An Evolutionary Game Model of the Supply Decisions between GNPOs and Hospitals during a Public Health Emergency

Abstract

The distribution of medical supplies tied to the government-owned nonprofit organizations (GNPOs) is crucial to the sustainable and high-quality development of emergency response to public health emergencies. This paper constructs a two-sided GNPO–hospital game model in a Chinese context, and explores the strategies and influencing factors of medical supply distribution in public health emergencies based on evolutionary game theory. The results show that: (1) GNPOs, as the distributor of medical supplies, should choose strategies that balance efficiency and equity as much as possible. (2) Hospitals, as the recipient of medical supplies, should actively choose strategies that maximize the total benefit to society and strengthen trust in GNPOs. Meanwhile, hospital managers need to pay attention to reducing the impact of communication and coordination costs and strive for the reduction of conflicts between different values. (3) The government should strengthen supervision to avoid conflicts between medical distributors and receivers during a public health emergency and ensure the rescue efficiency. This study provides some reference for the sustainable development of emergency relief in public health emergencies.

1. Introduction

The outbreak of Corona Virus Disease 2019 (COVID-19) that began in late 2019 continues to threaten the lives of many people today. It is a public health emergency with worldwide repercussions. We define the infectious disease epidemics, mass illnesses of unknown origin, food and occupational poisoning with significant impacts, and other events that cause serious damage to public life and health as public health emergencies [1]. Public health emergencies can cause serious casualties, economic losses, and other social hazards and require emergency management measures to respond to them.

Since the outbreak of COVID-19 at the end of 2019, hospitals have faced challenges such as a shortage of medical supplies and insufficient human resources in terms of patient screening, referral and treatment, medical supplies, and security work [2]. Hospitals have experienced varying degrees of medical supply shortage. The shortage is greatest in areas with a high incidence of the epidemic. As the first line of epidemic prevention and control, if infections within the hospital occur due to an acute shortage of medical supplies, the consequences may be tragic. Moreover, the increasing demand for medical supplies from patients and the public also increase the pressure of the gap in hospitals due to the unclear spread pattern of the epidemic, the asymmetry of supply and demand information, the panic psychology of the public, and the unknown risk of mutation [3]. Therefore, the question of how to solve the scarcity of medical supplies in public health emergencies and the large demand for medical supplies in hospitals has become a major concern for the public, scholars, and government departments.

When there is a sudden outbreak of COVID-19 and a sharp increase in patients, the demand for medical supplies in hospitals, which are the main battlefield for treating patients, far exceeds the daily level. The supply of medical supplies in hospitals is seriously affected. With a limited production capacity in the short term, accepting social donations can alleviate the shortage of medical supplies. In addition, government departments should play a leading role in the resource allocation process of medical supplies in public health emergencies, which can ensure that medical supplies reach the most urgent places immediately and that the supply of medical supplies can be as continuous and sufficient as possible to achieve a dynamic equilibrium point between the total supply and total demand. In the public health emergency of COVID-19, the Ministry of Civil Affairs of China designated the Red Cross Society of China Hubei Branch, the Hubei Charity Federation, the Red Cross Society of China Wuhan Branch, and other nonprofit organizations with a government background to receive funds collected for the prevention and control of the epidemic in Wuhan, Hubei Province, while the abovementioned government-owned nonprofit organizations have the right to draw up the distribution plans for the non-directed donations of supplies. This gives them the sole right to receive and distribute emergency donated resources, which also helps to alleviate the conflict between the shortage of medical supplies and the huge demand for supplies in hospitals.

In China, a government-owned nonprofit organization (GNPO) is a very important part of nonprofit organization (NPO), which is an organization with Chinese characteristics that is founded by the government through a top-down approach and relies heavily on the resources provided by the government. Therefore, the positioning, functions, and internal structure of GNPOs are similar to government departments [4]. They are an extended form of state organizations, which are initiated, funded, and managed by the government agencies that fund them [5]. Moreover, they are also non-governmental organizations that do not aim at profit and engage in certain social management functions, with funding sources that partially rely on government appropriations. Therefore, they are also called quasi-governmental organizations and nonprofit organizations of quasi-governmental nature [6]. Meanwhile, they are governmental background NPOs, and are social organizations with legal authorization and certain social management functions [7]. GNPOs have the characteristics of non-government, nonprofit, quasi-governmental or quasi-profit “intermediate organizations”, with two social roles of buffer and innovation [8]. The characteristic of legitimacy allows them to respond quickly to emergencies. They can mobilize and distribute a wide range of social resources according to simple procedures. In public health emergencies, GNPOs also play an important rescue role under the leadership of the government by distributing the medical supplies mobilized from the society [9], and the rescue process of GNPOs is characterized by continuous and intensive involvement [10].

GNPOs have a natural advantage in the process of resource mobilization and distribution in emergencies, especially public health emergencies, due to their governmental background, and are more efficient in resource mobilization, integration, and distribution [4]. Hence, the distribution of medical supplies linked by GNPOs has become a crucial link in public health emergencies. The distribution of emergency resources such as medical supplies can directly determine the efficiency and effectiveness of emergency management, which is important for saving lives and reducing property damage after the outbreak of a public health emergency. However, during the outbreak of COVID-19, the GNPOs, which were the distributors of medical supplies, did not distribute medical supplies as well as they should have, and a lot of medical masks and other medical supplies donated from home and abroad were reserved in the temporary storage of the Red Cross Society while the frontline healthcare staffs were extremely short of medical supplies. The GNPOs’ strategy for distributing medical supplies during the public health emergency, especially for hospitals on the front lines of the emergency, has been widely questioned by the community.

In the public health emergencies, the conflict between supply and demand of medical supplies is mainly reflected in the interaction and game between GNPOs with limited medical supplies and medical institutions with a huge demand for supplies. Scholars have conducted a lot of research in related fields, mainly focusing on realizing the distribution of emergency resources by different emergency subjects, constructing emergency decision-making systems with information technology methods, and adopting different quantitative research methods to optimize the decision-making of emergency solutions. However, research work on the distribution of medical supplies between GNPOs with limited medical supplies and hospitals with a huge demand for supplies are relatively rare.

At the same time, in the process of the emergency management of emergencies, emergency management departments must consider many factors before making initial decisions, and, in the process of making decisions, they must also make timely adjustments based on feedback from other groups on the decisions made by themselves. There has been a lot of research on the problem of emergency resource distribution, which can be mainly divided into two categories: constructing stochastic optimization models or game models to solve the problem of emergency resource distribution. The stochastic optimization model is employed to solve the overall planning or dynamic collaborative planning problems made under the information uncertainty following a single objective or multiple objectives, rather than seeking optimal strategies for players with different concerns and objectives, as in the game model. However, the process of distributing medical supplies between GNPOs and hospitals in public health emergencies mainly involves the cooperation and conflict and interaction and game between two players, with the aim of seeking the optimal strategy to maximize the interests of both players. Therefore, the process can be regarded as a game process. Classical game theory relies on the assumption of entire rationality. In addition, behavioral game theory is the empirical study based on behavioral experiments, and the analysis results are built on the experimental results. However, the public health emergencies are characterized by chance and sudden change, incomplete information, the complexity of emergency policies, and differences in the motivation of interests and decision-making purposes among game players. Hence, the emergency management departments and affected groups show the characteristics of the bounded rationality. Meanwhile, they cannot obtain the optimal strategy of the game through iterative reasoning, but instead seek the optimal strategy through the dynamic process of continuous imitation and learning in multiple games. The theory of evolutionary games based on bounded rationality transforms the model of the behavior of game subjects into a progressive evolutionary process of adaptive adjustment of subjects with the ability to learn by imitation under the combined effect of various influences [11]. The game subject learns by imitation, rather than by the inductive projection of the optimization strategy by an entirely rational subject who does not conform to the actual situation, which better overcomes the limitations and shortcomings of classical game theory [12]. This is also more in line with the evolutionary characteristics of progressive learning in GNPOs and hospitals regarding medical supply distribution behavior. In recent years, the evolutionary game method has become an effective research tool for analyzing the complex behavioral interactions between biological populations and social human actors. However, research applying the evolutionary game method to GNPOs and hospitals for the distribution of medical supplies are rare, and there has been little research performed on the game mechanisms between them.

