The National Health Commission and the other relevant departments in China have started to test the diagnosis-related groups (DRGs) system in 30 pilot locations since 2019. DRGs refer to a price standardization mechanism based on patients' classification. Patients can be classified based on their demographic and medical information, such as age, gender, length in hospital, types of diseases, the severity of diseases and comorbidity, complications, the necessity of surgery, etc (1). DRGs aim to categorize patients who have similar clinical features and medical resource consumption into the same groups, which can standardize prospective payment to hospitals or other medical institutions (2).

Currently there are various issues that exist in medical institutions around the world as a result of the price-setting in DRGs, which include the fact that medical institutions tend to choose patients, the payment standard for complex cases cannot reasonably compensate the cost, and medical service providers increase the amount of outpatient services in order to reduce hospitalization expenses (3). High price-setting in DRGs leads to the difficulty of the government controlling medical expenses, while low price-setting is not able to fairly compensate the costs in medical institutions (1), and even leads to hospitals refusing to accept patients in critical conditions. This conflict in price-setting can be understood as a problem of adverse selection (4). Therefore, it is imperative to explore an optimal pricing system that helps to resolve the problem of adverse selection.

Many scholars apply the game theory to the pricing of medical services, which solves the conflict of interests of the stakeholders in the pricing of medical services (5–10). For example, Weingarten (11) applies a generalized game theory model to study the price-setting strategy of competitive medical institutions. Westhoff et al. (5) and Agee et al. (8) apply game theory to study cooperation and competition in disease treatment payment and compensation. Nagurney (12) focus on the supply chain of blood products in an asymmetric information environment. Their study determines a generalized Nash equilibrium model by adopting sharing constraints. This model discusses the price-setting problem in the blood products supply chain, which finds that participants nearer the termination of the supply chain will have better access to information and stronger bargaining power. However, most of these studies focus on the linkages between price makers and medical institutions' behavior, which ignore the impact on medical buyers' behavior and patient preferences with regard to medical service quality.

The present study thus aims to address the above issues by providing some improvements in the game theoretical model assumptions and the model design. First, the quality of medical service is introduced as a variable in a medical service pricing game model. Second, in China, medical institutions are monopolistic in the provision of medical services, and they are in a dominant position in the negotiation of price and quality. Patients as individuals, their bargaining power is scattered. Moreover, the lack of professional knowledge and complete information also weaken patients' bargaining power. Therefore, it is necessary for the government, as the co-payer of medical services, to balance the power of the supplier and the purchasers.

We design a DRG pricing and compensation model as a composite model of two Stackelberg games and bargaining games. In the first round of the game, with the alliance of government and patients as the leader of the game and the medical institutions as the game follower, we get an equilibrium solution about the price and quality of medical services. In the second round of the game, with the medical institutions as the leader of the game and the alliance of government and patients as the follower of the game, we get another equilibrium solution about the price and quality of medical services. In the third round of the game, the two equilibrium solutions obtained above are used as the initial solution of the bargaining game, and the government and patients share the medical expenses.

In the early days, scholars generally considered the competitiveness of the medical service market and established a game pricing model. The Shleifer pricing mechanism is a type of yardstick regulation developed by Shleifer (13). They show that the specific Nash equilibrium is the point where medical institutions achieve optimal marginal cost. Ellis and McGuire (14) explore an optimal neutral risk situation for medical insurance paying to medical institutions. Allen and Gertler (15) use game theory to study the influence of price-setting strategy on patients' satisfaction with services. Weingarten (11) applies a generalized game theory model to study the price-setting strategy of competitive medical institutions. Westhoff et al. (5) and Agee et al. (8) apply game theory to study cooperation and competition in disease treatment payment and compensation. Robinson (16) adopt a Stackelberg game theory model in discussing the relationship between cost transfer and payment methods for medical institutions. They reveal a direct connection between the cost-shifting and medical service payment policy, which shows paying by classification of diseases is better than paying by medical service items.

Many other scholars develop game models on price-setting and negotiation by multiple participants in the medical services. Most of these studies are based on information asymmetry between participants (17, 18). Yaesoubi and Roberts (19) develop a game model that considers two types of constraints. The needed constraint is the bargaining powers of medical service purchasers, and the soft constraint is the cost reimbursement for service providers. These two types of constraints help the model better determine the pricing in an information asymmetric environment. De (20) adopts game theory to examine the impact of information transactions on the relationship between medical institutions and patients, and finds that there will be a Nash equilibrium when patients can restrict medical suppliers' induced demand.

In recent years, scholars have focused on the influence of various factors, such as government intervention (21, 22), medical service quality (17, 23–25), and payment mode of medical services (26, 27), on the results of pricing games (2, 28–30). Behzad and Jacobson (31) investigate the impact of government price intervention and duopoly competition on medical price-setting. Their study applies the asymmetric Bertrand-Edgeworth model to study this impact within a duopoly market. A similar study developed by das Chagas Moura et al. (32), who apply queue theory to improve the Stackelberg game model, focuses on the problems of medical service pricing and medical quality. Koenecke (33) analyses the non-extreme equilibrium between two different payment methods in medical insurance and medical institutions by applying a Stackelberg game model. These two payment methods include a charge on a per-service basis and a per-patient basis. Assuming medical insurance can offer non-linear bonuses, the analysis shows that charges on a per-patient basis can better contribute to sharing medical service information between medical institutions and patients.

