition, given that $C{R}_{hp}$ is 0.75 in the current setting, the threshold of preference bias is $1-\frac{C{R}_{hp}}{2}=0.625$ . Previous estimates of $\beta$ (or $\frac{{\delta }_o}{{\delta }_o+{\delta }_u}$ ) in the literature are $0.6044$ (Ho et al., 2010) and $0.7384$ (Zhao et al., 2020), either lower or higher than the threshold. Hence, there is no clear prediction for the preference bias in the $hp$ setting.

The next hypothesis concerns the asymmetric preference biases existing between the profit conditions. Because the previous estimates of $\beta$ are both larger than 0.50, we follow Proposition 4a and hypothesize that the preference bias for promotion effort is stronger in the $lp$ condition than in the $hp$ condition, ${\overline{U}}_{lp}>{\overline{U}}_{hp}$ . Given that the values of $\mathrm{\Delta }{\mathrm{\Pi }}_P$ are the same for both profit conditions, such an asymmetry between biases is reflected by the differences in proportions of investing the promotion effort. We hypothesize:

Hypothesis 3. Across all nine tasks, the value of $P{r}_P$ in the low-profit condition is higher than that in the high-profit condition, that is, $P{r}_{P,lp}>P{r}_{P,hp}$ .

5. Data Analysis

In this section, we present the experimental results. We take the mean promotion effort investment of an individual subject as the unit of analysis. The Wilcoxon signed-rank test is used for one-sample tests, and the Mann–Whitney test for two-sample tests, using two-tailed p-values for all tests. In Section 5.1, we report the preliminary analysis and show the performance of the effort investments. For some tests, we examined whether the proportions differed significantly from 50% because testing $P{r}_P=50\mathrm{\%}$ is equivalent to testing $P{r}_P=P{r}_C$ , given that $P{r}_P+P{r}_C=100\mathrm{\%}$ . In Section 5.2, following our behavioral model, we estimate $\overline{U}$ —the additional utility induced by reference dependence to the retailer's relative profit $\mathrm{\Delta }{\mathrm{\Pi }}_P$ —based on the observed choices made by the subjects. This approach allows us to rigorously show the strength of the preference and test our hypotheses in addition to the results in Section 5.1. Finally, in Section 5.3, we structurally estimate our behavioral model and assess the consistency of the estimates.

5.1 Performance of Decisions

First, we calculated the proportion of optimal decisions in the $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ and $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ scenarios to provide a sense of the magnitude of the effort investment bias. For this part of the analysis, we excluded the scenario $\mathrm{\Delta }{\mathrm{\Pi }}_P=0$ , as either effort type is optimal. Table 2 shows the results for each profit condition. The results for each of the nine tasks can be found in Table D1 in Section D of the Supplemental Material. All numbers are significantly lower than 100%, indicating that subjects did not consistently invest the optimal effort as predicted by the normative model.

Table 2.

The proportions investing the optimal effort type (%; excluding $\mathrm{\Delta }{\mathrm{\Pi }}_P=0$ ).

Scenario	Overall	$hp$ condition	$lp$ condition
$\mathrm{\Delta }{\mathrm{\Pi }}_P>0$	${70.83}^{***}$	${65.98}^{***}$	${76.25}^{***}$
$\mathrm{\Delta }{\mathrm{\Pi }}_P<0$	${61.05}^{***}$	${62.20}^{***}$	${59.77}^{***}$
both	${65.94}^{***}$	${64.09}^{***}$	${68.01}^{***}$

We tested whether the proportions were significantly different from 100%.

$***p<0.01$ .

Upon examining the overall data, the proportion of investing the optimal effort is significantly larger in scenario $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ than in scenario $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ ( $70.83\mathrm{\%}$ vs. $61.05\mathrm{\%}$ , $p=0.0523)$ . Such a difference is not significant in the $hp$ condition ( $65.98\mathrm{\%}$ vs. $62.20\mathrm{\%}$ , $p=0.9022$ ) but is significant in the $lp$ condition ( $76.25\mathrm{\%}$ vs. $59.77\mathrm{\%}$ , $p<0.0111$ ). This finding implies that the subjects, on average, made better decisions in scenario $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ than in scenario $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ . To better understand the decision patterns, we calculated the proportions of promotion effort in each scenario, $\mathrm{\Delta }{\mathrm{\Pi }}_P=0$ , $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ , and $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ , under each profit condition. Table 3 presents these results.

Table 3.

The proportions investing promotion effort (%).

	$hp$ condition	$lp$ condition
$\mathrm{\Delta }{\mathrm{\Pi }}_P=0$	43.99	63.60**
$\mathrm{\Delta }{\mathrm{\Pi }}_P>0$	65.98***	76.25***
$\mathrm{\Delta }{\mathrm{\Pi }}_P<0$	37.80	40.2
Overall	49.26	60.03***

We tested whether $P{r}_P$ was significantly larger than 50%.

$*p<0.1$ , $**p<0.05$ , $***p<0.01$ .

First, we note that the proportion in the $lp$ condition is significantly larger than $50\mathrm{\%}$ when $\mathrm{\Delta }{\mathrm{\Pi }}_P=0$ ( $63.60\mathrm{\%}$ , $p=0.0178$ ), but not for the $hp$ condition ( $43.99\mathrm{\%}$ , $p=0.1038$ ). Next, for $\mathrm{\Delta }{\mathrm{\Pi }}_P\ne 0$ , the sum of values from the two scenarios, $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ and $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ , should ideally equal one as per the normative prediction, given the symmetric nature between the two. However, in the $lp$ condition, this sum significantly surpasses one, whereas, in the $hp$ condition, it is significantly not different from one. Accordingly, the proportion of investing promotion effort across the three scenarios shows that $P{r}_{P,lp}$ is significantly larger than 50% ( $60.03\mathrm{\%}$ , $p=0.0050$ ), while $P{r}_{P,hp}$ ( $49.26\mathrm{\%}$ , $p=0.1548$ ) is not significantly different from 50%. The results are equivalent to $P{r}_{P,lp}$ being significantly larger than $P{r}_{C,lp}$ and $P{r}_{P,hp}$ not being significantly different from $P{r}_{C,hp}$ . Hence, we reject Hypothesis 1 only for the $lp$ condition and accept Hypothesis 2.