Therefore, this paper adopts the evolutionary game model to deal with the subject game problem in public health emergencies based on the relevant studies. In this paper, we first construct a mathematical model to analyze the distribution of medical supplies between GNPOs and hospitals in public health emergencies. Then, we discuss the evolutionary stabilization strategy (ESS) and analyze the evolutionary path stimulated by different factors. We expect that the results of our study will provide a corresponding reference and help with the sustainable supply of medical supplies during public health emergencies.

This research work makes contributions both theoretically and practically. The theoretical contribution has the following two main parts: The first one is that we try to improve the sustainable emergency response capacity of GNPOs and hospitals during public health emergencies from the perspective of evolutionary game theory. Secondly, the study of the game behavior between GNPOs and hospitals fills a gap in this field. This paper analyzes the game mechanism between GNPOs and hospitals for the distribution of medical supplies in public health emergencies, and the results fill a gap in this field. The practical contribution has the following three main parts: Firstly, immediate actions to improve the response are quite necessary in public health emergencies. Secondly, we examine the important factors that could affect the efficiency of the medical supply distribution in the early stage of public health emergencies. Thirdly, although GNPO is a unique organization in a Chinese context, the results could also help other organizations who play the same role as GNPOs achieve the sustainable supply of medical supplies in public health emergencies all over the world.

The rest of the paper is organized as follows: Section 2 presents a literature review in two areas: emergency resource distribution and evolutionary game theory and its applications. In Section 3, an evolutionary game model and a payoff matrix are constructed to study the distribution of medical supplies by GNPOs and hospitals. Section 4 analyzes the ESS of the evolutionary game model based on replicated dynamic equations and Jacobi matrices. Section 5 discusses the numerical simulation to verify the analytical results in Section 4. Finally, Section 6 presents conclusions and future perspectives.

2. Literature Review

The distribution of emergency resources in public health emergencies has currently been a pressing topic of research. Since the outbreak of COVID-19, this issue has become a focus of attention for the public, scholars, and government departments. The literature review in this section mainly consists of two parts: emergency resource distribution and evolutionary game theory and its application.

2.1. Emergency Resource Distribution

As the key link in emergency response, the distribution of emergency resource is an important task to ensure a smooth rescue and safeguard the lives and livelihoods of people in the affected areas. Compared with other ordinary resources, the distribution of emergency resources has specific operational characteristics. Firstly, the distribution of emergency resource has an instant nature [13]. The uncertainty of the emergency resource distribution exists throughout the rescue activities. As the rescue cycle progresses and the rescue information becomes more and more complete, uncertainties are gradually reduced, and the strategy of emergency resource distribution should be adjusted and changed accordingly. Secondly, the distribution of emergency resource should meet the needs of the affected areas as much as possible and achieve the optimization of resource utility. Finally, the distribution of emergency resources needs to achieve the coordination of stakeholders [14]. In the face of different levels of disaster, limited emergency resources are competing among multiple demand parties, which is similar to those in a business environment, and fair resource distribution requires coordination of the interests of all parties at the same time. On the basis of ignoring the cost of disaster relief, better coordination of the interests of all parties should be achieved, in addition to achieving fair distribution [15].

To address the problem of the emergency resource distribution plan, many scholars have attributed the distribution of emergency resources to a stochastic optimization problem under uncertain demand, considering the uncertainty of the environment and the information incompleteness. Accordingly, they have constructed stochastic programming models to reach the goal of an emergency resource distribution. For example, Yuxuan He [16] made time-varying forecasts of medical demands at disaster sites based on a modified susceptibility–exposure–infection–recovery model, and distributed supplies through a linear optimization approach. Some scholars also believe that the dynamic process when distributing emergency resources needs to be divided into multiple stages for research, and that the dynamic distribution of supplies is implemented based on the real-time updated demand at each stage. Yanyan Wang [17] proposed a multi-objective emergency resource distribution model that can balance efficiency and equity, and used a particle swarm algorithm to optimize the model with minimal total distribution cost and total loss of resources. Jiuh-Biing Sheu [18] proposed a dynamic relief demand management model under incomplete information after natural disasters, and the test results of different experimental scenarios showed that the model could meet the prediction of dynamic relief demand and resource distribution under incomplete information conditions. From existing literature, it can be seen that most studies of emergency resource distribution are of the overall planning or dynamic collaborative planning made under information uncertainty following a single objective or multiple objectives.

Since the process of emergency resource distribution involves many stakeholders with different concerns and objectives, some scholars have simulated the competition among multiple affected areas in the process of emergency resource distribution from a game perspective [19,20]. Gupta and Ranganathan [21] developed a noncooperative game model that considered multiple emergency centers as game players and studied the competition between multiple parties to find the optimal resource distribution strategy under the resource maximization objective. Adida [22] constructed a noncooperative game model to study the stocking of supplies in the event of a public emergency in several hospitals under demand uncertainty and showed that the noncooperative game can reduce inventory costs. Amiya K. and Chakravarty [23] developed a hybrid reactive and proactive resource distribution response game model to reduce the cost of emergency response.

There has been a lot of research on the problem of emergency resource distribution, which can be mainly divided into two categories: constructing stochastic optimization models or game models to solve the problem of emergency resource distribution. The research work of constructing stochastic optimization models mainly focus on the overall planning or dynamic cooperative planning for single or multiple objectives under the uncertainty of demand, while the research work of constructing game models mainly focus on the premise of seeking the optimal strategy of the game players by constructing cooperative game models or noncooperative game models, and finally using the actual calculation cases to verify the simulation. However, there are few quantitative studies on the game between emergency resource distributors and receptors for public health emergencies in the Chinese context. Therefore, this paper focuses on the mismatch between the supply and demand of emergency medical supplies for hospitals and GNPOs with Chinese characteristics in public health emergencies and investigates it by constructing a game model.

2.2. Evolutionary Game Theory and Its Application

Game theory provides effective decision-making guidelines for game players. In order to match the predicted outcome of the game with the actual situation, classical game theory requires that the game players should satisfy the assumptions of full rationality and complete information. This requires that each player in the game knows the behavior and characteristics of all other players, and that there is complete information symmetry between the players in the game. However, this cannot always be satisfied in reality. In other words, a game player cannot know all the information of other game players and their motives, and he cannot always make entirely rational decisions calmly. In order to solve this problem, the classical game theory can only be extended by abandoning the assumption of full rationality.

As a result, behavioral game theory and evolutionary game theory, which break away from the assumption of full rationality in classical game theory, have emerged. Behavioral game theory integrates behavioral and experimental economics with standard game theory and aims to provide more consistent explanations and predictions of individual or group actions under various strategic conditions. It uses data and psychological materials as analytical tools, constructs theories on experiments and their results, and designs the whole process from experimental objects and experimental methods. Finally, scholars conduct extended research work by controlling and measuring relevant factors. Behavioral game theory is based on the laws of psychology to weaken the assumption of rationality. The structural relationship between behavioral motivation, cognitive ability, and reasoning process is studied through experimental methods, and the experimental simulation results are used to correct the deviation between theory and reality. Hence, it is concerned with how the participants act in reality rather than theoretical logical reasoning.

