Cooperative and Non-Cooperative Strategies in Product Warranty Pricing: A Hierarchical Game Approach

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper proposes a game-theoretic hierarchical model to study product warranty pricing issues, involving three main stakeholders: manufacturers, independent service agents, and consumers. By combining cooperative and non-cooperative games, a hierarchical game-theoretic model is constructed to better reflect the complexity of multi-party interactions in real-world scenarios. The Shapley value is used to fairly allocate the total revenue of the alliance, thereby reflecting the respective contributions of manufacturers and agents. Overall, this paper provides theoretical support and practical guidance for product warranty pricing issues through an innovative game theory model and rigorous analytical methods. However, there are still some shortcomings that require further revision:

1. The logical coherence of the introduction is insufficient, with transitions between paragraphs feeling unnatural and lacking a clear overarching thread to connect the entire text. For example, the transition from the importance of warranty policies directly to the issue of information asymmetry lacks necessary bridging content. The introduction should be reorganized according to the logical sequence of “background → problem → limitations of existing research → contributions of this paper.”

2. Figure 1 does not display all content and should be moved to the next page for better presentation.

3. The literature review merely states phenomena without delving into how these studies support the innovation of this paper or what unresolved issues remain.

4. Sentences are overly complex: Some sentences are too convoluted (e.g., “The pricing strategies in competitive markets, such as this one, represent a complex factor of cooperative and non-cooperative behaviors...”), impairing readability.

5. The sources of parameters are not specified. For example, are the failure rate (λ = 0.0008/hour) and maintenance rate (μ = 0.02/hour) based on actual data? The data sources should be clarified.

6. The conclusion section lacks structural coherence, logical flow, and clear articulation of key findings. Therefore, the conclusion section should strengthen logical consistency and clearly articulate theoretical innovations. Avoid generalizations and focus on closely aligning with the core issues of the entire paper.

7. References should be updated to include the latest research findings from the past three years.

Author Response

Reviewer: 1

This paper proposes a game-theoretic hierarchical model to study product warranty pricing issues, involving three main stakeholders: manufacturers, independent service agents, and consumers. By combining cooperative and non-cooperative games, a hierarchical game-theoretic model is constructed to better reflect the complexity of multi-party interactions in real-world scenarios. The Shapley value is used to fairly allocate the total revenue of the alliance, thereby reflecting the respective contributions of manufacturers and agents. Overall, this paper provides theoretical support and practical guidance for product warranty pricing issues through an innovative game theory model and rigorous analytical methods. However, there are still some shortcomings that require further revision:

 

A: Thank you for the comment. We believe we have addressed all the comments and suggestions, but we remain available for any additional concerns.

 

1. The logical coherence of the introduction is insufficient, with transitions between paragraphs feeling unnatural and lacking a clear overarching thread to connect the entire text. For example, the transition from the importance of warranty policies directly to the issue of information asymmetry lacks necessary bridging content. The introduction should be reorganized according to the logical sequence of “background → problem → limitations of existing research → contributions of this paper.”

 

A: Thank you. Please refer to the following new introduction section as suggested. The background can be found from lines 24 to 44, background from 45 to 69, limitations of existing research from 70 to 77, and contributions of this paper from lines 78 to 97. Please see the introduction as follows:

 

Background:

 

“Price and warranty are two variables that affect companies' profitability, as well as consumers' purchase decisions. Warranty serves as an incentive for the customer to buy a product (Esmaeili et al., 2014; Rahman & Chattopadhyay, 2006) and it can generate an additional revenue for companies (Zhou et al., 2025). Consequently, this environment is characterized by the interaction of multiple decision-makers (players) who face several decision problems and pursue different goals (Murthy & Jack, 2014).

 

The base warranty is an integral part of the sale and is factored into the product’s price (Murthy, 2007). A common type of warranty is the Free Replacement policy, under which consumers obtain free replacements or repairs for failures within the warranty period ((Thomas & Rao, 1999; Rao, 2021). In certain situations, manufacturers may also offer money-back warranty in addition to offering free replacement (Boom, 1998).

 

Beyond the base warranty provided by the manufacturer, other companies, often referred to as service agents, extend coverage plans by assuming responsibility for post-warranty maintenance under a maintenance service contract (Murthy et al., 2015). This aftermarket warranty is illustrated by third-party plans available for Apple products. For example, Apple sells its products with a one-year Limited Warranty, starting from the purchase date. Consumers can then purchase an extended protection plan for their devices through SquareTrade.”

 

Problem:

 

“The interaction between these players creates a complex problem. Firstly, the cost of a warranty policy is inherently uncertain. Because failures are modeled as a stochastic process, the future cost of providing this service is not a deterministic event, since it depends on the failure time of the product (Ben-Daya et al., 2016). Given this uncertainty, developing effective pricing models becomes a critical strategic decision for manufacturers and service agents to maximize their profits.