The effort decisions in the $lp$ condition exhibit a significant preference bias for promotion effort, while no particular preference is observed in the $hp$ condition. This asymmetric performance is supported by subject-level data, with 66% of subjects (57 out of 87) in the $lp$ condition investing promotion effort more than 50% of the time. In contrast, fewer subjects in the $hp$ condition (42% or 41 out of 97) invested promotion effort more than 50% of the time. Furthermore, the proportions for investing promotion effort ( $P{r}_P$ ) should be the same between profit conditions in each scenario of $\mathrm{\Delta }{\mathrm{\Pi }}_P$ because of the same values of $|\mathrm{\Delta }{\mathrm{\Pi }}_P|$ . When combining all scenarios, the experimental data show that $P{r}_{P,hp}$ is significantly lower than $P{r}_{P,lp}$ (49.26% vs. 60.03%, $p=0.0039$ ). On a scenario basis, the values of $P{r}_P$ are significantly lower in the $hp$ condition than in the $lp$ condition in two of the three scenarios: $\mathrm{\Delta }{\mathrm{\Pi }}_P=0$ ( $43.99\mathrm{\%}$ vs. $63.60\mathrm{\%}$ , $p=0.0002$ ) and $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ ( $65.98\mathrm{\%}$ vs. $76.25\mathrm{\%}$ , $p=0.0318$ ). In scenario $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ , we see directional support for this relationship ( $37.80\mathrm{\%}$ vs. $40.23\mathrm{\%}$ ; $p=0.7008$ ). As a result, Hypothesis 3 is supported. The biased preference for promotion effort is significantly higher in the $lp$ condition than in the $hp$ condition.

5.2 Measuring the Strength of the Preference Bias

In Section 5.1, the experimental results indicate a significant preference bias toward promotion effort. Since decision deviations are influenced by both bounded rationality and the theorized preference bias, comparing the proportions of promotion effort alone does not reveal the magnitude of the bias. To further demonstrate the existence of and measure the strength of the observed preference bias, we follow the behavioral model described by Equation (9) to estimate promotion-effort preference bias $\overline{U}$ . This approach also allows us to test our hypotheses more rigorously. Following a logit choice framework, the probability of investing promotion effort is given by:

$$\begin{array}{r}\text{Pr}(I_P=1)=\frac{e^{(\mathrm{\Delta }{\mathrm{\Pi }}_P+\overline{U})/\tau }}{1+e^{(\mathrm{\Delta }{\mathrm{\Pi }}_P+\overline{U})/\tau }}.\end{array}$$

(10)

First, we formulated $\overline{U}={\alpha }_0$ across all profit conditions (Model 1). Second, we formulated $\overline{U}={\alpha }_0+{\alpha }_{lp}\cdot lp$ , including a dummy variable for our focal variable, the profit condition ( $lp=1$ for the $lp$ condition; Model 2). Finally, we formulated a full model $\overline{U}={\alpha }_0+{\alpha }_{lp}\cdot lp+{\alpha }_{\text{Loss}}\cdot \text{Loss}+{\alpha }_{\upsilon }\cdot \upsilon$ , including control variables such as loss aversion ( $Loss$ ) from the loss-aversion elicitation task and the control-effort factor $\upsilon$ (Model 3). Model 3 allows us to test the dependency of the additional utility $\overline{U}$ induced by reference dependence on the profit condition (between-profit comparisons) and the loss aversion while controlling for the influence of the control-effort factor. All models were estimated using maximum likelihood, and standard errors were clustered at the subject level.

Table 4 presents the estimation results. In Model 1, ${\alpha }_0$ is significant and positive ( $p=0.0230$ ), indicating that subjects, on average, overinvest in promotion effort relative to the normative prediction. In Model 2, the coefficient ${\alpha }_{lp}$ is significantly positive ( $p=0.0050$ ) while ${\alpha }_0$ is not significant ( $p=0.7660$ ), suggesting a preference bias for promotion effort in the $lp$ condition but not in the $hp$ condition. In parallel to Model 2, we re-estimated Model 1 separately for $lp$ and $hp$ conditions under those scenarios when $\mathrm{\Delta }{\mathrm{\Pi }}_P\le 0$ and $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ . The results, detailed in Table D2 in Section D of the Supplemental Material, reveal biases consistent with the observations in Model 2. In Model 3, the significance of the coefficient ${\alpha }_{lp}$ ( $p=0.0030$ ) further confirms the different preference biases for promotion effort between the $lp$ and $hp$ conditions ( ${\overline{U}}_{lp}>{\overline{U}}_{hp}$ ). The parameter ${\alpha }_{\text{Loss}}$ is not significant ( $p=0.6020)$ , indicating that loss aversion does not play a role in effort investments.

Table 4.

Maximum likelihood estimates for the promotion-effort preference bias measure $\overline{U}$ .

	Model 1	Model 2	Model 3
${\alpha }_0$	10.3711**	−1.7524	0.5705
	(4.5646)	(5.8840)	(14.2395)
${\alpha }_{lp}$		25.6490***	25.8385***
		(9.0549)	(8.7516)
${\alpha }_{\text{Loss}}$			−1.6410
			(3.1425)
${\alpha }_{\upsilon }$			0.0509
			(0.3746)
$\tau$	55.2619***	54.5543***	53.9075***
	(6.1363)	(6.0573)	(6.7505)
$LpL$	−1081.5181	−1071.1000	−1070.6460
$AIC$	2167.0360	2148.2000	2151.2920
$BIC$	2177.8610	2164.4360	2178.3530
Obs. (Cluster)	1656	1656	1656
Obs. (Cluster)	(184)	(184)	(184)

Robust standard errors are reported in parentheses. $LpL$ = log pseudolikelihood; $\text{AIC}$ = Akaike information criterion; $\text{BIC}$ = Bayesian information criteria.

