Designing a Smart Health Insurance Pricing System: Integrating XGBoost and Repeated Nash Equilibrium in a Sustainable, Data-Driven Framework

Abstract

Designing fair and sustainable pricing mechanisms for health insurance requires accurate risk assessment and the formulation of incentive-compatible strategies among stakeholders. This study proposes a hybrid framework that integrates machine learning with game theory to determine optimal, risk-based premium rates. Using a comprehensive dataset of insured individuals, the XGBoost algorithm is employed to predict medical claim costs and calculate corresponding premiums. To enhance transparency and explainability, SHAP analysis is conducted across four risk-based groups, revealing key drivers, including healthcare utilization and demographic features. The strategic interactions among the insurer, insured, and employer are modeled as a repeated game. Using the Folk Theorem, the conditions under which long-term cooperation becomes a sustainable Nash equilibrium are explored. The results demonstrate that XGBoost achieves high predictive accuracy (R² ≈ 0.787) along with strong performance in error measures (RMSE ≈ 1.64 × 10⁷ IRR, MAE ≈ 1.08 × 10⁶ IRR), while SHAP analysis offers interpretable insights into the most influential predictors. Game-theoretic analysis further reveals that under appropriate discount rates, stable cooperation between stakeholders is achievable. These findings support the development of equitable, transparent, and data-driven health insurance systems that effectively align the incentives of all stakeholders.

1. Introduction

Universal Health Coverage (UHC), as a core principle of sustainable development, plays a vital role in promoting public health and improving social welfare indicators [1]. This concept highlights the importance of ensuring access to quality healthcare services without imposing financial burdens on individuals, regardless of their economic, social, or geographic status [2,3]. In this context, health insurance systems serve as key pillars of public health policy, responsible for creating effective financing mechanisms to lower the costs of medical treatments and promote health equity [4,5,6]. In advanced economies, risk is conceptualized and traded as a financial commodity within capital markets [7]. Insurance companies, in exchange for premium payments, assume and manage the risks associated with diverse economic activities [8].

However, one of the most complex and essential challenges in these systems is setting premium rates that accurately reflect individual or group risk levels [9] while maintaining a dynamic and stable balance between the three main stakeholders in the ecosystem—insurer, insured, and employer. Premiums are not just economic figures; they also mirror the strategic behaviors of stakeholders facing uncertainty, incomplete information, and economic incentives [10,11]. This issue becomes even more important in health insurance due to the wide variety of demographic, behavioral, and health-related characteristics among individuals [12]. When insurers aim to cover potential risks while employers and the insured seek fair and affordable premiums, misaligned interests can lead to market failures, adverse selection, and system instability. Recent studies by Obermeyer and Emanuel (2016) and by Chen et al. (2020) underscore the potential of machine learning to enhance health insurance operations—in areas such as fraud detection, claims prediction, risk adjustment, and transparent premium-setting—thereby supporting both efficiency and fairness in coverage [13,14].

UHC, as a foundational concept, plays a vital role in improving public health and promoting societal well-being [1]. It emphasizes that all individuals, regardless of their socioeconomic position, must have access to mechanisms that mitigate the financial harm caused by healthcare expenses and ensure access to essential services [2,3]. In this regard, the health insurance industry, as a cornerstone of social welfare systems, plays a key role in providing financial security and facilitating healthcare access [4,5,6].

Nonetheless, one of the most complex challenges in health insurance is setting premiums that are not only aligned with individuals’ actual risks [9] but also maintain a sustainable and equitable balance between the system’s three primary players—insurers, policyholders, and employers. A premium is not merely a numerical figure; it also reflects the strategic behavior of stakeholders in the face of risk, incomplete information, and financial motives [10,11]. As noted by Einav and Finkelstein (2018), individuals’ decisions in health insurance markets are shaped not only by economic incentives but also by behavioral biases and risk perceptions [15].

This complexity is magnified in health insurance, due to the vast diversity in demographics, health conditions, and behavioral traits [12]. On one hand, insurers seek premiums that adequately cover potential losses; on the other hand, employers and policyholders demand reasonable and fair rates. If this inherent tension is not managed properly, it can lead to phenomena such as adverse selection and market failure. As Cutler and Zeckhauser (1998) explain, adverse selection arises when high-risk individuals are more likely to purchase insurance, driving up costs and reducing overall market efficiency [16].

This study presents a novel framework for health insurance pricing by integrating game theory and machine learning. It leverages the concept of Nash equilibrium in repeated games and the predictive power of the XGBoost algorithm. For the first time, the three key players of this ecosystem are modeled in a strategic space where each actor’s decision is not only based on their own state but also depends on the likely reactions of other stakeholders over time. The research design deliberately integrates XGBoost with repeated game theory to address both predictive accuracy and strategic stability. XGBoost is selected for its proven superiority in handling nonlinear relationships, high-dimensional features, and imbalanced health datasets, while offering interpretability through SHAP values. Repeated Nash equilibrium modeling is employed because health insurance markets are inherently dynamic and involve ongoing interactions between multiple stakeholders. Static models fail to capture this temporal dimension, whereas repeated games enable the identification of long-term cooperative strategies that are economically viable and behaviorally stable.