The above section presents the bargaining game model on pricing and government compensation between the government, patients, and medical institutions. In the first stage, the alliance formed by the government and patients leads the price as the price maker. The alliance will determine a pricing strategy bases on social welfare maximization. Medical institutions as price takers will make a service quality strategy in response to the pricing strategy from the alliance. This will form a Stackelberg equilibrium solution between prices of disease treatment and medical service quality.

In the second stage, medical institutions will be the price maker in the game. Based on profit maximization, medical institutions will create an optimal service strategy for providing the highest level of medical quality. While based on the social welfare maximization, the alliance will be the price taker, to make an optimal disease treatment price in response to the optimal service strategy made by the medical institutions. This will create another Stackelberg equilibrium solution between prices of disease treatment and medical service quality.

Taking these two equilibrium solutions as the initial solution for the bargaining game between the alliance and medical institutions, and based on the bargaining power of the alliance and medical institutions, we can achieve an equilibrium price of Rubenstein's bargaining game through developing Rubenstein's game. In the final stage, based on the minimization of deadweight social loss, the government and patients will share the payment of medical costs. To analyze PED on diseases, the coefficient of patients' preferences on service quality, the coefficient of patients' absolute risk aversion, and the relationship between disease pricing and government compensation, this study provides several specific values to simulate a series of impacts on the equilibrium price. Firstly, the study analyzes the PED on the equilibrium price. It assumes that the coefficient of the demand function a = 0.3, the coefficient of disease treatment quality cost γ = 10, and the coefficient of patients' preferences on treatment quality c = 0.6. Figure 1 shows the relationship between the equilibrium price and disease PED.

The relationship presented in Figure 1 is in a scenario where the optimal pricing strategy made by the alliance as a price maker will call for medical institutions to respond with their service quality strategy. It is a positive correlation between the treatment equilibrium price and disease treatment PED, i.e., the higher the PED in the treatment of the disease, the higher the price of the treatment. This is the opposite of Ramsey pricing. Medical services can be categorized as basic medical services and special medical services. The PED of basic medical services is low, and it is largely for non-profit-making purposes. In this regard, the pricing for this type of medication will be low. While special medical services have more significant commodity attributes and their high PED value leads the pricing to be higher than that of basic medical services.

To analyze the relationship between the treatment pricing and the coefficient of patients' service quality preference, it is necessary to set the elasticity of treatment demand b = 0.3, the parameter o‘f the demand function a = 0.3, and the coefficient of treatment quality cost γ = 10.

Figure 2 shows a scenario that reflects the relationship between the equilibrium price of diseases and the coefficient of patients' quality preferences. In this scenario, medical institutions act as price makers. Based on the objective of maximizing profit, their priority is to optimize disease quality control decisions. Furthermore, the alliance of government and patients as the price taker will make the optimal price decision according to the goal of maximizing social welfare. In this model, when the coefficient of patients' quality preferences can be controlled in a certain range, the relationship between disease equilibrium price and patients' quality preference can be positive, i.e., the higher the coefficient of patients' quality preference, the more disease equilibrium price increases, and vice versa.

The curve WF₁ in Figure 3 shows that the total social welfare increases first and then decreases with the increase of social medical insurance co-payment ratio, where the patients' absolute risk aversion coefficient is R≥σ2b+12aδδ2a. The curve WF₂ presents another scenario in which the patients' absolute risk aversion coefficient is R<σ2b+12aδδ2aand the total social welfare increases with the increase of social insurance co-payment ratio.

The above scenarios reflect the patient's risk aversion attitude toward health threats. When patients have a higher absolute risk aversion coefficient, they will experience more medical moral hazard, and when the co-payment rate of social insurance increases to a certain extent, it will lead to a loss on total social welfare. Opposite to this, when patients' absolute risk aversion coefficient reduces, their medical moral hazard decreases. This can lead to a continuous increase in the total social welfare while the social insurance co-payment ratio increases.

In the process of DRGs operation in many countries, especially in developing countries, such as India and China, unreasonable pricing and government compensation of medical services under the DRGs operation environment will lead to two major problems. On the one hand, underpricing will lead to medical institutions shirking the responsibility of critically ill patients and decomposing the DRGs category; on the other hand, overpricing will lead to a sharp increase in or even out of control medical expenses. To balance the interests of the government, patients, and medical institutions, the improved bargaining game model based on the Stackelberg game is introduced in order to balance this system and improve the total utility level of society.

The game pricing model developed in this study considers the relationship among the three roles including the government, medical institutions, and patients. The interests and behaviors of these roles have been studied through the models which present a suggestion on price negotiation and setting to the government and medical institutions, and the conducted result can guide medical institutions with a better price-setting strategy. The model also classifies price-setting according to the price elasticity of demand for medical services. To be more realistic, the study has considered price-setting under the impact of patients' preferences for medical service quality. As a result, we find that the absolute risk aversion coefficient of patients affects the relationship between the co-payment rate of medical expenses and total social welfare. In addition, with more investment in public health education, the government can control the moral risk of over-treatment of patients during the operation of DRGs by influencing the absolute risk aversion coefficient of patients and limiting the absolute risk aversion coefficient of patients in a certain range. The future of this study is to generalize these models to DRGs pricing systems with the effects of competition of medical institutions, including considering differences in the compensation of DRGs systems between infirmaries and hospitals, in order to promote the goal of hierarchical diagnosis and treatment.

The original contributions presented in the study are included in the article/supplementary materials, further inquiries can be directed to the corresponding author/s.

JD conceiving the algorithm, designing the model, and writing—original draft. ZL revising and editing. FJ writing and editing. All authors contributed to the article and approved the submitted version.

This work was partially supported by the Innovation Strategy Research Project of Fujian Province under grant 2020R0090.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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