$*p<0.1$ , $**p<0.05$ , $***p<0.01$ .

As additional analysis to verify the theoretical formulation of $\overline{U}$ in Section 3.2, we re-estimated Model 1 with $\overline{U}$ as a function of the control-effort factor $\upsilon$ , $\overline{U}(\upsilon )={\alpha }_{\upsilon }\cdot \upsilon$ (Model 1 $\upsilon$ ), and also separately under the $lp$ and $hp$ conditions (Models 1 ${\upsilon }_{lp}$ and 1 ${\upsilon }_{hp}$ ). This estimation is reasonable because, according to $\overline{U}=\eta p[CR\cdot \upsilon (2(1-\beta )-CR)]$ , $\overline{U}$ has no intercept term when $\upsilon =0$ . The estimates for ${\alpha }_{\upsilon }$ are significantly positive for Models $1\upsilon$ and $1{\upsilon }_{lp}$ (Model $1\upsilon$ : ${\alpha }_{\upsilon }=0.3260$ , $p=0.0160$ and Model $1{\upsilon }_{lp}$ : ${\alpha }_{\upsilon }=0.6147$ , $p<0.001$ ) but not for Model 1 ${\upsilon }_{hp}$ ( ${\alpha }_{\upsilon }=-0.0657$ , $p=0.7780$ ). These results align with those in the previous paragraphs and confirm the theoretical representation of $\overline{U}$ as an increasing function of $\upsilon$ when $\overline{U}>0$ .

Overall, the experimental data support the predictions of the reference-dependent behavioral model. These findings provide evidence to directly measure the preference bias, rejecting Hypothesis 1 and supporting Hypothesis 2. They also support Hypothesis 3, indicating a stronger preference bias toward promotion effort in the $lp$ condition than in the $hp$ condition. Furthermore, in the $lp$ condition, we show that the strength of the promotion-effort preference bias does increase as the control-effort factor increases. This finding supports the integration of reference-dependence theory into the behavioral model for effort decisions. It suggests that the preference bias for promotion effort intensifies as control effort becomes more attractive, resulting in a higher likelihood of making mistakes when control effort is optimal. Finally, the estimation shows that loss aversion does not have a significant effect on promotion effort investment. This result provides preliminary evidence supporting the loss neutrality assumption in our behavioral model. We provide further analysis of this assumption in Section 7.1.1.

5.3 Analysis of the Behavioral Model

The above analysis demonstrates that actual effort investments align with our model's predictions. In this section, we provide further analyses to validate this proposed model. First, we estimate the behavioral parameters and test the predictive power of the behavioral model, further validating the theoretical underpinnings of the observed preference bias. Next, using the behavioral estimates, we conduct a sensitivity analysis to illustrate the influence of the preference bias on profit.

5.3.1 Validity of Proposed Behavioral Theory

We structurally estimated our behavioral model to validate the proposed behavioral theory further. We pooled all experimental data and specified that the behavioral parameters were common across all treatments and subjects (Becker-Peth et al., 2013; Duhaylongsod et al., 2023; Ho et al., 2010) because the reference effect affects all individuals across both profit conditions. Similar to Equation (10), the probability of investing promotion effort ( $I_P=1$ ) is

$$\begin{array}{r}P{r}_P=\frac{e^{(\mathrm{\Delta }{\mathrm{\Pi }}_P+\overline{U})/\tau }}{1+e^{(\mathrm{\Delta }{\mathrm{\Pi }}_P+\overline{U})/\tau }},\end{array}$$

(11)

where $\overline{U}=\eta p[CR\upsilon (2(1-\beta )-CR)]$ . The behavioral parameters in $\overline{U}$ were estimated by maximizing the likelihood value, that is, the product of $P{r}_P$ across all subjects and all tasks. The standard errors were clustered at the subject level. By setting $\eta =0,$ we estimated the nested normative model and used its results as an evaluation benchmark for the behavioral model. Table 5 summarizes the estimation results.

Table 5.

Structural estimation results.

	Proposed behavioral model	Normative model (no reference effect)
$\beta$	0.6258***
	(0.0222)
$\eta$	0.4623***
	(0.1402)
$\tau$	50.8741***	55.7240***
	(5.4784)	(6.1920)
Obs. (Cluster)	1656	1656
	(184)	(184)
$LpL$	−1073.8620	−1088.2630
$AIC$	2153.7240	2178.5250
$BIC$	2169.9610	2183.9370
Wald test (test against the proposed behavioral model)		${\chi }^2(2)=49.03$ $p<0.001$

Robust standard errors are reported in parentheses. $LpL$ = log pseudolikelihood; $AIC$ = Akaike information criterion; $BIC$ = Bayesian information criteria. The Wald test was used instead of the traditional likelihood ratio test because of the standard error clustering.

$*p<0.1$ , $**p<0.05$ , $***p<0.01$ .

The proposed behavioral model (column 1) surpasses the normative model (column 2) in performance. The behavioral parameters, $\beta$ ( $0.6258$ , $p<0.001$ ) and $\eta$ ( $0.4623$ , $p=0.001$ ), are both significantly different from 0. The estimated value of β is also consistent with previous estimates reported in the literature (Ho et al., 2010; Zhao et al., 2020). Furthermore, the behavioral model's $LpL$ is higher, and its AIC and BIC values are lower than those in the normative model. The Wald test shows that the proposed behavioral model explains the data better than the normative model ( $\chi (2)=49.03$ , $p<0.001$ ). Overall, the results validate our proposed behavioral theory and indicate that it better captures actual effort investments than the normative model.