Within this framework, XGBoost, as an advanced boosting-based learning algorithm, plays a central role in accurately estimating healthcare risks. The output of this model—risk categorization and dynamic premium adjustments—is incorporated into a repeated game among the stakeholders. This intelligent mechanism, using long-term strategies such as the Folk Theorem, provides a stable path for sustained cooperation and discourages opportunistic behaviors. To enhance model interpretability, the SHAP method is employed to quantify the contribution of features in risk prediction. This analysis improves transparency and establishes a foundation for fair classification of insured individuals [17].

The empirical analysis, based on real health insurance data from employees of the Northeast Iran Oil Industry, indicates that the proposed hybrid framework not only achieves satisfactory predictive accuracy (R² ≈ 0.787), but also leads to premium rates that correct policyholder behavior while restoring economic fairness in cost allocation. Low-risk groups experience a reasonable reduction in premiums, whereas high-risk individuals fairly assume a greater share—without undermining the collaborative ties between players.

In sum, this study introduces a pioneering approach to “health insurance premium engineering” that, by relying on real-world data, machine learning algorithms, and game-theoretic logic, paves the way for designing intelligent, collaborative, and resilient insurance systems.

The remainder of the paper is organized as follows: Section 2 reviews the related literature on health insurance pricing, machine learning methods, and game theory. Section 3 outlines the theoretical foundations of Nash equilibrium and XGBoost. Section 4 presents the proposed model in detail, and Section 5 discusses the results obtained from applying the model to real insurance data from the North-East Oil Industry Health and Treatment Organization of Iran, a subsidiary of the National Iran Oil Company based in Mashhad.

2. Literature Review

This section provides a structured review of the academic literature on health insurance premium determination, covering four main themes: traditional actuarial approaches, the application of machine learning, the use of game theory, and emerging efforts to integrate these domains. The goal is to identify research trends, uncover existing gaps, and define the innovative contribution of the present study within this framework.

2.1. Foundational Approaches: Actuarial Science and Risk Pooling

Traditional actuarial models have laid the theoretical groundwork for initial premium-setting methods. The Bühlmann credibility model, by combining individual and group-level data, offers a mathematical structure for more accurate risk estimation [18]. In parallel, macro policies such as the Affordable Care Act in the United States have sought to introduce risk-sharing mechanisms among insurers. As McGuire and Van Kleef (2018) highlight, “Risk sharing and premium regulation are integral tools in health insurance markets to ensure that insurers are not disproportionately burdened by high-risk enrollees, which helps stabilize the market financially [19].” Despite their theoretical rigor, these models face significant challenges when applied to high-volume, imbalanced, and dynamically evolving datasets.

2.2. Machine Learning in Health Risk Analysis

Machine learning (ML) has become an indispensable tool in healthcare, particularly in enhancing risk assessment and optimizing patient care. The proliferation of digital health data has fueled an exponential increase in ML applications across diverse medical domains, enabling more precise diagnostics, personalized treatments, and efficient resource allocation [20].

Recent research has highlighted that ensemble tree-based methods, such as XGBoost, significantly outperform traditional statistical approaches in forecasting healthcare expenditures. For instance, Xie et al. (2022) showed that XGBoost delivers superior predictive accuracy in estimating hospital readmission costs [21]. Gradient boosting methods, originally introduced by Friedman (2001), have proven effective in capturing complex nonlinearities and enabling precise risk segmentation in healthcare analytics [22].

To improve the interpretability of these high-performing models, researchers have increasingly applied techniques such as SHAP (SHapley Additive exPlanations), which quantify the relative contribution of each feature to the model’s output [17]. This level of transparency is particularly valuable for insurers and policymakers seeking to align premium strategies with individual risk profiles. In addition, recent work emphasizes that interpretive ML techniques support more economically meaningful analyses of risk behaviors and enhance strategic decision-making in premium setting [23,24]. ML models have also proven effective in detecting insurance fraud [25].

2.3. Game Theory and Strategic Behavior Analysis

Game theory provides a rigorous analytical framework for examining strategic interactions within complex economic environments, offering valuable insights into the underlying dynamics of insurance markets. Backović et al. (2016) applied game-theoretic models to mutual insurance, analyzing non-cooperative games under conditions of complete information [26]. Similarly, Zhang et al. (2017) proposed a bi-level game model for cyber insurance, capturing the strategic interplay between users, attackers, and insurers to design attack-aware insurance policies and assess network effects [27]. Their findings emphasized key principles such as the zero-profit condition for insurers and the formulation of linear insurance contracts.