 

Secondly, under this uncertainty, manufacturers and service agents must anticipate how the consumer react to this pricing structure, which leads towards a game-theoretic formulation (Singpurwalla & Wilson, 1993). Players’ payoffs are interdependent, that is, the action of one affects the outcome of all (Parilina et al., 2016). For example, if the consumer does not buy the product, the manufacturer and the service agent obtain zero profit. Therefore, manufacturers and service providers must strike a balance between warranty coverage, service quality, and pricing to maximize profitability while ensuring customer satisfaction (Rahman & Chattopadhyay, 2006).

 

Most research on game theory applied in warranty problems focuses on two-party interactions: a maintenance provider (manufacturer or service agent) and a consumer (Esmaeili et al., 2014; Iskandar el al., 2022). To define the equilibrium strategies, these studies apply Nash Equilibrium (Jackson & Pascual, 2007; Hamidi et al., 2016; Taleizadeh & Sherafati, 2019), and more frequently, Stackelberg (leader-follower) structure (Murthy & Young, 1995; Ashgarizadeh & Murthy, 2000; Murthy & Ashgarizadeh, 1998; Murthy & Ashgarizadeh, 1999; Rinsaka & Sandoh, 2006; Hamidi et al., 2016; Arabi et al., 2017; Santos et al., 2017; Panda et al., 2020; Iskandar el al., 2022; Li et al.,2023; Zhang et al., 2023).”

 

Limitations of existing research:

 

“However, the dynamics of a more complex, three-player market remain largely underexplored. A few studies do consider this three-player view, modeling both non-cooperative and cooperative outcomes (Esmaeili et al., 2014; Gamchi et al., 2013; Taleizadeh & Sherafati, 2019). More importantly, a significant gap persists: these models do not analyze the marginal contribution of each player to the different possible coalitions. This omission prevents a deeper understanding of fair profit allocation and the stability of cooperative agreements.”

 

Contributions of this paper:

 

“This paper addresses this gap by developing and comparing solutions from both non-cooperative and cooperative game-theoretic frameworks. Our non-cooperative model is formulated as a sequential game with a two-leader game. At the upper level, the manufacturer and the service agent simultaneously set their prices; at the lower level, the consumer observes these prices and decides whether to purchase the product bundled with the extended maintenance service. The subgame-perfect Nash equilibrium for this interaction is derived.

 

For the cooperative framework, we introduce a novel coordinated equilibrium concept. Unlike traditional models that simply assume payoff redistribution, our approach models a structured coordination mechanism between the manufacturer and the service agent. This coordinated strategy jointly optimizes the pricing of the product and the maintenance service. To ensure a fair allocation of the gains from this cooperation, we employ the Shapley Value to quantify each player's marginal contribution (Sun & Sun, 2018; Petrosyan & Zenkevich, 2016). This hybrid approach captures the benefits of both cooperative stability and strategic autonomy, providing a more realistic and implementable framework for warranty pricing. A key methodological aspect of this work is the use of simulation to accurately estimate the maintenance costs for the underlying failure-repair process of the system.”

 

 

2. Figure 1 does not display all content and should be moved to the next page for better presentation.

 

A: We have adjusted Figure 1 as suggested.

 

3. The literature review merely states phenomena without delving into how these studies support the innovation of this paper or what unresolved issues remain.

 

A: Thank you for the comment. Nineteen new references were added to this last version of the manuscript, most in the introduction (see previous response 1) and Model Formulation. Please refer to the reference section.

 

4. Sentences are overly complex: Some sentences are too convoluted (e.g., “The pricing strategies in competitive markets, such as this one, represent a complex factor of cooperative and non-cooperative behaviors...”), impairing readability.

 

A: Thanks for the comment. We have proofread the entire manuscript for a better readability. We hope this is appropriate now. For this specific paragraph, we have reformulated as follows:

 

“Pricing strategies in competitive markets involve a complex mix of cooperative and non-cooperative behaviors, often modeled through game-theoretic frameworks that illustrate the hierarchical relationships among manufacturers, retailers, and consumers. In this context, manufacturers and service providers must strike a balance between warranty coverage, service quality, and pricing to maximize profitability while ensuring customer satisfaction. Additionally, estimating the costs associated with warranty servicing can be challenging due to uncertainties related to product usage and maintenance”.

 

5. The sources of parameters are not specified. For example, are the failure rate (λ = 0.0008/hour) and maintenance rate (μ = 0.02/hour) based on actual data? The data sources should be clarified.

 

A: Thank you for the comment. Parameters are based on simulated data. We have inserted this explanation through the lines 319 to 326 of the third section:

 

“The algorithm computes one replica. Therefore, to estimate the expected value of the penalty time, we use Monte Carlo approach, performing the loop on the algorithm. The following numerical application is carried out to illustrate the game equilibrium. The nominal values for the model's parameters are:  (hours),  (hours),  (per hour),  (per hour),  (hours), , , ( per hour), , and . Table 2 presents the expected value of the reliability-related performance measures. The Monte Carlo sample size for simulating the penalty time was 10,000.”