Next, the behavioral model's estimates (column 1) support the propositions from Section 3.2. Firstly, the significantly positive $\eta$ confirms the existence of the reference effect in our operational context. Secondly, the significantly positive estimate of $\beta$ lies within the range $(1-C{R}_{hp},1-C{R}_{lp})$ , specifically between 0.25 and 0.75, is significantly larger than 0.5 (Wald test, ${\chi }^2(1)=32.14$ , $p<0.001$ ).This finding explains the stronger promotion-effort preference bias in the $lp$ condition compared to the $hp$ condition. This result is also in line with Proposition 4a: the value of $\beta$ is significantly smaller than the threshold of preference bias on promotion effort for the $lp$ condition ( $1-\frac{C{R}_{lp}}{2}=0.875$ ; Wald test, $;{\chi }^2(1)=126.15,p<0.001$ ), while it is not significantly different from the threshold for the $hp$ condition ( $1-\frac{C{R}_{hp}}{2}=0.625$ ; Wald test, $;{\chi }^2(1)=0.00,p=0.9718$ ). These findings further elucidate the preference bias for promotion effort in the $lp$ condition and the insignificant bias in the $hp$ condition.

Finally, we compare the predictive power of the behavioral and normative models using the out-of-sample validation method proposed by Özer and Zheng (2018, p. 511). We randomly selected 2/3 of the data in each task as a training set for estimating the models, with the remaining 1/3 forming the testing set. After obtaining the parameter estimates from the training set, we assessed the models’ predictive power using the testing set. Specifically, we used two standard metrics (Brandt and Dlugosch, 2021; Kasapidis et al., 2021): the accuracy score (proportion of correct predictions over all predictions) and the F1 score (harmonic mean of prediction precision and prediction sensitivity), where the prediction precision in the current case is the proportion of correctly predicted promotion effort over all promotion-effort predictions, and the prediction sensitivity is the proportion of correctly predicted promotion effort over all observations of promotion-effort decisions. ² The results reveal that the predictive power of the behavioral model outperforms the normative model, with a higher accuracy score ( $65.39\mathrm{\%}$ vs. $59.38\mathrm{\%}$ ) and a higher F1 score ( $66.90\mathrm{\%}$ vs. $54.21\mathrm{\%}$ ).

In summary, the analyses demonstrate that incorporating reference-dependent preferences into the retailer's effort investment decisions is beneficial because it results in a more accurate fit or prediction of actual effort investments than the normative model. Moreover, we find sufficient evidence to validate our proposed behavioral theory: the estimates of the behavioral parameters are consistent with the parametric thresholds in the derived propositions based on the behavioral model.

5.3.2 Profit Sensitivity

After empirically demonstrating the existence of preference bias in effort decisions, it is essential to understand how this bias can affect profit performance. We conducted analyses based on both experimental and simulation data.

First, we calculated the profit difference based on the experimental data. Table 6 presents the absolute and relative profit differences between the actual effort decisions and the normative ones. We denote the expected profit of the actual effort decision by ${\mathrm{\Pi }}^b$ and the normative one by $\mathrm{\Pi }*$ . Then, the absolute difference is measured by ${\mathrm{\Pi }}^b-{\mathrm{\Pi }}^{*}$ and the relative difference is measured by $({\mathrm{\Pi }}^b-\mathrm{\Pi }*)/|\mathrm{\Delta }{\mathrm{\Pi }}_P|$ , where $|\mathrm{\Delta }{\mathrm{\Pi }}_P|$ is the maximum profit difference. Because the values of $|\mathrm{\Delta }{\mathrm{\Pi }}_P|$ are symmetric between $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ than $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ , as well as between the $hp$ and $lp$ conditions, we can rely on the measure of the absolute difference to compare profit performances. The relative difference provides a uniform measure between different values of $|\mathrm{\Delta }{\mathrm{\Pi }}_P|$ .

Table 6.

Absolute (relative) profit difference induced by biased decisions.

Scenario	Overall	$hp$ condition	$lp$ condition
$\mathrm{\Delta }{\mathrm{\Pi }}_P>0$	−9.73 (−29.17%)	−11.41 (−34.02%)	−7.86 (−23.75%)
$\mathrm{\Delta }{\mathrm{\Pi }}_P=27$	−9.02 (−33.42%)	−10.44 (−38.66%)	−7.45 (−27.59%)
$\mathrm{\Delta }{\mathrm{\Pi }}_P=54$	−11.15 (−20.65%)	−13.36 (−24.74%)	−8.69  (−16.09%)
$\mathrm{\Delta }{\mathrm{\Pi }}_P<0$	−13.89 (−38.95%)	−13.55 (−37.80%)	−14.28 (−40.23%)
$\mathrm{\Delta }{\mathrm{\Pi }}_P=-27$	−10.71 (−39.67%)	−10.30 (−38.14%)	−11.17 (−41.38%)
$\mathrm{\Delta }{\mathrm{\Pi }}_P=-54$	−20.25 (−37.50%)	−20.04 (−37.11%)	−20.48 (−37.93%)
Overall	−11.81 (−34.06%)	−12.48 (−35.91%)	−11.07 (−31.99%)

The results confirm that the biased effort decisions lead to significant loss of profit, that is, all the absolute profit differences are negative. The overall relative loss is $34.06\mathrm{\%}$ , implying biased decisions lead to losing around one-third of $|\mathrm{\Delta }{\mathrm{\Pi }}_P|$ . These results are unsurprising since any deviation from the normative decisions incurs a loss.

Moreover, the loss of profit is significantly more pronounced in scenario $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ than $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ ( $13.89$ vs. $9.73,p=0.0068$ ). The same results hold in the $lp$ condition ( $14.28$ vs. $7.86$ , $p=0.0043$ ), while only directional support is present in the $hp$ condition ( $13.55$ vs. $11.47$ , $p=0.4160$ ). Similar directional patterns are observed in the subgroups $|\mathrm{\Delta }{\mathrm{\Pi }}_P|=27$ and $|\mathrm{\Delta }{\mathrm{\Pi }}_P|=54$ . The preference bias for promotion effort explains these asymmetric observations, as it incurs a loss in scenario $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ where control effort is optimal. Concerning the differences between profit conditions, we observe that the loss of profit is significantly smaller in the $lp$ condition than in the $hp$ conditions when $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ ( $7.86$ vs. $11.41,$ $p=0.0285$ ), while there is no significant difference between profit conditions when $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ ( $14.28$ vs. $13.55,$ $p=0.7849$ ). These findings suggest that the preference bias for promotion effort in the $lp$ condition facilitates better performance when $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ .

Second, we conducted simulations based on the estimates of the behavioral model in Table 5. To isolate the effect of reference dependence, we assume bounded rationality for both the behavioral and normative models. Since we apply bounded rationality to explain the random error of actual decisions in the behavioral model, it is reasonable to include bounded rationality in the normative model as well. This approach provides a more realistic reflection of real-world decisions than the assumption that neither model is influenced by bounded rationality or that only behavioral decisions are influenced by it. Accordingly, we calculate the expected profits for the normative condition ( $\overline{U}=0$ ) by Equation (12) and the biased condition ( $\overline{U}=\eta p[CR\upsilon (2(1-\beta )-CR)]$ ) using Equation (13).

$$\begin{array}{r}E(\mathrm{\Pi })={\mathrm{\Pi }}_P\frac{e^{(\mathrm{\Delta }{\mathrm{\Pi }}_P)/\tau }}{1+e^{(\mathrm{\Delta }{\mathrm{\Pi }}_P)/\tau }}+{\mathrm{\Pi }}_C\frac{1}{1+e^{(\mathrm{\Delta }{\mathrm{\Pi }}_P)/\tau }}\end{array}$$

(12)

$$\begin{array}{r}E({\mathrm{\Pi }}^b)={\mathrm{\Pi }}_P\frac{e^{(\mathrm{\Delta }{\mathrm{\Pi }}_P+\overline{U})/\tau }}{1+e^{(\mathrm{\Delta }{\mathrm{\Pi }}_P+\overline{U})/\tau }}+{\mathrm{\Pi }}_C\frac{1}{1+e^{(\mathrm{\Delta }{\mathrm{\Pi }}_P+\overline{U})/\tau }}\end{array}$$

(13)

Then the absolute profit difference is defined by $E({\mathrm{\Pi }}^b)-E(\mathrm{\Pi })$ and the relative profit difference is $\frac{E({\mathrm{\Pi }}^b)-E(\mathrm{\Pi })}{|\mathrm{\Delta }{\mathrm{\Pi }}_P|}$ . We denote such relative profit difference with $\delta$ .

$\delta <0$ indicates that the biased decisions result in a loss of profit, while $\delta >0$ indicates a gain in profit when compared to the normative decisions. The reference strength $\eta$ does not influence the sign of $\overline{U}$ because it is nonnegative. Therefore, we only check the influence of the optimism parameter $\beta$ on the relative profit difference $\delta$ . We can prove that $\delta$ decreases with $\beta$ when $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ and increases with $\beta$ when $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ (details can be found in Section E of the Supplemental Material).

Figure 3 visualizes the relationship between $\delta$ and $\beta$ . When $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ , biased decisions in the $lp$ profit condition always lead to better performance than do the normative ones, while the $hp$ condition only benefits from these decisions when $\beta <1-\frac{C{R}_{hp}}{2}$ . Consistent with Propositions 4a and 4b, the bias leads to more advantage in the $hp$ condition than the $lp$ condition when $\beta <\frac{1}{2}$ , and conversely, it provides more advantage in the $lp$ condition when $\beta >\frac{1}{2}$ . The estimate of $\beta$ based on the experimental data is shown in Figure 3 by the vertical dotted line, indicating that profit in the $lp$ condition benefits from the bias, while profit in the $hp$ condition is not significantly influenced.

When $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ , biased decisions mostly lead to loss of profits except for in the scenario where the preference bias toward control effort occurs in the $hp$ condition, that is, $\beta >1-\frac{C{R}_{hp}}{2}$ . At the point of the $\beta$ -estimate from the experimental data ( $\beta =0.6258$ ), profit in the $lp$ condition suffers loss from the bias ( $\beta <1-\frac{C{R}_{lp}}{2}=0.875$ ), while there is no significant influence in the $hp$ condition ( $\beta \approx 1-\frac{C{R}_{hp}}{2}=0.625$ ). Furthermore, we demonstrate that the relative profit difference in the $hp$ condition exhibits greater sensitivity to changes in β, as indicated by $\left|\frac{\mathrm{\partial }{\delta }_{hp}}{\mathrm{\partial }\beta }\right|>\left|\frac{\mathrm{\partial }{\delta }_{lp}}{\mathrm{\partial }\beta }\right|$ . Meanwhile, when considering the same values of $\beta$ and $|\mathrm{\Delta }{\mathrm{\Pi }}_P|$ , the benefits associated with bias in the scenario where $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ are outweighed by the drawbacks in the scenario where $\mathrm{\Delta }{\mathrm{\Pi }}_P<0$ , as indicated by $|{\delta }_{\mathrm{\Delta }{\mathrm{\Pi }}_P>0}|<|{\delta }_{\mathrm{\Delta }{\mathrm{\Pi }}_P<0}|$ . Both characteristics (i.e., $\left|\frac{\mathrm{\partial }{\delta }_{hp}}{\mathrm{\partial }\beta }\right|>\left|\frac{\mathrm{\partial }{\delta }_{lp}}{\mathrm{\partial }\beta }\right|$ and $|{\delta }_{\mathrm{\Delta }{\mathrm{\Pi }}_P>0}|<|{\delta }_{\mathrm{\Delta }{\mathrm{\Pi }}_P<0}|$ ) can be observed in Figure 3.

Figure 3.

Relative profit difference between biased and normative decisions as a function of $\beta$ . (a) $\mathrm{\Delta }{\mathrm{\Pi }}_P=27>0,$ $\upsilon =28,\xi =30(10\text{for high profit}),$ $p=12,c=9(3\text{for high profit}),$ $\eta =0.462,\tau =50.874$ . (b) $\mathrm{\Delta }{\mathrm{\Pi }}_P=-27<0,$ $\upsilon =28,\xi =12(4\text{for high profit}),$ $p=12,c=9(3\text{for high profit}),$ $\eta =0.462,\tau =50.874$ .

6. Additional Analyses

We conducted additional analyses to further validate the behavioral model and observed regularities. Section 6.1 carries out an additional experiment that provides different decision support to demonstrate the robustness of preference bias toward promotion effort in the $lp$ condition. Section 6.2 extends our theoretical analysis by (a) allowing the decision maker to determine order quantity alongside the effort and (b) considering a general distribution for the initial demand.