2.4. Convergence of Approaches: Integrating ML and Game Theory

Recent literature suggests that while direct applications of machine learning (ML) in health insurance game-theoretic frameworks remain limited, ML has significant potential to enhance decision-making in repeated games, potentially leading to more stable and interpretable premium structures. For example, the application of deep reinforcement learning in intrusion detection systems has shown how ML can be combined with game theory to improve strategic decision-making in competitive environments, which may be analogous to health insurance markets [28]. Nevertheless, the structured and algorithmic integration of ML outputs—particularly in the context of Nash equilibrium analysis within health insurance—requires further exploration.

This study builds upon previous research on blockchain-based health insurance models (Shouri & Ramezani, 2025), where decentralized mechanisms, such as quadratic voting, were employed to enhance fairness and personalization [29]. In contrast, our approach leverages ML-driven risk stratification, integrated with the dynamics of repeated games, to establish a premium-setting framework that is both behaviorally stable and economically sustainable.

2.5. Research Gaps and Innovation of the Present Study

The literature review highlights several critical gaps that the current study seeks to address, despite significant advances in health risk modeling and strategic behavior analysis in insurance markets:

Lack of an Interactive Framework Based on Multilateral Nash Equilibrium:

Most game-theoretic models in health insurance are limited to static, bilateral structures, whereas real-world insurance ecosystems involve dynamic, multilateral interactions between insurers, policyholders, and employers. By employing the concept of generalized Nash equilibrium, this study introduces a model where each player’s strategy depends on the others’ decisions, enabling the analysis of equilibrium stability in a repeated framework. This approach provides a more realistic and sustainable model for premium allocation by incorporating mutual interests and decision-making dynamics.

Underutilization of the Folk Theorem in Stability Analysis:

Many studies settle for identifying a single equilibrium point, while the Folk Theorem suggests that in infinitely repeated games with complete information, a wide range of stable equilibria can be sustained—provided that players lack the incentive to deviate. This research applies the Folk Theorem to show how a set of premium rates can be both fair and strategically stable, without giving any party motivation to disrupt the equilibrium.

Need for Structured Integration of ML Outputs into Strategic Analysis:

Although some studies use ML models for risk prediction, the structured integration of their outputs—such as risk classification by XGBoost—into multilateral strategic relationships remains underexplored. This study bridges that gap by designing a clear data-transfer pathway from ML model outputs into the game-theoretic framework.

Focused Application to Real-World Health Insurance Markets:

In contrast to more generalized insurance studies, this research focuses on the real-world context of organizational health insurance, where complex three-way interactions are clearly observable. This specificity enables a more precise empirical validation of theoretical assumptions and proposed equilibria.

In sum, the present study takes a significant step toward developing stable, fair, and data-driven structures for determining health insurance premiums by proposing a hybrid interactive model based on XGBoost and repeated game theory with multilateral Nash equilibrium. This innovative approach offers a foundation for “insurance equilibrium engineering” in complex health markets.

3. Theoretical Foundations

This study is grounded in a unique theoretical framework that integrates game theory—specifically multilateral Nash equilibrium in repeated environments—with advanced machine learning algorithms, particularly XGBoost. This framework aims to design a model for determining health insurance premiums that is not only based on real-world data but also behaviorally stable, fair, and optimized. The theoretical structure is developed along three main axes:

3.1. Nash Equilibrium and Tripartite Dynamics in the Health Insurance Market

The concept of Nash Equilibrium, introduced by Nash (1950), is a cornerstone of non-cooperative game theory [30]. It occurs when no player has an incentive to unilaterally deviate from their strategy [30]. With the expanded role of employers as active participants in the design and provision of health insurance [31], interactions in the insurance market have evolved from a traditional bilateral structure into a more complex tripartite model involving insurers, insured individuals, and employers.

In this framework, the insurer designs pricing policies and incentive structures; employers, as contributors and facilitators of incentives, play a key role in influencing insured individuals’ behavior; and the insured choose their level of participation and healthcare consumption based on the perceived costs and benefits [15]. A Nash equilibrium exists when none of the players—given the strategies of the others—has any incentive to deviate [32]. These dynamics require rethinking conventional pricing models by integrating behavioral complexities into insurance design.

To assess the long-term strategic stability of interactions, this study adopts an infinitely repeated game framework, where a stage game is played in each period among the three actors, and the outcomes affect future strategies.

Let $u_i\left(s_t\right)$ denote the instantaneous utility of player i at time t, and $\delta$ ∈ (0,1) the discount factor, representing the present value of future payoffs. Then, the cumulative discounted utility over a finite horizon t is:

$$U_i^t=\sum _{k=0}^t{\delta }^k·u_i\left(s_k\right)$$

(1)

Here, $U_i^t$ represents the discounted cumulative utility over t periods.

In the case of an infinite time horizon, the discounted utility becomes:

$$U_i=\sum _{k=0}^{\infty }{\delta }^k·u_i\left(s_k\right)$$

(2)

In this study, the discount factor is set to $\delta$ = 0.87, which is sufficiently high to satisfy the conditions of the Folk Theorem for sustaining cooperation over time.

