1. Introduction

1 Introduction

In recent years, the integration of online and offline channels has become a prevailing trend in retail, driven by evolving consumer preferences and technological advancements. The Buy-Online-Pick-up-in-Store (BOPS) model, which allows consumers to purchase products online and collect them offline, represents a hybrid retail strategy that combines the convenience of e-commerce with the immediacy and trust of physical stores. Global retailers such as Walmart, Best Buy, and JD.com, have implemented BOPS systems to improve service efficiency, reduce last-mile logistics costs, and alleviate product return pressures. According to recent industry surveys, approximately 42% of retailers offer BOPS options, with 9% of orders being fulfilled through this channel ^[1].

Despite its operational advantages, BOPS faces several challenges. One of the most critical is the high product return rate caused by a mismatch between consumer expectations and received goods, referred to as product matching uncertainty ^[2]. To address this issue and enhance consumer confidence, many retailers offer money-back guarantee (MBG), which provide full refunds to unsatisfied customers ^[3]. While MBG can reduce perceived risk, its strategic effectiveness and profitability in BOPS settings remain underexplored, especially considering the decentralized nature of pricing and fulfillment decisions. Additionally, BOPS operations are often facilitated by e-commerce platforms (e.g., JD.com, Walmart), which may offer per-order subsidies or contractual incentives to offline stores to support in-store pickup. However, such platform-driven coordination mechanisms have received little attention in existing analytical models.

Recent studies have begun to explore the behavioral implications of return policies in omnichannel settings. For instance, ^[3] analyze how free return services affect repeat purchase intentions, concluding that return policies may both boost demand and increase operational costs. Meanwhile, ^[4] demonstrate that when consumers exhibit risk-averse behavior, MBG policies can significantly improve conversion rates, but this effect is evident only when the product matching rate is high. These insights highlight a critical trade-off: MBG increases consumer trust but may alter the competitive dynamics between channels, especially when pricing strategies are decentralized.

Given that online and offline channels often operate independently in BOPS models, strategic coordination becomes critical. The online retailer may benefit disproportionately from MBG due to reduced return friction and greater reach ^[3], while the offline channel bears increased costs from returns or reduced store visit frequency [^[5], ^[6]]. This asymmetry raises questions about optimal pricing, refund implementation, and profit distribution between channels. Stackelberg game models have proven to be an effective analytical tool for such hierarchical decision-making scenarios. Recent works by ^[7] and ^[8] apply Stackelberg frameworks to study price competition and channel leadership in omnichannel environments. However, few of these models incorporate MBG policies or consumer matching uncertainty explicitly.

Moreover, consumer choice in BOPS channel environments is heavily influenced by product matching rate and offline inconvenience cost ^[9]. While the matching rate reflects the degree to which a product meets consumer expectations, the inconvenience cost captures factors such as travel time, queuing, and waiting time during offline pickup ^[10]. These behavioral factors not only affect purchase likelihood but also influence the effectiveness of MBG policies. Studies by ^[11] emphasize the need to integrate behavioral economics parameters into pricing models, as ignoring them may lead to suboptimal policies and misleading managerial insights.

This paper examines how MBG policies influence pricing decisions and channel performance in a decentralized BOPS environment. Specifically, we develop a Stackelberg game involving an online retailer (leader) and an offline retailer (follower), incorporating two key behavioral parameters: product matching rate and offline inconvenience cost. In addition, we introduce a platform-mediated subsidy parameter that compensates the offline retailer for each BOPS transaction, representing a contractual coordination mechanism widely adopted in real-world platforms. By comparing scenarios with and without MBG, our model derives closed-form equilibrium solutions and evaluates outcomes in terms of channel profit allocation, consumer surplus, and social welfare.

This study addresses the following research questions:

(1) How does the implementation of MBG affect equilibrium pricing strategies across BOPS channels?

(2) Under what conditions does MBG enhance or undermine profitability and consumer welfare?

(3) Can MBG consistently improve overall social welfare, or might it induce distortions that reduce system efficiency?

Although the platform's per-order subsidy to offline retailers is not the central focus of our research questions, we incorporate this coordination mechanism into the model structure to better reflect practical platform support strategies in BOPS settings.

This study builds a Stackelberg game model featuring an online retailer as the leader and an offline retailer as the follower, reflecting the decentralized structure commonly observed in BOPS operations. On this foundation, it incorporates two key behavioral dimensions—product matching rate and offline inconvenience cost—to capture how consumers' perceived uncertainty and effort influence purchasing and return decisions. Based on this enriched framework, the paper further investigates how MBG policies shape strategic pricing outcomes across channels and derives practical insights for coordinating return guarantees and pricing incentives in omnichannel retail settings.

The remainder of the paper is structured as follows: Section 2 reviews the relevant literature on BOPS, MBG, and omnichannel pricing. Section 3 presents the model setup and core assumptions. Section 4 analyzes the equilibrium outcomes under MBG and non-MBG regimes. Section 5 conducts numerical simulations to test the theoretical insights, including a sensitivity analysis of the platform subsidy parameter. Section 6 discusses the theoretical and managerial implications of the results. Section 7 concludes with key takeaways and directions for future research.

2. Literature Review

2 Literature review

This section reviews the relevant literature on omnichannel retailing, with a particular focus on the BOPS (Buy-Online-Pick-up-in-Store) model and its associated pricing strategies. It further examines the role of money-back guarantees (MBGs) and the integration of behavioral factors such as product matching uncertainty and offline inconvenience cost in pricing models. These strands of research provide the theoretical foundation for constructing a behaviorally informed Stackelberg game framework in this study.

2.1. Omnichannel Retailing and BOPS Pricing Strategies

2.1 Omnichannel retailing and BOPS pricing strategies

The rapid advancement of digital technology and shifting consumer expectations have prompted retailers worldwide to adopt omnichannel strategies. Among these, the Buy-Online-Pick-up-in-Store (BOPS) model has gained substantial traction due to its ability to integrate the convenience of online shopping with the immediacy of offline fulfillment.