6.1 Robustness Experiment

In the main experiment, we provided subjects with plots of realized profits for both control and promotion efforts as decision support. The experimental results show that effort decisions demonstrated a significant preference bias for promotion effort in the $lp$ condition, consistent with the behavioral model. Nevertheless, the behavioral model assumes that subjects have full knowledge of profit distribution and the expected profit of each effort, which we did not provide in the experiment. To address this issue, we conducted a follow-up experiment specifically for the $lp$ condition, where the preference bias was significant in the main experiment.

In this follow-up experiment, subjects were given histograms of the profits under each effort investment and their respective profit means. In addition, we measured the subjects’ risk aversion using a Holt–Laury-type lottery choice game (Holt and Laury, 2002) to test whether risk attitudes affected the effort decisions. Screenshots of these configurations (i.e., decision support and risk elicitation) can be found in Figures C8 and C9 in Section C of the Supplemental Material. The experimental instruction and procedure remained the same as in the main experiment. We recruited 54 subjects. The power analysis based on the main experiment shows that a sample size of 486 (effort decisions by 54 subjects) is acceptable for a significance level of $\alpha =0.05$ . The average payment for the subjects was 12.47 euros.

The results are consistent with the main experiment. The overall proportion of promotion effort is significantly larger than 50% ( $70.28\mathrm{\%}$ , $p<0.001$ ), corroborating the preference bias for the promotion effort. Specifically, the proportions of promotion effort are significantly larger than 50% in two scenarios ( $\mathrm{\Delta }{\mathrm{\Pi }}_P=0$ : $72.84\mathrm{\%}$ , $p<0.001$ ; $\mathrm{\Delta }{\mathrm{\Pi }}_P>0$ : $81.48\mathrm{\%}$ , $p<0.001$ ), but only directional supported for scenario $\mathrm{\Delta }{\mathrm{\Pi }}_P<0(58.02\mathrm{\%}$ , $p=0.2203$ ). Furthermore, estimating $\overline{U}$ similar to Model 3 in Table 4 ( $\overline{U}={\alpha }_0+{\alpha }_{\text{Loss}}\cdot \text{Loss}+{\alpha }_{\text{Risk}}\cdot \text{Risk}$ , where $\text{Risk}$ is the elicited risk measure) showed that the ${\alpha }_0$ was still significantly positive ( ${\alpha }_0=62.9430$ , robust $se=28.9388$ , $p=0.0300$ ).

6.2 Robustness of Theoretical Findings

We made several assumptions in the normative and behavioral models to isolate the main effects and simplify the decision-making process for subjects in the experiment. To further validate our theoretical findings, which the experimental data corroborated, we extend the model in the following ways: a) allowing the retailer to place a biased order, instead of the optimal order, alongside the effort decision; and b) assuming the initial market demand follows a general symmetric distribution instead of a uniform distribution. The results of these extensions showcase the robustness of our theoretical findings.

6.2.1 Extension by Allowing Biased Orders

This extension only leads to changes in the behavioral model but not the normative model. In the updated behavioral model, the ordering quantity is determined by optimizing Equation (8) instead of being calculated by Equations (4) and (5). The analysis consistently demonstrates that the preference bias for promotional effort diminishes as the optimism level $\beta$ increases. This bias is significant in the $lp$ condition but not the $hp$ condition. Detailed analyses are found in Section F of the Supplemental Material.

6.2.2 Extension Under a General Demand Distribution

We follow Ren et al. (2017) and Ren and Croson (2013) to formulate the general demand distribution. Let $D=\mu +\sigma \mathrm{\Psi }$ be the original market demand, with mean $\mu$ and variance ${\sigma }^2$ . Thus, $\mathrm{\Psi }$ is a random variable following a standardized distribution with mean 0 and variance 1. We denote its probability density function (PDF) and cumulative distribution function (CDF) as $f(\cdot )$ and $F(\cdot )$ , respectively. We assume that it is unimodal and symmetric (i.e., $f(-x)=f(x)$ for all real numbers $x$ ), and $F(\cdot )$ is continuous and strictly increasing. We also assume that demand is nonnegative, $D\ge 0$ . Given the critical ratio $CR$ in a newsvendor model, we denote $b=F^{-1}(CR)$ .

If the retailer invests control effort ( $I_C=1$ ), the transformed demand is denoted by $D_C=\mu +{\mathrm{\Psi }}_C$ , where ${\mathrm{\Psi }}_C=\frac{\mathrm{\Psi }}{\sqrt{v}}$ (Kurtuluş et al., 2012). Control effort results in $D_C$ where demand mean $\mu$ remains unchanged, but demand variance is reduced to $\frac{{\sigma }^2}{\upsilon }$ . If the retailer invests promotion effort ( $I_P=1$ ), the transformed demand is denoted by $D_P={\mu }_P+\sigma \mathrm{\Psi }$ , where ${\mu }_P=\mu +\xi$ . Promotion effort results in $D_P$ where demand mean increases to $\mu +\xi$ but demand variance ${\sigma }^2$ is unchanged. A detailed analysis of the findings below is found in Section G of the Supplemental Material.

In the normative model, we can derive a threshold of the promotion-effort factor $\overline{\xi }$ under the general demand distribution similar to that in Section 3.1. Proposition 5 summarizes the results under the general distribution.

Proposition 5.

Given the initial market demand $D=\mu +\sigma \mathrm{\Psi }$ , there is a threshold of the promotion-effort factor, $\overline{\xi }=\frac{1}{CR}\cdot \sigma \left(\frac{\sqrt{\upsilon }-1}{\sqrt{\upsilon }}\right)\underset{b}{\overset{\mathrm{\infty }}{\int }}xf(x)dx$ , such that the retailer

1. (a)

invests control effort ( $I_C=1$ ) when $\xi <\overline{\xi }$ ; or

(b)

invests promotion effort ( $I_P=1$ ) when $\xi >\overline{\xi }$ .