According to the Folk Theorem [33], any feasible utility vector that exceeds the minmax payoff level can be sustained as a Nash equilibrium in an infinitely repeated game, provided that the discount factor is sufficiently close to 1 [34].

Therefore, a high discount factor ensures that the value of continued cooperation exceeds the short-term benefit of deviation, essential for maintaining equilibrium in the health insurance ecosystem. This incentivizes insured individuals to maintain responsible behavior while enabling insurers to implement effective reward and penalty strategies.

3.2. The XGBoost Algorithm and Redefining Health Risk Prediction

XGBoost, one of the most powerful gradient boosting algorithms, has gained significant attention in medical and insurance domains due to its high performance in identifying complex patterns [35,36]. Leveraging an iterative process of correcting decision tree errors, the algorithm exhibits a strong capability in accurately predicting individual risk.

In this study, XGBoost is used to cluster insured individuals based on biomarkers, medical history, health-related behaviors, and socioeconomic variables. This stratification enables the assignment of differentiated insurance premiums aligned with the actual risk profile of each cluster [37]. The high accuracy of this approach not only reduces unexpected costs for insurance companies but also enhances transparency and fairness in premium calculations, ultimately strengthening policyholders’ trust in the health insurance system.

3.3. The Synergy of Nash Equilibrium and XGBoost: A Novel Framework for Health Insurance Pricing

The theoretical innovation of this study lies in the structured integration of two powerful approaches: data-driven analysis via XGBoost and strategic reasoning through game theory. This synergy enables the design of insurance premium rates that are not only precisely data-informed but also strategically stable and equitable. Within this model:

• XGBoost is employed to extract precise individual risk profiles;
• Multilateral Nash equilibrium is used to model the strategic interactions between the insurer, employer, and insured party;
• And ultimately, a structure is formed in which no actor has an incentive to deviate unilaterally from the agreed strategy.

According to the Folk Theorem, if the discount factor δ is sufficiently high, any payoff points within the cooperative utility space that exceeds the minimum guaranteed (minmax) level for each player can be sustained as a stable equilibrium in a repeated game. As a result, multiple equilibria and long-term cooperative behaviors are not only feasible but also optimal and desirable.

This model not only mitigates adverse selection and promotes stable participation but also leverages the principles of the Folk Theorem in repeated games to support a range of sustainable equilibria in which both economic efficiency and social fairness are achieved simultaneously.

Ultimately, the proposed framework serves not merely as a computational tool but also as a policy-oriented model for redesigning the architecture of modern health insurance systems. It offers novel insights into the development of sustainable and intelligent mechanisms in health economics.

4. Methodology

4.1. Strategic Modeling Framework

This study aims to develop a sustainable, adaptive, and data-driven framework for health insurance pricing by integrating two complementary approaches:

• XGBoost machine learning model, employed for accurate prediction of medical claims and classification of insured individuals based on their risk levels;
• Repeated Game Theory, using long-term cooperation strategies (Folk Theorem) to analyze behavioral stability among actors and derive a sustainable Nash equilibrium within insurance contracts.

In this structure, interactions between the three key players—the insured (employee), the insurer, and the employer—are modeled as an asymmetric repeated game. Each player’s behavior is designed based on long-term economic incentives and aligned strategies.

Specifically:

• Employee/Insured: The insured individual can adopt either healthy or risky strategies in lifestyle and healthcare service utilization.
• Insurer: The insurance company makes decisions regarding premium setting and risk management.
• Employer: As a facilitator and partial contributor to the insurance premium, the employer plays a critical role in contract stability and in discouraging risky behaviors.

To ensure reproducibility, the methodology follows a clearly defined pipeline:

Step 1—Data Preparation: The dataset, consisting of $\left[\mathrm{X}\right]$ records from the North-East Oil Industry Health and Treatment Organization of Iran [38], was cleaned by removing incomplete entries and normalizing numerical variables. Categorical features were encoded using one-hot encoding.

The simulations were executed on a local machine equipped with an Intel^® Core™ i7-1185G7 @ 3.00 GHz processor and 16 GB of RAM, using a personal internet connection. The average computation time for model training and equilibrium analysis was approximately 10 min per simulation cycle, ensuring scalability for larger datasets.

Step 2—Model Training with XGBoost: The dataset was split into 80% training and 20% testing. The model parameters were adjusted to achieve optimal prediction accuracy, and performance was evaluated using standard regression metrics, including $R^2$, RMSE, and MAE.

Step 3—SHAP Analysis: Feature importance was computed using SHAP values to interpret individual predictions and aggregate effects.

Step 4—Game-Theoretic Modeling: The XGBoost risk outputs were mapped into four risk categories, each associated with a premium rate. These rates formed the basis of a stage game between the insurer, employer, and insured. Payoff functions were defined in terms of net benefits:

$$U\_insurer=P-C,U\_employer=-\alpha P+\beta H,U\_insured=H-P$$

where $P$ is the premium, $C$ the expected claim cost, $H$ the health benefit index, and $\alpha$, $\beta$ weighting parameters.