The analysis method of evolutionary game provides a great possibility to study people’s continuous institutional innovation at the margin by imitation–learning–adaptation–growth behavior logic in a constant environment [24]. The concept of the evolutionary game was originally derived from Darwin’s term evolution. Initially, evolutionary game was used to analyze the behavior of different species and the competitive behavior between populations by following the outcome of population competition over time, which was called the optimal strategy of the population. Thus, evolutionary game was applied to the field of ecology very early, and an ecologist named Fisher found that in many cases it is possible to explain the evolutionary outcomes of animals or plants using game theoretic approaches based on the assumption of bounded rationality. Price [25] and Smith [26] further extended the concept of evolutionary equilibrium by proposing evolutionary stabilization strategy (ESS) in their articles. Natural selection is replaced by the principle that bounded rational individuals seek to maximize their benefits as a determinant of the probability that an individual will choose a certain strategy. That is the basic idea of evolutionary stabilization strategy. Game players continuously adjust and optimize their strategies according to the magnitude of the benefits they can obtain until all game players retain only one strategy in the process of completing the adjustment and optimization, and this retained strategy is the ESS.

In the last few decades, evolutionary game theory has flourished and been applied to many areas of research. Christoph Adami [27] proposed an agent-based approach to the evolutionary game model that predicts problems that cannot be solved by purely mathematical methods. Keke Sun [28] developed an evolutionary game model for focal and marginal firms based on evolutionary game theory and exponential random graph models to analyze the dynamic evolutionary process of players and changes in evolutionary stabilization strategies, which promoted the diffusion of green innovation in the tourism industry. Yingying Shi [29] proposed an agent-based evolutionary game model to study the adaptation of firms to low-carbon policy subsidies and then explored the impact of policy interventions on the diffusion behavior of low-carbon technologies among firms.

At the same time, evolutionary game theory has become an effective method for analyzing complex interactions among social agents, and it is widely used in social science fields such as the study of public health emergencies. Kazuki Kuga and Jun Tanimoto [30] constructed a two-sided evolutionary game model to study the degree of defense against infectious diseases for healthy and infected individuals with or without vaccination. What is more, they validated the model with a multi-agent numerical. Zhiqi Xu [31] constructed a tripartite evolutionary game model of local government–business–public for the public health emergencies such as COVID-19 to explore the evolutionary stabilization strategies under different conditions and the influence of different factors on players’ decisions. MingChu Li proposed a new model on the impact of vaccination strategies on the transmission of infectious diseases based on evolutionary game theory and fully accounted for subsidies for vaccination failure and vaccination incentives to increase vaccination rates in the model [32]. In addition, some scholars had also used the evolutionary game approach to study the impact of different prevention and control strategies adopted by the government and the public on the spread of an epidemic in the context of a public health emergency [33], the impact of different measures of epidemic prevention and control adopted between countries on the spread of an epidemic [34], and so on.

By combing the relevant literature, it is found that, compared with classical game theory, evolutionary game theory is an effective method to manage interorganizational relationships, especially under the condition of information asymmetry, and it can effectively reflect the complex relationship between stakeholders. In addition, behavioral game theory is the empirical study based on behavioral experiments, and the analysis results are built on the experimental results. However, the public health emergencies are sudden, therefore it is impossible to design experiments in advance and obtain experimental results. Meanwhile, due to the mismatch between supply and demand of medical supplies, information asymmetry between GNPOs and hospitals, and the different starting points of their interests in the context of public health emergencies, the game behavioral decisions of GNPOs and hospitals show the characteristics of bounded rationality. Therefore, it is more difficult for game players to find the optimal strategy in the short term, but they need to find the optimal strategy through continuous imitation, learning, adaptation, and growth in multiple dynamic games. Therefore, in order to analyze the interaction behavior of GNPOs and hospitals in the process of medical supply distribution, and to explore the factors influencing the choice of a certain behavior of GNPOs and hospitals, this paper introduces evolutionary game theory into the study of medical supply distribution between GNPOs and hospitals, and constructs an evolutionary game model between “GNPO-hospital”, exploring how GNPOs can better meet hospitals’ demand for medical supplies in public health emergencies.

3. Model Construction

3.1. Problem Description

From the literature review above, we can find that, after an outbreak, GNPOs and hospitals take different behavioral decisions randomly due to their different identities and different starting points of interests, therefore we need to analyze the interactive behavioral decisions between GNPOs and hospitals in public health emergencies.

When a public health emergency occurs, the demand for medical supplies from all walks of life suddenly increase. At the same time, the public health emergency causes many restrictions on the supply chain of supplies, which results in labor shortages, soaring transportation costs, and other problems. These problems can lead to the failure of medical supply manufacturers’ timely production. At this time social donations become an important way to ensure the supply of supplies in the prevention and control of the epidemic, and the distribution of socially donated medical supplies is mainly the responsibility of GNPOs such as the Red Cross Society. However, the sudden and far greater than daily level of donated supplies poses a great challenge to the GNPO’s ability to allocate resources for scheduling. In addition, patients requiring care by healthcare workers increase exponentially because of the ongoing spread of the outbreak [35]. Therefore, it is likely that, even if the GNPO tries their best to distribute medical supplies fairly and reasonably, it will not be able to meet the hospital’s needs, which will result in two types of behavioral decisions for the GNPO: able to distribute by demand or unable to distribute by demand.

In the public health emergency, hospitals, as the first battlefield in the fight against the disaster, need to meet the increasing demand for the medical supplies due to increasing patients in a timely manner. Because infectious diseases are generally urgent and require timely treatment in order to turn a critical situation into a safe one, at the same time, hospitals must also meet the medical supplies of health care workers in time. Once cross-infections within the hospital occur, the consequences are unimaginable. The situation in which many healthcare workers were infected due to SARS was a very painful lesson in 2003. Many studies have shown that, at the beginning of an epidemic, the demand for medical supplies in hospitals greatly exceeds the daily supply of the community [36,37,38,39]. Hence, some scholars have studied how social media can be used to appeal to people for donations to meet the huge gap in medical supplies for hospitals brought about by the epidemic [40]. In times of public health emergencies, social donations can greatly alleviate the pressure of medical supply shortages in hospitals, and GNPO, as the distributor of medical supplies, is responsible for distributing the donated supplies according to the needs of hospitals. Therefore, hospitals can also make two types of behavioral decisions regarding the GNPO’s distribution of medical supplies: agreeing with the distribution plan or disagreeing with the distribution plan and fighting for more medical supplies. It can be simplified as the strategy of “agree” or “disagree and fight”.

It is worth noting that the hospitals do not know whether the distribution strategy of GNPO is able to distribute by demand or unable to distribute by demand, and the GNPO does not know whether the hospitals will agree with their plan of distribution and any other information, similarly. In the meantime, the interaction between the GNPO and hospital is a dynamic game process, but both the GNPO and the hospital make behavioral decisions that are related to their own interests at the same time. Therefore, they are both bounded rational.

3.2. Model Construction

According to evolutionary game theory, the model constructed in this paper needs to satisfy the following assumptions:

Assumption 1.

Both GNPO and hospital are game subjects that are bounded rational, which is between entirely rational and entirely irrational;

Assumption 2.

The behaviors of GNPO and hospital are mutually influential, and, since the game subjects can imitate and learn autonomously, they can adjust their strategies at any time according to the actual situation when they appear interact behavior in the game;

Assumption 3.

The hospital bears the greater loss in both cases as follows:

(1) 

They disagree with the distribution plan and fight for more medical supplies when the GNPO adopts the strategy of “able to distribute by demand”;

(2) 

They agree with the distribution plan considering the maximization of the social benefits and the overall stability of epidemic prevention and control when the GNPO adopts the strategy of “unable to distribute by demand”;

Assumption 4.

There is an opportunistic behavior of “free-riding” by the GNPO when it adopts the strategy of “unable to distribute by demand”, which brings it additional benefits [41,42];

Assumption 5.