 

6. The conclusion section lacks structural coherence, logical flow, and clear articulation of key findings. Therefore, the conclusion section should strengthen logical consistency and clearly articulate theoretical innovations. Avoid generalizations and focus on closely aligning with the core issues of the entire paper.

 

A: Thank you. We have rewritten the conclusion as suggested. Please refer to the new conclusion section:

 

“This paper presented a three-level service contract defined under a hierarchical structure for a manufacturer, an agent, and a consumer. The provision of the base warranty serves to encourage the consumer to purchase the product, and maintenance services provided by the agent after the base warranty period aim to ensure the consumer retains the item throughout its lifetime. The proposed model illustrates how the manufacturer and the agent can collaboratively establish pricing strategies to capture consumer surplus. In this approach, the equilibrium prices for the sale of the product, including base warranty costs and maintenance services, are determined using the Shapley value to provide a unique solution that highlights the marginal contribution of each decision-maker.

 

The game theory approach was applied due to the presence of many participants with different goals and strategies. The cooperative game established a coalition between the manufacturer and the agent that competes against the consumer (non-cooperative approach). For a given price structure , the consumer replies, buying or not buying the product with maintenance services. As a result, the non-cooperative approach is via a sequential game whose solution is via the subgame-perfect Nash equilibrium. Combining cooperative and non-cooperative game theory elements characterized equilibrium strategies that supported allocating the gains from coalition formation. The results demonstrate how coordination between the manufacturer and the agent can lead to optimized pricing strategies and improved profit-sharing outcomes, offering a useful theoretical contribution to the literature on warranty pricing and service contracts.

 

Some of the limitations in this work are discussed as follows. First, the model assumes complete and symmetric information. Extending the framework to incorporate asymmetric information on product quality or repair costs can allow for the exploration of screening or signaling contracts. Second, our model assumes a homogeneous consumer with fixed willingness-to-pay. Future work can introduce consumer heterogeneity and competing service offerings, potentially embedding an endogenous demand function to enrich the pricing dynamics. Third, the current model is static and does not account for dynamic reliability degradation or pricing adjustments over time. A multi-period or repeated game formulation could capture these aspects more realistically.

 

Moreover, the model employs a simplified reliability structure based on a homogeneous Poisson process. Future research may adopt more advanced reliability models, and consider condition-based maintenance strategies. In addition, the sensitivity analysis on individual parameters requires more comprehensive robustness assessments to further strengthen the model’s applicability. Additionally, empirical calibration using real warranty and cost data would be valuable for validating the model and benchmarking predicted outcomes against observed market behaviors. Finally, the policy and welfare implications of coordinated versus competitive pricing regimes remain an open question that can inform regulatory discussions and consumer protection frameworks.

 

Other interesting extensions can be modelling  the manufacturer and the agent interaction with only one warranty option or additional warranty options with different coverage mechanisms and decision variables - such as and consider the risk parameter to model the players' preferences as seen in Asgharizadeh and Murthy (2000), Esmaeili et al. (2014), Murthy and Asgharizadeh (1998, 1999). We view these limitations not as shortcomings, but as opportunities to extend the theoretical foundation laid in this paper. Each of these directions would require distinct modelling techniques, richer datasets, and a shift in analytical focus, and are therefore left for future investigation.

 

The decision-making process can be further enriched by incorporating two distinct maintenance agents, allowing the consumer to select one for the provision of maintenance services following the expiration of the base warranty period. This extension can transform the model into a four-player framework, comprising a manufacturer, two maintenance agents, and a consumer. Such a configuration should offer deeper insights into competitive dynamics between maintenance agents, fostering a more comprehensive understanding of pricing strategies, service quality differentiation, and the resulting consumer behavior, also facilitating the exploration of equilibrium outcomes under varying market conditions and service contracts, contributing to consumer-oriented service policies.”

 

 

7. References should be updated to include the latest research findings from the past three years.

 

A: Thank you. We have updated many references (please see the answer to comment 3). We remain available for any additional comments if needed.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Please check the PDF file.

Comments for author File: Comments.pdf

Author Response

A: We appreciate the reviewer's detailed and thoughtful suggestions. We fully recognize the value of each proposed extension, including relaxing information assumptions, introducing consumer heterogeneity, incorporating dynamic interactions, replacing the homogeneous Poisson process with a non-homogeneous rate, sensitivity analysis of parameters and uncertainty quantification, grounding the model empirically with real warranty cost data, refining exposition, among others. However, we respectfully note that these directions require substantial additional data, modelling structures, or analytical frameworks that fall beyond the scope of the present work. Given the theoretical and foundational nature of our model, we have chosen to focus on establishing the core hierarchical game framework and its equilibrium properties. Nevertheless, we have acknowledged the reviewer’s recommendations in the revised conclusion section as promising avenues for future research. We hope this response clarifies our position and appreciate the opportunity to improve the manuscript accordingly.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The author has answered all questions and thoroughly revised the paper. I am satisfied that the paper has reached publication standard and therefore recommend its acceptance and publication.

Reviewer 2 Report

Comments and Suggestions for Authors

Thank you for the updates.