Step 5—Repeated Game Simulation: The game was iterated for T = 50 periods with $\delta$ = 0.87.

The goal is to reach a stable, long-term Nash equilibrium where no player has an incentive to unilaterally deviate from their agreed strategy, even in the presence of short-term temptations or reduced cooperation.

Figure 1 illustrates the overall process framework of the proposed hybrid model. This flowchart presents the key stages from data preprocessing and risk clustering using XGBoost to strategic interaction analysis within the repeated game structure. As shown, the outcomes of the machine learning algorithm serve as inputs for strategic decision-making. Then, through the implementation of long-term cooperative strategies (Folk Theorem), adjusted premium rates are generated to reinforce Nash equilibrium, ensure behavioral stability, and sustain cooperation at the systemic level of the health insurance ecosystem.

As shown in Figure 1. Updated proposed framework for the smart health insurance pricing system. The flowchart introduces a decision point after the risk segmentation step, determining whether an individual’s risk score exceeds the high-risk threshold. High-risk individuals enter the premium adjustment and behavioral incentive process, while low- and medium-risk individuals follow the standard premium update process. All paths feed into the repeated game module to ensure stable, cooperative strategies (via the Folk Theorem), ultimately producing a sustainable premium structure. The proposed model benefits from the intelligent integration of the predictive power of the XGBoost algorithm [35] with strategic analysis based on repeated game theory [33,34].

To address the model’s conceptual evolution and its implications,

Table 1. Conceptual changes in the proposed framework and associated complexities.

Conceptual Change in Model	Description	Complexity Level	Computational Impact	Strategic/Behavioral Impact
Integration of ML (XGBoost) with Game Theory	Combining predictive analytics with strategic modeling	High	Increased computation time due to ML training	Improved accuracy in premium setting and stability of equilibrium
Risk Stratification Using SHAP Analysis	Interpretable feature contributions for classification	Medium	Minor overhead for SHAP value computation	Enhanced transparency and fairness
Multilateral Nash Equilibrium	Modeling three stakeholders instead of bilateral	High	More simulation iterations for equilibrium convergence	Captures realistic tripartite dynamics
Application of Folk Theorem	Sustaining cooperation in repeated games	Medium	Requires simulation over multiple time periods	Promotes long-term stability and cooperation
Dynamic Premium Adjustment Mechanism	Premiums updated based on real-time risk profiles	Medium-High	Periodic recalculation needed	Aligns incentives and discourages opportunism

Source: Developed by the author based on the conceptual structure of the proposed model.

summarizes the key modifications introduced in the proposed framework, along with their associated complexities, computational requirements, and strategic impacts. These changes reflect the integration of machine learning with game-theoretic reasoning, the use of interpretable risk stratification, and the implementation of dynamic mechanisms that enhance both predictive performance and behavioral realism.

As shown in Table 1, while some conceptual changes such as multilateral Nash equilibrium and XGBoost integration significantly increase computational requirements, they also enhance the realism, accuracy, and long-term stability of the proposed health insurance pricing framework.

4.2. Data and Risk Classification

To assess individual health risk and predict future medical claims, we utilized a comprehensive dataset comprising 29,785 insured employees from the North-East Oil Industry Health and Treatment Organization of Iran. Each insured individual is described by a structured set of features, including Gender, Age, Specific diseases, Number of visits, and Past claims. These variables were selected for their clinical relevance and predictive value in actuarial risk modeling.

Using the Extreme Gradient Boosting (XGBoost) algorithm, policyholders were automatically stratified into four distinct risk categories—ranging from low to high—based on their health profiles and historical utilization patterns. The XGBoost model achieved a strong predictive performance, with an R² score of approximately 0.787, indicating that nearly 79% of the variance in future claim costs could be explained by the input features.

This stratification not only enhances the precision of risk-adjusted premium pricing but also enables the identification of high-risk subpopulations for targeted interventions. By combining behavioral, demographic, and clinical indicators, the proposed approach supports the development of data-driven, personalized insurance strategies aligned with modern principles of actuarial fairness and health economics.

4.3. Premium Rate Modeling Steps

Step 1: Risk Prediction Using XGBoost

The XGBoost model functions as the predictive engine for estimating annual medical expenses per individual. Based on the predicted loss ratio, individuals are classified into four risk categories, as shown in

Table 2. Risk-based classification of policyholders based on claims-to-premium ratios.

Category	Description	Count	% of Population	% of Total Claims (IRR)
Cat. 1	No claims submitted	156	0.52%	0.00%
Cat. 2	Claims less than the paid premium	20,459	68.69%	17.84%
Cat. 3	Claims equal to the paid premium	333	1.12%	1.13%
Cat. 4	Claims greater than the paid premium	8837	29.67%	81.00%
Total	–	29,785	100%	100%

Source: Authors’ calculations based on internal dataset (2023).