The GNPO’s strategy set is (able to distribute by demand, unable to distribute by demand. The strategy of “able to distribute by demand” means that GNPO can make the distribution plan of supplies meet the emergency needs of the hospital as much as possible, which is based on the needs of medical supplies reported by the hospital and combined with the degree of urgency of the supplies for the hospital with actual research. The strategy of “unable to distribute by demand” means that the GNPO, considering the severe shortage of medical supplies during the epidemic and the level of urgency of the epidemic in the hospital, has developed a distribution plan of medical supplies that does not meet the demand for medical supplies of the hospital in order to optimize the total benefit to society. The hospital’s strategy set is (agree, disagree and fight). The strategy of “agree” means that the hospital chooses to accept all of the GNPO’s distribution plans, even if the GNPO’s distribution plans do not meet the hospital’s demand for medical supplies. Considering the maximization of the social benefits and the overall stability of epidemic prevention and control, hospital chooses to agree to the distribution plan. The strategy of “disagree and fight” means that hospital is unsatisfied with the GNPO’s distribution plan of medical supplies and chooses to fight for more medical supplies or exposes the GNPO’s distribution plan to the society;

Assumption 6.

The game players select the probability parameter of the behavior strategy. We assume that the probability of choosing the strategy of “able to distribute by demand” for GNPO is x, where 0 ≤ x ≤ 1, and the probability of choosing the other strategy is 1 − x. Similarly, the probability of choosing the strategy of “agree” for hospital is y, where 0 ≤ y ≤ 1, and the probability of choosing the other strategy is 1 − y.

In the early stage of public health emergencies, the GNPO and hospital have different identities and interests. At the node of epidemic emergency prevention and control, when GNPO makes efforts to reasonably distribute supplies to hospitals, there are some hospitals that still cannot meet the demand after receiving supplies from the GNPO and will choose to fight for getting more supplies. This will result in the GNPO incurring losses such as social reputation being affected (denoted as M₁). At the same time, due to the fighting, the hospitals will receive additional supplies as compensation from the GNPO for the purpose of calming the dispute, resolving the conflict, and maintaining social stability and unity (denoted as S₁), but the hospitals will be punished by the government at this time (denoted as F₁) [4]. In the process of fighting, it costs the hospitals’ time and effort to coordinate with the GNPO (denoted as C₁), and the GNPO also has to pay for the cost of coordination (denoted as C₂) [43]. This paper denotes the degree of the hospitals’ fighting as β [44].

When the GNPO does not reasonably distribute supplies to hospitals, it will result in a certain additional loss to the hospital (denoted as L₁) due to accepting the GNPO’s distribution plan for the maximization of the social benefits and the overall stability of epidemic prevention and control. Meanwhile, the government will reward the hospital (denoted as J₁) and the government will punish the GNPO (denoted as P₂) [4]. At the same time, the GNPO will gain additional benefits (denoted as B) [41] due to the opportunistic behavior of distributing supplies irrationally. However, GNPO will also be punished by the government for the hospital’s fighting actions (denoted as P₁) [4]. The hospitals’ action of disagreeing and fighting will lead to the exposure of the GNPO’s unreasonable behavior. It will cause the social group to be dissatisfied with the GNPO leading to a greater loss in the GNPO’s credibility and damage to the public’s trust in it (denoted as M₂, M₂ > M₁). At the same time, the hospitals that disagree with the distribution plan and fight for getting more supplies will receive additional supplies compensation (denoted as S₂) from the GNPO (S₁ < S₂). Although the hospitals’ behavior of fighting exposes the GNPO’s unreasonable distribution plan of supplies, and the hospitals’ behavior also causes certain adverse effects in the emergency time of epidemic prevention and control. Hence, the government will also punish the hospital for the overall stability of epidemic prevention and control (denoted as F₂, F₁ > F₂) [45,46].

The related parameters and their descriptions are summarized in

Table 1. Parameters and descriptions.

Item	Parameter	Description
For hospital	L1	Additional losses incurred by the hospital during the pre-epidemic period because government-owned nonprofit organization (GNPO) adopts the strategy of “unable to distribute by demand”;
	D	Additional losses incurred by the hospital in the early stage of epidemic period due to the impact of the sudden epidemic that is indeed difficult to ensure the supply of supplies (L1 > D);
	β	The degree of hospital fighting;
	S1	Benefits from additional GNPO’s supplies generated by hospitals that do not agree the GNPO’s supplies distribution plan and fight when GNPO adopts the strategy of “able to distribute by demand”;
	S2	Benefits from additional GNPO’s supplies generated by hospitals that do not agree the GNPO’s supplies distribution plan and fight when GNPO adopts the strategy of “unable to distribute by demand” (S1 < S2);
	C1	Coordination costs incurred by hospitals that disagree with the distribution plan and fight for more supplies;
	J1	Reward from the government due to the hospital’s agreement with the GNPO’s supplies distribution plan when GNPO adopts the strategy of “unable to distribute by demand”;
	F1	Punishment from the government due to the hospital’s disagreement with the GNPO’s supplies distribution plan and fighting when GNPO adopts the strategy of “able to distribute by demand”;
	F2	Punishment from the government due to the hospital’s disagreement with the GNPO’s supplies distribution plan and fighting when GNP adopts the strategy of “unable to distribute by demand” (F1 > F2);
For GNPO	P1	Punishment from the government due to the hospital’s fighting;
	P2	Punishment from the government because GNPO adopts the strategy of “unable to distribute by demand”;
	M1	Losses such as damage to GNPO’s social reputation due to the hospital’s disagreement with the GNPO’s supplies distribution plan and fighting when GNPO adopts the strategy of “able to distribute by demand”;
	M2	Greater losses such as social discontent and loss of reputation due to the hospital’s disagreement with the GNPO’s supplies distribution plan and fighting when GNPO adopts the strategy of “unable to distribute by demand” (M2 >M1);
	B	Additional benefits to GNPO of opportunistic behavior due to adopting the strategy of “unable to distribute by demand”;
	C2	Coordination costs incurred by GNPO as a result of hospitals’ disagreeing with the distribution plan and fighting for more supplies.

below. They are defined as real numbers and are all non-negative.

Based on the setting of the above descriptions, we can obtain the evolutionary game payoff matrix of the GNPO and the hospital in the process of medical supplies distribution during the shortage of supplies in the pre-epidemic period, as shown in

Table 2. The payoff matrix of the game.

		Hospital
		Agree	Disagree and Fight
GNPO	Able to distribute by demand	0; -D	-P1-βM1-βC2; βS1-D-βF1-βC1
	Unable to distribute by demand	B-P2; J1-L1	B-βM2-βC2-P1-P2; βS2-L1-βF2-βC1

.

4. Replicated Dynamic Differential Equations and Equilibrium Point Analysis

4.1. Replicated Dynamic Differential Equations

According to the assumptions of the game model above and the summary of previous literatures, it is found that the evolutionary game theory often uses the replicative dynamic evolutionary game approach to solve.

For GNPOs, the expected returns that adopt the strategy of “able to distribute by demand” are:

$${\mathrm{E}}_{G1}=y*0+\left(1-y\right)\left(-P_1-\mathsf{\beta }{M}_1-\mathsf{\beta }{C}_2\right)$$

(1)

The expected returns that adopt the strategy of “unable to distribute by demand” are:

$${\mathrm{E}}_{G2}=y\left(B-P_2\right)+\left(1-y\right)\left(B-\mathsf{\beta }{M}_2-\mathsf{\beta }{C}_2-P_1-P_2\right)$$

(2)

Therefore, the average expected returns of GNPOs under the mixed strategies can be denoted by:

$$\begin{array}{cc}\hfill {\mathrm{E}}_G=x{\mathrm{E}}_{G1}+(1& -x){\mathrm{E}}_{G2}\hfill \\ & =x\left(1-y\right)\left(-P_1-\mathsf{\beta }{M}_1-\mathsf{\beta }{C}_2\right)+y\left(1-x\right)\left(B-P_2\right)+\left(1-x\right)\left(1-y\right)(B-\mathsf{\beta }{M}_2-\mathsf{\beta }{C}_2-P_1\\ & -P_2)\hfill \end{array}$$

(3)