Otherwise, the retailer is indifferent between control and promotion efforts. Additionally, $\overline{\xi }$ decreases in $CR$ .

In the behavioral model, we can calculate the threshold ${\overline{\xi }}^{Beh}$ under the general demand distribution as well. Proposition 6 shows the preference bias under different values of $\beta .$

Proposition 6.

Given the initial market demand $D=\mu +\sigma \mathrm{\Psi }$ , if $\beta <1-CR-\frac{{\int }_b^{\mathrm{\infty }}xf(x)dx+F^{-1}(0)CR}{b-F^{-1}(0)}$ , $\overline{U}=R_P(Q_P)-R_C(Q_C)>0$ and ${\overline{\xi }}^{Beh}<\overline{\xi }$ , implying a preference bias for promotion effort; if $\beta >1-CR-\frac{{\int }_b^{\mathrm{\infty }}xf(x)dx+F^{-1}(0)CR}{b-F^{-1}(0)}$ , $\overline{U}<0$ and ${\overline{\xi }}^{Beh}>\overline{\xi }$ , implying a preference bias for control effort; otherwise, $\overline{U}=0$ and ${\overline{\xi }}^{Beh}=\overline{\xi }$ , implying no preference bias.

Our analysis also confirms the presence of a preference bias toward promotion effort in the $lp$ condition, and the direction of the preference bias is not singular in the $hp$ condition.

Proposition 7.

In the low-profit condition, effort investments exhibit a preference bias for promotion effort, that is, ${\overline{U}}_{lp}>0$ , because $\beta <1-C{R}_{lp}<1-C{R}_{lp}-\frac{{\int }_{b_{lp}}^{\mathrm{\infty }}xf(x)dx+F^{-1}(0)C{R}_{lp}}{b_{lp}-F^{-1}(0)}$ .

7. Discussion and Conclusion

This study examines the preferences between control and promotion efforts. We utilized a reference-dependent model to analytically specify the behavioral conditions under which effort decisions exhibit a preference bias for a particular effort type and characterize a measure for this bias. The experimental data confirm the predictions of the behavioral model, demonstrating the different preference biases between the $lp$ and $hp$ conditions; that is, there is a significant bias toward the promotion effort in the $lp$ condition and no significant bias in the $hp$ condition.

In this section, we extend the study by discussing alternative behavioral factors that might explain these preferences. We also present the managerial implications and theoretical relevance of our findings and outline potential paths for future research.

7.1 Discussion of Other Potential Explanatory Factors

In this section, we first provide additional evidence to validate our model's loss neutrality assumption. Next, we discuss other explanatory drivers inspired by those influencing newsvendor decisions, such as risk aversion, overconfidence, and a reference-dependent model with other reference points.

7.1.1 Evaluation of the Loss Neutrality Assumption

Our behavioral model was developed under the assumption of loss neutrality. We validated this assumption using the experimental data. First, the estimate of ${\alpha }_{\text{Loss}}$ from the main experimental data shows that the effort decisions do not depend on the loss attitude elicited by the task in the experiment (see Table 4). A similar analysis in the follow-up experiment (see Section 6.1) further corroborates this finding. The estimation of $\overline{U}$ in Section 6.1 revealed no significant relationship between effort decisions and loss aversion ( $p=0.1040$ ).

It is important to note that the loss aversion collected from the elicitation task takes the status quo as the reference point, which is conceptually different from the reference point in our behavioral model. Hence, to further validate loss neutrality in our setting, we re-estimate our model by including loss aversion parameter $\lambda$ , resulting in the model by Long and Nasiry (2015). The estimation results (see Table H1 in Section H of the Supplemental Material) reveal no significant difference in the $LpL$ values between the models with and without $\lambda$ ( $Lp{L}_{Beh}=Lp{L}_{Long\mathrm{\&}Nasiry}=-1073.86$ ). Notably, the estimate $\lambda =1.1117$ is not significantly different from 1 (Wald test, ${\chi }^2(1)=0.18,p=0.6753$ ). In summary, these results support our assumption of loss neutrality in our model.

These analyses reveal that including loss aversion is less effective than our proposed reference dependence model in fitting the experimental data. This finding does not imply that loss aversion plays no role in the decision-making process. Instead, it suggests that such a parameter does not contribute additional positive effects to the fit of the behavioral model in this context. Some studies demonstrate that adapting a particular reference point can influence decision-making performance in a manner similar to loss aversion (Kahneman et al., 1991).

7.1.2 Risk Aversion

One possible model for explaining effort decisions is risk aversion. This reasoning stems from the fact that retailers may increase their willingness to invest control effort because such efforts reduce the supply-demand mismatch risk or profit-risk derived from the expected profit's random nature. Hence, the control effort's informational value (Delquié, 2008) is associated with a possible increase in utility.

To investigate the abovementioned predictions, we use the constant absolute risk aversion function (CARA), $u(\pi )=-e^{-\alpha \pi }$ (Davis, 2015; Eeckhoudt et al., 1995), where $\alpha$ represents the level of risk aversion, to analyze the subject's utility function and run numerical analyses (see Section H of the Supplemental Material for details). The CARA measure for risk has been well accepted in various studies in supply chain management (Bonzelet, 2022; Kremer and Van Wassenhove, 2014). According to our analysis, this model predicts control-effort bias under both profit conditions, under our experiment parameters.

Next, we used two other approaches to further investigate the risk aversion framework. First, we employed a structural approach to show the performance of this model in fitting the main experimental data (see Table H1 in Section H of the Supplemental Material). The results show that this model provides a poor fit for the data. Second, in the follow-up experiment, the estimation of $\overline{U}$ in Section 6.1 indicated no significant relationship between effort decisions and risk aversion ( $p=0.1260$ ). In summary, these approaches demonstrate that risk aversion is not apt for characterizing the observed effort decisions.