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Step 2: Base Rate Calculation and Risk-Based Adjustments

The base premium rate is calculated using the expected value of claims and Wald’s Principle. Based on the risk group classification, the following adjustments are applied:

• Groups 1 & 2: Premium reduction to encourage healthy behavior
• Group 3: No change (base rate applied)
• Group 4: Premium increase to compensate for higher risk
• The base premium rate is computed using Wald’s Principle, derived from the expected value of future claims [39].

$$\pi \left(X\right)=E\left(S\right)=Q=\overline{X}.\overline{n}$$

(3)

where:

$\pi \left(X\right)$ represents the base premium.

$E\left(S\right)$ is the expected total claim.

$\overline{X}$ is the average claim amount.

$\overline{n}$ is the average number of claims.

This formulation follows from Wald’s Identity, which states that if X1, X2, … are i.i.d. claim amounts with finite mean E[X], and N is the number of claims (independent of $\left\{Xi\right\}$ with finite mean E[N], then:

$$E\left[S\right]=E\left[E\left[S|N\right]\right]=E[E[\sum _{i=1}^N{X}_i|N]]={\sum }_{n=0}^{\mathrm{\infty }}\left(N-n\right)E[\sum _{i=1}^n{X}_i]=\sum _{n=0}^{\infty }P\left(N=n\right)nE\left[X\right]=E\left[X\right]E\left[N\right]=\overline{X}·\overline{n}$$

(4)

The resulting value from this calculation is 18,297,364 IRR for the base year corresponding to March 2023–March 2024.

4.4. Nash Equilibrium in Repeated Games with Folk Strategy

To assess the long-term stability of optimal behaviors, the tripartite interaction between the insurer, the insured, and the employer is modeled as a repeated game. It is assumed that:

• The game repeats annually;
• In the case of defection by any player (e.g., unjustified reduction of insurance coverage or filing fraudulent claims), the other party applies a permanent punishment strategy based on the Folk Theorem;
• The players are rational and prefer long-term benefits over short-term gains.

For each risk group, the adjusted premium, the Net Present Value (NPV) of long-term cooperation, and the short-term payoff from defection were calculated. The results are summarized in

Table 3. Evaluation of cooperative stability in repeated game across risk groups.

Category	Adjusted Premium (IRR)	Long-Term Cooperation Value	Initial Defection Value	Result
Category 1 (No Claims)	15,552,759	−186,633,113	−200,630,596	Sustainable Cooperation
Category 2 (Low Risk)	17,382,496	−208,589,950	−224,234,196	Sustainable Cooperation
Category 3 (Medium Risk)	18,297,364	−219,568,368	−236,035,996	Sustainable Cooperation
Category 4 (High Risk)	23,786,573	−285,438,878	−306,846,794	Sustainable Cooperation

Source: Calculations based on simulation of repeated Nash strategies.

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Table 3 presents the results of the repeated game simulation for each risk category, showing the adjusted premium determined from the XGBoost risk segmentation, the Net Present Value (NPV) of cooperation over the long term, and the short-term payoff from a one-time defection. The “Long-term Cooperation Value” represents the present value of sustained collaboration, calculated using the discount factor δ as described in the methodology, while the “Initial Defection Value” reflects the immediate gain (or avoided cost) from breaking the agreement in the first period. Across all four categories, the long-term cooperation value is higher (less negative) than the defection value, which indicates that cooperative behavior is economically rational for all players when δ is sufficiently high. This outcome is consistent with the Folk Theorem, which predicts that in infinitely repeated games with high discount factors, cooperation can be sustained as an equilibrium strategy. The final column confirms that for every group—whether low-risk or high-risk—sustainable cooperation is the dominant strategy.

Numerically, even for the high-risk group (Category 4), the gain from cooperation outweighs the short-term temptation to defect, ensuring that the equilibrium remains stable.

4.5. Model Conclusion

The integration of machine learning prediction models with behavioral frameworks from game theory demonstrates that:

• Optimal and data-driven adjusted premium rates can be derived;
• Long-term cooperation remains stable from both economic and behavioral perspectives across all risk groups;
• The combination of XGBoost + Repeated Games + Folk Strategy results in a model that benefits from both predictive accuracy and behavioral stability at the policy-making level.

4.6. Fundamental Controlling Mechanism of the Model

The core controlling mechanism of the proposed framework operates through an iterative feedback loop between risk prediction and strategic interaction. First, the XGBoost algorithm segments insured individuals into risk-based clusters, producing dynamic premium adjustments. These outputs are then fed into the repeated game model, where strategic responses of the insurer, employer, and insured are evaluated under the Folk Theorem conditions. The resulting equilibrium strategies inform subsequent pricing updates, creating a continuous cycle of prediction–interaction–adjustment. This closed-loop structure ensures that premiums remain both actuarially fair and behaviorally sustainable over time.

5. Results and Discussion

5.1. Claim Prediction Model Performance

In this study, the XGBoost algorithm was employed as the primary machine learning model to predict medical insurance claims. Utilizing demographic features, medical history, and past claim data, the model demonstrated satisfactory performance in forecasting insurance risk.