Based on the evolutionary game theory, if the expected returns of one player adopting a strategy are higher than the average expected returns of the population, the strategy will spread in the population, and the number of players adopting it will increase. This paper uses the replicator dynamics differential equation to describe the frequency of the specific strategy in population [47]. Thus, the replicator dynamics equation for GNPO can be denoted as:

$$\mathrm{F}\left(\mathrm{x}\right)=\frac{d\left(x\right)}{d\left(t\right)}=\mathrm{x}({\mathrm{E}}_{G1}-{\mathrm{E}}_G)=x\left(1-x\right)\left[\left(\mathsf{\beta }{M}_1-\mathsf{\beta }{M}_2\right)y-\left(B-P_2+\mathsf{\beta }{M}_1-\mathsf{\beta }{M}_2\right)\right]$$

(4)

Similarly, the expected returns for the hospitals that adopt the strategy of “agree” are:

$${\mathrm{E}}_{H1}=x\left(-D\right)+\left(1-x\right)\left(J_1-L_1\right)$$

(5)

The expected returns for the hospitals that adopt the strategy of “disagree and fight” are:

$${\mathrm{E}}_{H2}=x\left(\mathsf{\beta }{S}_1-\mathrm{D}-\mathsf{\beta }{F}_1-\mathsf{\beta }{C}_1\right)+\left(1-x\right)\left(\mathsf{\beta }{S}_2-L_1-\mathsf{\beta }{F}_2-\mathsf{\beta }{C}_1\right)$$

(6)

Therefore, the average expected returns of the hospitals under the mixed strategies can be denoted by:

$$\begin{array}{cc}\hfill {\mathrm{E}}_H=y{\mathrm{E}}_{H1}+\left(1-y\right){\mathrm{E}}_{H2}=xy\left(-D\right)+y& \left(1-x\right)\left(J_1-L_1\right)+x\left(1-y\right)\left(\mathsf{\beta }{S}_1-\mathrm{D}-\mathsf{\beta }{F}_1-\mathsf{\beta }{C}_1\right)+\left(1-x\right)(1-\hfill \\ & y)\left(\mathsf{\beta }{S}_2-L_1-\mathsf{\beta }{F}_2-\mathsf{\beta }{C}_1\right)\hfill \end{array}$$

(7)

The replicator dynamics equation for the hospitals can be denoted as:

$$\mathrm{F}\left(y\right)=\frac{d\left(y\right)}{d\left(t\right)}=\mathrm{y}({\mathrm{E}}_{H1}-{\mathrm{E}}_H)=y\left(1-y\right)\left[\left(\mathsf{\beta }{F}_1-\mathsf{\beta }{F}_2+\mathsf{\beta }{S}_2-\mathsf{\beta }{S}_1-J_1\right)x-\left(\mathsf{\beta }{S}_2-\mathsf{\beta }{F}_2-\mathsf{\beta }{C}_1-J_1\right)\right]$$

(8)

From the stability principle of the differential equation, we know that, when the strategy is in stable equilibrium, we have $F\left(x\right)=0$ and $F\left(y\right)=0$. Therefore, from Equations (4) and (8), we can obtain five equilibria, namely (0, 0), (0, 1), (1, 0), (1, 1) and $\left(x^{*},y^{*}\right)$, where:

$$x^{*}=\frac{\mathsf{\beta }{S}_2-\mathsf{\beta }{F}_2-\mathsf{\beta }{C}_1-J_1}{\mathsf{\beta }{F}_1-\mathsf{\beta }{F}_2+\mathsf{\beta }{S}_2-\mathsf{\beta }{S}_1-J_1},y^{*}=\frac{B-P_2+\mathsf{\beta }{M}_1-\mathsf{\beta }{M}_2}{\mathsf{\beta }{M}_1-\mathsf{\beta }{M}_2}$$

(9)

D. Friedman [24] proposed that the stability of the equilibrium point of the dynamic differential system can be derived from the local stability analysis of the Jacobi matrix. Thus, the Jacobian matrix of the replicator dynamics system is defined by Equation (10).

$$\mathrm{J}=\left[\begin{array}{cc}\frac{\partial F\left(x\right)}{\partial x}& \frac{\partial F\left(x\right)}{\partial y}\\ \frac{\partial F\left(y\right)}{\partial x}& \frac{\partial F\left(y\right)}{\partial y}\end{array}\right]=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$$

(10)

where:

$$a_{11}=\frac{\partial F\left(x\right)}{\partial x}=\left(1-2x\right)\left[\left(\mathsf{\beta }{M}_1-\mathsf{\beta }{M}_2\right)y-\left(B-P_2+\mathsf{\beta }{M}_1-\mathsf{\beta }{M}_2\right)\right]$$

(11)

$$a_{12}=\frac{\partial F\left(x\right)}{\partial y}=x\left(1-x\right)\left(\mathsf{\beta }{M}_1-\mathsf{\beta }{M}_2\right)$$

(12)

$$a_{21}=\frac{\partial F\left(y\right)}{\partial x}=y\left(1-y\right)\left(\mathsf{\beta }{F}_1-\mathsf{\beta }{F}_2+\mathsf{\beta }{S}_2-\mathsf{\beta }{S}_1-J_1\right)$$

(13)

$$a_{22}=\frac{\partial F\left(y\right)}{\partial y}=\left(1-2y\right)\left[\left(\mathsf{\beta }{F}_1-\mathsf{\beta }{F}_2+\mathsf{\beta }{S}_2-\mathsf{\beta }{S}_1-J_1\right)x-\left(\mathsf{\beta }{S}_2-\mathsf{\beta }{F}_2-\mathsf{\beta }{C}_1-J_1\right)\right]$$

(14)

If the following conditions can be satisfied:

(1)

the determinant of the Jacobi matrix: $\det \left(\mathrm{J}\right)=a_{11}{a}_{22}-a_{12}{a}_{21}>0$;

(2)

the trace of Jacobi matrix: $\mathrm{tr}\left(\mathrm{J}\right)=a_{11}+a_{22}<0$.

Then, the equilibrium point of the replicator dynamic equations is locally stable, and we can regard the equilibrium point as the evolutionary stabilization strategy (ESS).

4.2. Equilibrium Point Analysis

The analysis of the model and the Jacobi matrix shows that the five equilibrium points for replicator dynamic system are (0, 0), (0, 1), (1, 0), (1, 1), $\left(x^{*},y^{*}\right)$, where $x^{*}=\frac{\mathsf{\beta }{S}_2-\mathsf{\beta }{F}_2-\mathsf{\beta }{C}_1-J_1}{\mathsf{\beta }{F}_1-\mathsf{\beta }{F}_2+\mathsf{\beta }{S}_2-\mathsf{\beta }{S}_1-J_1},y^{*}=\frac{B-P_2+\mathsf{\beta }{M}_1-\mathsf{\beta }{M}_2}{\mathsf{\beta }{M}_1-\mathsf{\beta }{M}_2}$.

When $0<y^{*}<1$, $0<\mathsf{\beta }<1,$ we can get $0<P_2<B$ and $0<\frac{B-P_2}{M_2-M_1}<\mathsf{\beta }<1$.

Scenario 1: when $0<P_2<B$, $0<\frac{B-P_2}{M_2-M_1}<\mathsf{\beta }<\frac{J_1}{S_2-F_2-C_1}$, $F_1>S_1-C_1$, the results of the local stability analysis of the game equilibrium points of the replicator dynamic system are shown in

Table 3. Local stability analysis of the equilibrium points in the condition of Scenario 1.

Stability Balanced Point	Det (J)	Tr (J)	State
(0, 0)	+	+	Unstable
(1, 0)	−	*	Saddle point
(0, 1)	+	−	ESS
(1, 1)	−	*	Saddle point
(x*,y*)	−	0	Saddle point

Note: “+” means the value is greater than 0, “−” means the value is less than 0, “0” means the value is equal to 0, and “*” means the value is uncertain.