7.1.3 Other Theories Explaining Newsvendor Behavior

The literature has proposed several theories to predict newsvendor decisions, such as the anchoring-and-adjustment heuristic (Schweitzer and Cachon, 2000), minimizing ex-post inventory error (Kremer et al., 2014), overprecision (Ren et al., 2017; Ren and Croson, 2013), bounded rationality (Su, 2008), and reference-dependence (Ho et al., 2010; Jammernegg et al., 2022; Long and Nasiry, 2015; Uppari and Hasija, 2019). We integrated bounded rationality into our model. Regarding other models, we believe that they also have the potential to explain effort decisions with appropriate adjustments to this decision setting. For example, we find that overprecision can predict promotion-effort bias, control-effort bias, or no bias, depending on how its influence on demand is modeled (see Section H of the Supplemental Material for the proof and details). ³

Next, there could be other candidate reference points in our operational context. We previously elaborated on the rationale behind our choice of reference point in Section 3.2. Now, we examine the suitability of a reference point proposed by Uppari and Hasija (2019). They demonstrate that a stochastic reference point can explain the pull-to-center effect and subjects’ heterogeneity in the newsvendor context. Such a reference point is computed as the payoff evaluated at ordering the mean demand $\mu ,$ $\pi (x,Q=\mu ).$ Accordingly, the reference utility is calculated by $R(Q)=\eta ({\int }_{x\in D{}_{\ge 0}}(\pi (x,Q)-\pi (x,\mu ))dF(x)-\lambda {\int }_{x\in D_{<0}}(\pi (x,\mu )-\pi (x,Q))dF(x))$ , where $\lambda$ is the loss aversion parameter. Using the formulation in Uppari and Hasija (2019), we can similarly have $\mathrm{\Delta }{U}_P=\mathrm{\Delta }{\mathrm{\Pi }}_P+\overline{U}$ , where $\overline{U}$ is the difference between $R_P(Q_P)$ and $R_C(Q_C)$ (details can be found in Section H of the Supplemental Material). We can show that this model predicts the preference bias toward promotion-effort (control-effort) preference when loss aversion is below (above) a threshold $\overline{\lambda }$ . However, the thresholds for the $lp$ and $hp$ conditions ${\overline{\lambda }}_{lp}$ and ${\overline{\lambda }}_{hp}$ are identical when $C{R}_{lp}=1-C{R}_{hp}$ . Thus, this model indicates a consistent preference bias across profit conditions, contrasting with the observations from the main experiment. Furthermore, the structural estimation results also show that this model cannot properly fit the experimental data (see Table H1 in Section H of the Supplemental Material).

We also use structural estimation based on the experimental data to test other alternative reference points, such as “MaxMin” (the maximum realized payoff evaluated at the minimum demand $\underset{\_}{D}$ , $(p-w)\underset{\_}{D}$ ) and “Status Quo” (zero payoff) reference points, as discussed in Uppari and Hasija (2019). Our model also outperforms these models (see Table H1 in Section H of the Supplemental Material).

7.2 Managerial and Theoretical Implications and Future Research

The role of retailers’ efforts in shaping demand information is crucial for supply chain performance, yet little behavioral research has been conducted in this area. This study provides valuable insights for managers, highlighting potential mistakes and corresponding profit consequences of effort choices under different profit conditions. A subsequent challenge in practice is to manage preference bias in effort decisions to enhance profit performance. The behavioral model highlights a distorted understanding of the impact of demand variance reduction due to reference-dependent preferences. The effect of reference-dependence on the preference bias largely depends on optimism factor β.

To control the preference bias, one can either mitigate the strength of reference dependence (denoted by $\eta$ in the model) or manipulate the optimism factor. Since the preference bias can also have positive effects when it aligns with the optimal effort, maintaining reference dependence and adjusting the optimism factor may be a more practical approach. This finding suggests that managers should monitor their optimism levels during decision-making. Low optimism should be encouraged when promotion effort is optimal, while high optimism should be discouraged in such cases, and vice versa for situations where control effort is optimal. We particularly advise less optimistic managers to exercise caution when making effort decisions in the control-effort-expected situation, as biased choices may lead to considerable losses. Such manipulation of the optimism attitude, or positive/negative thinking, has been well studied using framing methods (Levin et al., 1998) and applied in practice, such as health decisions (Krishnamurthy et al., 2001). Implementing these strategies can help managers better align their effort decisions with optimal outcomes and improve overall performance.

From a theoretical perspective, this study is the first attempt to apply the reference-dependence framework to demand-shaping decisions in an operational context. The normative model provides insight into the focal operational problem, while the behavioral model explores the drivers of preference bias and accurately captures decision-making patterns. These models serve as building blocks for further studies, such as bargaining or contract design with demand-shaping decisions. Another contribution is the empirical support for applying the reference-dependence framework from the newsvendor context to demand-shaping efforts. We validated this application by testing our analytical predictions, providing model extensions, and discussing other potential explanatory drivers. This study emphasizes that a unified behavioral theory can capture various retailer decisions.

Finally, we addressed Uppari and Hasijas’ (2019) call for understanding how reference points are chosen through theoretical and empirical analysis. We adopted the reference point proposed by Long and Nasiry (2015) and Kirshner and Ovchinnikov (2018)—a convex combination of the maximum and minimum payoffs—and compared it with three alternative reference points using empirical data. Few articles have provided empirical evidence to support the theoretical foundation of the model and the selection of reference points (Uppari and Hasija, 2019). Therefore, our study is a valuable contribution in this regard.

Our study provides potential avenues for future research. The current empirical observations could benefit from validation across various scenarios, including alterations in demand distribution or the supply chain contracts in use. An intriguing possibility is to evaluate various contracts or craft one that specifically aims to elicit optimal demand-shaping efforts, considering the identified preference biases. While our current setting focuses on the retailer's effort decisions, future iterations that account for both the supplier's and retailer's motivations in demand-shaping efforts stand to offer deeper insights.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478231224965 - Supplemental material for Characterizing the Retailer's Preference for Demand–Control and Demand–Promotion Efforts

Supplemental material, sj-pdf-1-pao-10.1177_10591478231224965 for Characterizing the Retailer's Preference for Demand–Control and Demand–Promotion Efforts by Jose Benedicto Duhaylongsod and Yingshuai Zhao in Production and Operations Management