To evaluate the effectiveness of the proposed XGBoost model in predicting insurance claims, it was compared against several commonly used methods, including Linear Regression, Decision Tree, Random Forest, and Support Vector Machine (SVM).

Figure 2 compares evaluation metrics—R-squared (R²), Mean Absolute Error (MAE), and Root Mean Square Error (RMSE)—across the mentioned machine learning methods.

As shown in Figure 2, the XGBoost model outperforms all other models across the three evaluation metrics:

• Coefficient of Determination (R²): The XGBoost model achieves the highest R² value (≈0.787), indicating superior ability to explain the variance in medical claim costs. While models like Linear Regression and Random Forest also show reasonable performance, they fall short compared to XGBoost. The SVM model demonstrates the weakest performance in this metric.
• Mean Absolute Error (MAE) and Root Mean Square Error (RMSE): XGBoost achieves the lowest prediction error across both MAE and RMSE, reflecting its strong capability in accurately estimating actual claim amounts. In contrast, the high error rates of the SVM model on both metrics highlight its inefficiency when dealing with nonlinear and imbalanced datasets.

These results underscore XGBoost’s strong capability in capturing complex, nonlinear patterns present in real-world health insurance data. Its boosted tree-based structure enables it to effectively manage imbalanced datasets and intricate variable interactions, resulting in superior predictive performance.

Table 4. Performance metrics of the XGBoost model (R², MAE, RMSE).

Evaluation Metric	XGBoost Algorithm
R2	0.787
RMSE	16,352,672
MAE	1,080,986

Source: Model output generated by authors using internal dataset (2023).

below presents the model performance metrics based on common evaluation criteria:

The scatter plot between actual and predicted medical claims (Figure 3) indicates a high degree of alignment, particularly in the low and medium claim ranges. The R² value shows that approximately 79% of the variance in claim amounts is explained by the model, which is a reasonable and acceptable figure given the diversity in medical and insurance datasets.

The RMSE and MAE values, especially when compared to the broad range of claim amounts and premium levels, also fall within acceptable thresholds. Overall, these results suggest that the model demonstrates a strong degree of accuracy in predicting insured individuals’ claims.

To further evaluate the performance of the XGBoost model in predicting healthcare costs, a scatter plot is drawn comparing actual claim amounts to those predicted by the model. This visualization effectively shows how well the model captures the structure of real-world data.

Each point in the plot represents an individual insured person, with the horizontal axis showing the actual claim amount and the vertical axis representing the predicted claim.

If the points are densely clustered around the 45-degree diagonal line, this indicates high model accuracy and strong agreement between predictions and actual outcomes.

As shown in Figure 3, the distribution of points around the diagonal line indicates the acceptable performance of the XGBoost model in predicting medical claim costs. This high level of alignment is particularly important in high-risk groups, where healthcare expenses are significantly higher. The model’s coefficient of determination (R² ≈ 0.78) supports this finding, demonstrating that the model can explain approximately 78% of the variance in actual claim amounts. This degree of accuracy is both acceptable and highly valuable, especially given the inherent uncertainty and noise in health insurance pricing.

Overall, the figure provides a visual representation of the model’s accuracy, and when considered alongside other statistical indicators such as RMSE and MAE, offers strong evidence supporting the validity of using XGBoost for data-driven insurance pricing.

5.2. Analysis of Repeated Game with Long-Term Cooperation Strategy (Folk Theorem)

Continuing the strategic modeling, a Nash equilibrium framework was developed in the form of a repeated game between the three main stakeholders—the insured individual, the insurance company, and the employer. This game is played over a long-term horizon, based on annually renewed insurance contracts.

Using Folk Theorem equilibrium strategies, the dynamics of interactive behavior over time were modeled. The core idea of this strategy is that if each player is aware that any defection (e.g., false claims or reduction in coverage) will result in punitive actions by the others in the future (such as increased premiums or contract termination), then cooperative behavior becomes more beneficial in the long run.

Basis of the Calculations:

• Profit/loss for each player is expressed in monetary units (IRR), comparing continued cooperation with early defection.
• A reasonable discount rate is applied over time to calculate the present value in the repeated game.
• The decision-making algorithm combines data from the XGBoost model with dynamic game scenarios.

To visualize this, a three-dimensional surface chart (or contour plot) is used, illustrating the benefits of cooperation for each player (insurer, insured, employer) across different points in this two-dimensional space. This chart serves as a visual tool to highlight the conditions under which the repeated Nash equilibrium remains stable, and cooperative strategies are both rational and economically justified for all parties involved. Figure 4 presents the predictive performance of the XGBoost model, where estimated claim values are plotted against their actual counterparts to assess model accuracy. The three payoff surfaces in Figure 4 demonstrate how each stakeholder’s outcome varies with discount factor and risk level. The color variations in the insurer and employer plots highlight nonlinear changes in payoff, while the insured’s surface remains uniform.