. The system equilibrium point will converge to the point (0, 1) after multiple evolutions. Therefore, the point (0, 1) is the ESS, the points (1, 0), (1, 1), and $\left(x^{*},y^{*}\right)$ are the saddle points, and the point (0, 0) is an unstable node. When the GNPO chooses the strategy of “unable to distribute by demand”, the hospitals choose to agree to this plan. At this point, the reward benefits that the hospitals receive from the government will be greater than the difference between the benefit of additional supplies compensated after the fighting and the cost of being punished by the government during the process of fighting. Therefore, the strategy set (unable to distribute by demand, agree) in the Scenario 1 condition is a stable strategy outcome.

Scenario 2: when $0<P_2<B$, $\frac{J_1}{S_2-F_2-C_1}<\mathsf{\beta }<1$, $0<\frac{B-P_2}{M_2-M_1}<\mathsf{\beta }<1$ and $F_1>S_1-C_1$, the results of the local stability analysis of the game equilibrium points of the replicator dynamic system are shown in

Table 4. Local stability analysis of the equilibrium points in the condition of Scenario 2.

Stability Balanced Point	Det (J)	Tr (J)	State
(0, 0)	−	*	Saddle point
(1, 0)	−	*	Saddle point
(0, 1)	−	*	Saddle point
(1, 1)	−	*	Saddle point
(x*,y*)	+	0	Saddle point

Note: “+” means the value is greater than 0, “−” means the value is less than 0, “0” means the value is equal to 0, and “*” means the value is uncertain.

. The points (0, 0), (0, 1), (1, 0), (1, 1), and $\left(x^{*},y^{*}\right)$ are both saddle points. This indicates that the replicator dynamic system has been in an evolutionary steady state.

Scenario 3: when $0<P_2<B$, $0<\frac{B-P_2}{M_2-M_1}<\mathsf{\beta }<\frac{J_1}{S_2-F_2-C_1}$ and $F_1<S_1-C_1$, the results of the local stability analysis of the game equilibrium points of the replicator dynamic system are shown in

Table 5. Local stability analysis of the equilibrium points in the condition of Scenario 3.

Stability Balanced Point	Det (J)	Tr (J)	State
(0, 0)	+	+	Unstable
(1, 0)	+	−	ESS
(0, 1)	+	−	ESS
(1, 1)	+	+	Unstable
(x*,y*)	−	0	Saddle point

Note: “+” means the value is greater than 0, “-” means the value is less than 0, “0” means the value is equal to 0, and “*” means the value is uncertain.

. It can be seen that the strategy adopted by the two players is determined by the area of the quadrilateral OACD and OBCD from Figure 1. If S_OACD is greater than S_OBCD, the system has a higher probability of adopting the strategy set (able to distribute by demand, disagree and fight). If S_OBCD is greater than S_OACD, the system has a higher probability of adopting the strategy set (unable to distribute by demand, agree). Moreover, if S_OACD is equal to S_OBCD, it means that the system has an equal probability of adopting both strategies sets. The factors influencing the choice of strategy sets by both sides of the game and the specific mechanisms of influence are discussed in detail in the following section.

Scenario 4: when $0<P_2<B$, $\frac{J_1}{S_2-F_2-C_1}<\mathsf{\beta }<1$, $0<\frac{B-P_2}{M_2-M_1}<\mathsf{\beta }<1$ and $F_1<S_1-C_1$, the results of the local stability analysis of the game equilibrium points of the replicator dynamic system are shown in

Table 6. Local stability analysis of the equilibrium points in the condition of Scenario 4.

Stability Balanced Point	Det (J)	Tr (J)	State
(0, 0)	−	*	Saddle point
(1, 0)	+	−	ESS
(0, 1)	−	*	Saddle point
(1, 1)	+	*	Unstable
(x*,y*)	+	0	Saddle point

Note: “+” means the value is greater than 0, “−” means the value is less than 0, “0” means the value is equal to 0, and “*” means the value is uncertain.

. The system equilibrium point will converge to the point (1, 0) after multiple evolutions. Therefore, the point (1, 0) is the ESS, the points (0, 0), (0, 1), and $\left(x^{*},y^{*}\right)$ are the saddle points, and the point (1, 1) is an unstable node. When the GNPO chooses the strategy of “able to distribute by demand”, the hospitals choose the strategy of “disagree and fight”. At this point, the punishment that the hospitals receive from the government for fighting is less than the difference between the additional supplies benefit received from the GNPO after fighting and the cost of fighting. Meanwhile, if the hospitals choose to receive the distribution plan, the incentive or benefit from the government is less than the difference between the benefit of the additional supplies compensated after the fighting and the cost of being punished by the government during the process of fighting. Therefore, the strategy set (able to distribute by demand, disagree and fight) in the Scenario 4 condition is a stable strategy outcome.

5. Numerical Simulation Experiments

5.1. Computational Case

The emergence of the COVID-19 outbreak is a “black swan” event. Since the outbreak of COVID-19 in early December 2019, tens of millions of people have been infected worldwide, and the damage caused is incalculable. On 30 January 2020, the Director-General of the World Health Organization (WHO) officially made a statement at a press conference in Geneva, declaring it a Public Health Emergency of International Concern (PHEIC) [48]. The public health emergency has had an extremely serious impact on the lives of people all over the world. On the evening of 7 July 2021, U.S. time, U.N. Secretary-General Guterres issued a message on COVID-19 that has caused the cumulative death toll to top 4 million worldwide, and warned that the world must face this painful lesson head-on and take positive action [49].

COVID-19 has also had a huge impact on the lives of the Chinese people, causing incalculable damage to the industrial chain, social development, the survival and development of small- and medium-sized enterprises, and so on. For example, the Wuhan government suspended all urban transportation and closed all public entertainment venues. In the meantime, the offline operations of enterprises and institution were also suspended. Wuhan residents had persisted at home for three months in a closed environment [43]. Meanwhile, hospitals, the front line of the frontal fight against the outbreak of COVID-19, had also been hit hard by the outbreak. According to the national medical service situation in January–February 2020 and national patient costs in public hospitals above the second level in January–February 2020 published by the Statistical Information Center of the National Health Commission of the People’s Republic of China, the national hospital treatment volume (excluding data from Hubei Province) shrank by a quarter due to the COVID-19 pandemic, and the number of treatment visits in tertiary hospitals dropped the most, by 26.9%. At the same time, the number of discharges from medical and health institutions nationwide (excluding data from Hubei Province) fell by 17.2% year-on-year, and the number of hospital discharges fell by 16.8% year-on-year. The utilization rate of wards in tertiary hospitals fell the most, by as much as 28.4% [50].

Wuhan, as the most affected place of the COVID-19 in China at that time, was extremely short of medical protective materials. With the influx of a large number of donated supplies into Wuhan, the gap in the demand for protective supplies was slowly decreasing, and at the same time, with the expansion of the scale of hospital beds, there was still a large gap in the demand for protective supplies. The Hubei Provincial Epidemic Prevention and Control Command designated the distribution of socially donated medical supplies was handled by the GNPOs such as the Red Cross Society of China Hubei Branch. However, by tracing the use of 17 donations published on the official website of the Red Cross Society of China Hubei Branch, it was found that Wuhan Renai Hospital, which mainly focuses on infertility treatment and does not receive fever patients during that period, had received 16,000 N95 donated masks distributed by the Red Cross Society of China Hubei Branch. However, Union Hospital, Tongji Medical College of Huazhong University of Science and Technology, one of the 61 fever clinics in Wuhan, had received only 3000 masks. This situation quickly sparked a public debate about whether the Red Cross Society of China Hubei Branch was unreasonable in its distribution of donated supplies. The Red Cross Society of China Hubei Branch was also deeply involved in the public opinion storm.

In the early stage of the epidemic, due to the lack of technical means and staff, GNPOs such as the Red Cross Society of China Hubei Branch could not take up the huge workload of unified coordination and distribution of medical supplies. Thus, it caused a large backlog of medical supplies in warehouses, and, on the other hand, GNPOs did not have timely information of hospitals’ needs, and lacked scientific methods of supplies distribution, which also led to the unreasonable distribution of medical supplies. Meanwhile, it also created the distrust of the public towards the GNPOs. Therefore, this paper uses the distribution of medical supplies to hospitals by GNPOs in Hubei Province during the COVID-19 as a case study in order to provide a more reasonable reference for practical problems while validating and demonstrating the results of a richer theoretical analysis.

In this case, the randomness of the parameter value assignment does not affect the simulation results as the study focuses on the sensitivity of the game players’ behavior to the parameters in the model [44]. The inpatient medical revenue of public tertiary hospitals generally accounts for about 50% of the revenue of tertiary hospitals, coupled with the fact that the volume of outpatient services also declined in the same proportion or even a larger proportion during the same period. Hence, the decline in inpatient medical revenue basically reflects the decline in overall hospital revenue. The inpatient medical revenue of hospitals in February 2020 accounts for approximately 61.7% of February 2019 revenue (approximately USD 14,941,400); thus, we set the initial parameter of D to 500 in this paper. The value of β (the degree of hospital protest) ranges from 0 to 1, i.e., β ∈ [0, 1] [44]. To investigate the effect of the initial behavior of the game players, we use the value greater than 0.5 to indicate high desire to fight and less than 0.5 to indicate low desire to fight. The value of 0 for β means that the hospital fully agrees with the GNPO’s distribution plan, and the value of 1 for β means that the hospital disagrees with the GNPO’s distribution plan and demands that the GNPO should meet its demands. For the two parameters of coordination costs incurred as a result that hospitals disagree with the distribution plan and fighting, we set the initial parameters of C₁ to 90 and C₂ to 80 with reference to Li et al.’s study [51], and we set the initial parameter of S₁ to 150, the initial parameter of S₂ to 300, and the initial parameter of P₂ to 100 in this paper. By using the PYTHON software, we can study the interaction of different behavioral strategies between the GNPO and the hospital intuitively by setting different initial parameters. According to the reality of our country and the hypothetical conditions of the model, the initial values for the relevant parameters are shown in

Table 7. The values of parameters.

Parameter	L1	D	β	C1	C2	S1	S2	J1	F1	F2	P1	P2	M1	M2	B
Value	600	500	0.6	90	80	150	300	50	220	90	300	100	140	300	120

.

5.2. Parameter Sensitivity Analysis

5.2.1. Impact of Initial Parameters

Figure 2 and Figure 3 show the path diagram and phase diagram of the dynamic evolutionary game system of the game players with different initial probabilities, respectively. This paper aims to discuss the impact of the behavioral changes of the game players on the evolutionary results when x and y are assigned different values.

The results of the dynamic evolutionary game system path simulation show that the system is always in a dynamic evolutionary process after sufficient time of evolution; in other words, neither GNPO nor the hospital has reached a dynamic equilibrium point, and the phase diagram supports this result. In the early stage of the epidemic, the whole society was extremely short of medical supplies and the order of rescue was relatively chaotic. Therefore, in the face of the large demand for medical supplies from hospitals, GNPO, as the distributor of medical supplies, has been coordinating the medical demands of different hospitals and trying to meet the demands of designated hospitals as much as possible. However, there are also a lot of situations where GNPOs try their best to distribute medical supplies in a fair and reasonable way, but still cannot meet the demand for medical supplies from medical staff and the increasing number of patients. Both the simulation results and the actual situation show that GNPOs have not yet formed a long-term medical supplies distribution strategy at this time. Meanwhile, in the context of increasing patient pressure and protection pressure on medical and nursing staff, a similar dynamic exploration process exists in hospitals that are the main recipient of medical supplies out of concern for the maximization of the social benefits and the overall stability of epidemic prevention and control.

5.2.2. Impact of β

To represent the effect of the degree of the hospitals’ fighting on the evolutionary results, this paper varies the value of β from 0.1 to 0.9 sequentially, with the initial values of x, y both being 0.5, and the results are shown in Figure 4. From the graph, we can see that, when β is greater than 0.3, the probability of GNPO increases for a short time, but eventually converges to 0. As the value of β gradually increases from 0.1 to 0.9, GNPOs are more and more willing to choose the strategy of “able to distribute by demand” for medical supplies in the early stage, and hospitals eventually tend to choose the strategy of “agree”. However, when the evolution time is long enough, GNPO will eventually evolve to choose the strategy of “unable to distribute by demand” and the model will eventually evolve to a stable state of (unable to distribute by demand, agree).

This indicates that, as the degree of hospital’s fighting increased, the GNPO is willing to choose the strategy of “able to distribute by demand” for medical supplies at the beginning of the outbreak, in order to meet the medical needs of the hospitals for the overall stability of epidemic prevention and control. However, as the situation of epidemic prevention and control continues to stabilize and social concern increases, GNPO will eventually pursue the maximization of the social benefits based on the coordination of the needs and urgency of each hospital and will not be able to meet all the demands. Therefore, it will choose the strategy of “unable to distribute by demand”, and the hospital will choose the strategy of “agree”.

5.2.3. Impact of C₁

The parameter C₁ is another major factor affecting the players’ game behavior. The coordination cost of the hospital due to disagreeing with the GNPO distribution plan and fighting has an important impact on the expected benefits. We simulated the evolutionary trend of GNPO and hospital under different values of C₁. It can be found that the value of C₁ is gradually changed from 9 to 81, with the initial values of x, y both being 0.5, and the results are shown in Figure 5. With the increasing of C₁, the probability of GNPO will eventually converge to 0. In other words, it will gradually tend to choose the strategy of “unable to distribute by demand”, and, when C₁ is larger than 36, the speed of convergence of GNPO to 0 will become faster. Meanwhile, hospitals will gradually tend to choose the strategy of “agree”. The model eventually evolves to a stable state of (unable to distribute by demand, agree).

This indicates that hospitals will eventually choose the strategy of “agree” as the coordination cost increases due to the fight of the hospital, and that GNPOs choose the strategy of “unable to distribute by demand” in order to maximize the total benefits to society based on the fairness of supplies distribution.

5.3. Discussion

In the process of the disaster relief of public health emergencies with uncertainty, medical supplies occupy a very important position. Therefore, the reasonable distribution of medical supplies to the disaster site is an important guarantee to achieve rapid and effective rescue, which has great significance to save lives and reduce property losses after the occurrence of emergencies. In the early stage of an epidemic, medical relief is in great disorder and there is a serious shortage of medical supplies. How the supplies distributor and the supplies recipient reach the dynamic equilibrium point between total supply and total demand depends on whether the demand is satisfied. Based on the assumption of bounded rationality, this paper constructs a two-sided game model between GNPO as the distributor of medical supplies and the hospital as the recipient of medical supplies during the break of COVID-19 regarding the distribution of medical supplies based on evolutionary game theory.

In the case of the COVID-19 pandemic of Wuhan presented in this paper, we analyze, from a practical point of view, how the strategies of the GNPO and the hospital would change in a situation where the GNPO has limited medical supplies and the hospital has huge demand for medical supplies due to the continued deterioration of the epidemic. The analysis also focuses on the influencing factors β (the degree of hospital’s fighting) and C₁ (the coordination costs incurred by hospitals’ disagreeing the distribution plan and fighting) that produce change. The simulation case in this section captures the behavioral changes of the GNPO and hospital, and effectively verifies the impact of key influencing factors. Combined with the urgency and quantity of hospital requirements, our goal is to pursue the maximization of the social benefits based on the fairness of supplies distribution. The results of the simulation show that GNPO chooses the strategy of “unable to distribute by demand” and the hospital chooses the strategy of “agree”, taking the increase in coordination costs (C₁) and the degree of fighting (β) into account in order to reach a state of dynamic evolutionary equilibrium.