In this chart, the relative benefits of cooperation between the three main actors (insurer, insured, and employer) are illustrated against two key parameters: the discount rate (δ) and the behavioral risk level of the insured.

As δ increases (indicating a stronger preference for long-term cooperation), the stability of the Nash equilibrium is enhanced, and cooperation becomes economically viable for all actors, even at higher levels of risk.

These results align with the theoretical foundations of the Folk Theorem in game theory (Fudenberg & Maskin, 1986; Mailath & Samuelson, 2006), which states that when a game is repeated over time and future payoffs are valued, cooperation becomes a stable equilibrium [33,34].

5.3. Feature Importance Analysis Using SHAP Plots

To explain the decision-making logic of the XGBoost model and enhance transparency in the learning process, the SHAP (SHapley Additive exPlanations) tool was used to analyze the importance of input features.

This method, grounded in game theory, numerically and visually represents the contribution of each feature to the model’s output.

Given the main objective of the study—which is to design insurance premiums tailored to individual risk levels—SHAP plots were generated and presented separately for the four risk categories, as shown in Figure 5:

The results are summarized in

Table 5. Comparative SHAP analysis of key features across risk groups using the XGBoost algorithm.

Feature	Category 1: No Claims	Category 2: Low Risk	Category 3: Medium Risk	Category 4: High Risk
Number of Visits	Almost no impact—low frequency is common	Most influential variable—more visits increase risk	Strong impact—risk increases with more visits	Very strong effect—risk rises with a high number of visits
Specific disease	Almost irrelevant—patients are generally healthy	Mild effect—most individuals do not have a Specific disease	Negligible—the model does not give it much weight	Clear impact—presence of a Specific disease raises the risk significantly
Age	Minor effect—model not highly sensitive	Moderate effect—older age slightly increases risk	Significant effect—notable variation in both directions	Relatively strong—higher age is associated with increased risk
Gender	No noticeable effect	Very slight impact	Minor role in model decision-making	Low but visible influence

Source: SHAP output generated by authors using the trained XGBoost model on internal insurance dataset (2023).

, comparing the impact of key variables on the XGBoost model output across different risk categories.

As shown in Table 5, the number of medical visits is the most influential feature across all risk groups, with a markedly stronger effect in higher-risk individuals. Features such as age and specific disease also gain more importance in higher-risk groups. On the other hand, gender plays a relatively minor role in premium estimation across all segments.

These findings indicate that the model is not only capable of distinguishing between various risk levels but also aligns well with clinical and human reasoning—frequent visits and specific disease conditions are well-known contributors to increased healthcare costs.

6. Conclusions and Future Work

This study aimed to develop a novel framework for determining health insurance premiums by proposing a hybrid model that combines advanced machine learning algorithms (XGBoost) with repeated game theory strategies focused on long-term interactions (based on the Folk Theorem). The findings demonstrated that this approach not only achieved high predictive accuracy for medical claims (R² ≈ 0.787) from a technical perspective but also led to a stable Nash equilibrium in the tripartite interaction between the insurer, the insured, and the employer from a behavioral standpoint. Moreover, as shown in Section 5, the integration of XGBoost-based risk stratification with repeated game dynamics directly influenced premium adjustments, resulting in quantifiable reductions for low-risk groups and fair increases for high-risk individuals. This direct alignment between model outputs and stakeholder incentives, grounded in the empirical results, reinforces the robustness of the proposed framework.

The analysis of repeated games revealed that across all risk clusters, the present value of long-term cooperation outweighed the short-term benefits of opportunistic behaviors such as fraud or contract termination. This finding theoretically legitimizes reward or penalty mechanisms based on past behavior and underscores, from a policy perspective, the importance of adaptive and learning-based mechanisms in maintaining the behavioral and financial sustainability of the health insurance ecosystem.

Furthermore, incorporating risk-adjusted pricing into the model achieved actuarial fairness by aligning premiums with actual risk levels. Simultaneously, this provided an incentive structure to promote health-conscious behavior among the insured. The role of employers as facilitators and motivators was also found to be critical in enabling financial balance and data-driven collaborative decision-making.

In summary, the proposed framework represents an innovative step toward smart premium architecture in health insurance. The conceptual and technical flexibility of this model allows for its extension to other branches of insurance, such as life, accident, and liability insurance. In a broader perspective, this approach could serve as a foundation for developing dynamic pricing systems based on big data, continuous learning, and predictive behavioral analysis in health insurance policymaking. Future research directions will include the integration and benchmarking of more cutting-edge regression methods, such as artificial neural networks (ANNs), backpropagation (BP) models, probabilistic neural networks (PNNs), random forests (RFs), temporal convolutional networks (TCNs), extreme learning machines (ELMs), recurrent neural networks (RNNs), as well as advanced architectures like Transformer-based regression models and Kolmogorov–Arnold Networks (KANs). Incorporating these methods will enable a more comprehensive performance comparison and may further enhance the predictive and strategic robustness of the proposed framework.