5.4. Management Implications

Firstly, GNPOs, as the distributor of medical supplies, should choose strategies that balance efficiency and equity as much as possible. In the early stage of public health emergency, social donations become a necessary means to ensure supplies for epidemic prevention and control due to the severe shortage of medical supplies owned by disaster relief departments, labor shortages, and sharp increases in costs owing to supply chain constraints. Due to the special nature of medical supplies and the relative psychological vulnerability of the affected people in the epidemic environment, if medical supplies are unfairly distributed or unevenly dispatched, it is very easy to lead to public anger and may even turn into a mass incident with serious consequences. GNPO, as the distributor of medical supplies, should choose a strategy that balances efficiency and fairness to defuse the fighting and maintain social stability before the mass outbreak of hospitals’ fighting. This also requires GNPOs to improve their ability to utilize resources and maximize the value of scarce resources. At the same time, GNPOs can coordinate with government departments to increase the cost of penalties and incentives for hospitals and explore an effective incentive mechanism to turn conflicts into symmetrical incentives, which can help reduce the incentive for hospitals to fight. Our evolutionary game model depicts the dynamic paths of each game player and provides theoretical guidance to managers in order to maximize the total social benefits using the optimal game strategy.

In addition, GNPO, while implementing the distribution of supplies, should publicly release the information related to supplies in a timely and efficient manner to protect the right-to-know of the supply recipients and the public, and at the same time do a good job of supervising and managing the work. On the one hand, the hospital’s own benefits should be fully considered, and on the other hand, the troublemaking parties who ignore the overall stability of epidemic prevention and control and take advantage of the irrational climbing mentality to maximize their own interests in the event of public health emergencies should be dealt with seriously. The disclosure of information by the government administration, which itself has a very high credibility, can accelerate the promotion of mutual trust between organizations, which is conducive to the formation of long-term stable relationships, and it will also promote better decision-making and more coordinated management by both sides in urgent and critical situations. For example, after the public outcry about the distribution of medical supplies, the Red Cross Society of China Hubei Branch and other GNPOs promptly updated the source and destination of each donation in detail, and everyone could check it online at any time, which greatly promoted the trust of GNPOs-hospitals-society, and also promoted a more reasonable distribution of medical supplies. This has played an extremely important role in the effective control of a public health emergency such as COVID-19.

Secondly, the results of the study show that in public health emergency, the hospital as the recipient of medical supplies should actively choose the strategies that maximize the total social benefits and strengthen trust in GNPOs so that a win–win situation can be achieved and sustainable emergency relief can be realized. Even the hospitals may not receive as much medical supplies as they need from GNPOs during a public health emergency. Considering the fight costs and negative effects to themselves and the society, they should try to seek other channels to obtain the medical supplies, rather than spending time on fighting. Meanwhile, the modernization of the emergency response system and its capabilities for public health emergencies should be advanced to ensure the sustainable development of society.

The findings suggest that the managers of hospitals need to pay attention to reducing the impact of communication and coordination costs and strive for reducing conflicts between different values, which coincides with the findings of some studies. For example, some studies have shown that a rational division of labor, optimization of organizational processes, and clarification of specific responsibilities and rights can significantly reduce management costs and improve efficiency within the organization [52], and that maintaining the timely exchange of information can also improve the efficiency of communication and avoid misunderstandings [53]. In the meantime, barriers between languages and cultures are the main cause of communication problems between partners, which significantly increases the cost of coordination. Moreover, political values and identities can easily undermine collective efforts in a crisis [54]. Some scholars have proposed a solution to this problem by involving international collaborative tasks with locals who are more familiar with the local culture and laws, which helped to reduce additional mistakes and expenditures [55]. Eliminating cultural conflicts and policy constraints can also significantly reduce the cost of coordination [56], and, if there is a consistent culture of values between organizations, it will significantly reduce the cost of communication and coordination.

Thirdly, the government should strengthen supervision to avoid conflicts between the distributors and receivers of medical supplies during a public health emergency and ensure the rescue efficiency. As the coordinator and supervisor, the Chinese government authorizes GNPOs to have the sole right to distribute medical supplies in the public health emergency. While the GNPOs might not be able to guarantee enough medical supplies for each hospital, in this case, the government could consider extending the authorization to other NPOs to mobilize medical supplies from the society as much as possible during the public health emergency to ensure the sustainability of the medical supplies distribution.

Last but not least, the research results have certain guiding significance for the sustainable supply of medical supplies in epidemic areas in the case of public health emergencies. Public health emergencies are quite different from other emergencies, which are always companied with characteristics of complex transmission channels, a wide range, and difficult prevention and control. Therefore, it is particularly important to construct a sustainable medical supplies system. To achieve this goal, depends not only on the mutual coordination and cooperation between GNPO, the distributor of medical supplies, and the hospital, the receiver of medical supplies, but also requires the support and supervision of all sectors of society for the whole process of emergency medical supplies distribution. Only in this way can we achieve the sustainable supply of medical supplies. Nowadays, there will still be a sudden outbreak of the public health emergency all around the world. Due to the restrictions on the free flow of personnel and medical supplies during the sudden outbreak of the public health emergency, the local medical system, as well as the local authority’s capability to distribute the medical supplies, are facing a huge challenge. The research findings could implicate the organizations with the same role as distributors in other countries, as with GNPOs in China, to respond to the public health emergency timely and ensure the sustainable supply of medical supplies during the outbreak of public health emergencies.

6. Conclusions and Future Research

Based on evolutionary game theory, this study explores the sustainable distribution of medical supplies between GNPOs and hospitals in the situation of disaster relief for public health emergencies with uncertainty. Under the assumption of bounded rationality, the interests of the game players will conflict. Based on previous studies, this paper quantitatively discusses the impact of initial strategies and key factors on the dynamic game model. The results show that, as the degree of fighting(β) and the coordination cost of hospitals due to fighting (C₁) increase, GNPO chooses the strategy of “unable to distribute by demand” and hospitals choose the strategy of “agree”, and then the dynamic evolutionary equilibrium can be reached. At the same time, the reaction of game players can change the strategy of other players.

6.1. Research Contributions

The research makes contributions both theoretically and practically. On the one hand, we try to improve the sustainable emergency response capacity of GNPOs and hospitals during public health emergencies from the perspective of evolutionary game theory. The timely sharing of resource information and non-confrontation between organizations are necessary in the disaster relief process, which will make the relief activities more sustainable and avoid the waste of limited resources by repetitive work. Meanwhile, we use a quantitative approach to fill the research gap in the related field, and the results portray the interaction mechanism between players and show the optimal strategy choice under different parameters.

On the other hand, public health emergencies are inevitable, which could suddenly outbreak anywhere around the world. Therefore, immediate actions to improve the response are quite necessary. We examine the important factors that could affect the efficiency of medical supplies distribution in the early stage of public health emergencies, so that the distributor and the receiver could improve the efficiency of medical supplies distribution more quickly when facing public health emergencies. Although GNPO is the unique organization in a Chinese context, the results could also help the organizations who play the same role as GNPOs achieve the sustainable supply of medical supplies in public health emergencies all over the world.

6.2. Research Limitations

The limitations of this paper can be summarized as follows. Firstly, during the process of disaster relief in public health emergencies with uncertainty, the problem of medical supplies distribution involves many subjects, and this paper only focuses on the two-party game problem of the GNPO and hospital as the game subject, which has certain limitations. Secondly, this paper mainly studies the game problem in the early stage of public health emergencies but does not study the game problem in the whole process of the event, which is the lack of the complete picture of the game mechanism in the whole process.

6.3. Future Perspectives

In the future, this study can be extended in several directions. Firstly, as future research progresses, we can introduce more game players to explore the sustainable mechanism of medical supplies distribution during the process of disaster relief in public health emergencies with uncertainty, such as introducing the government as a game player to build a three-party evolutionary game model of government–GNPO–hospital. Secondly, we can discuss the game problem in the whole process of a certain type of emergencies and analyze whether there is a difference between the game behaviors in the early and late stages. Thirdly, we can extend the evolutionary game approach to more types of emergencies to explore how different players make optimal strategy choices under the premise of bounded rationality.