Introduction

In the landscape of global agriculture, agricultural insurance plays a crucial role in ensuring the stability and growth of the agricultural sector. Policy - based agricultural insurance, recognized as a significant “green box” policy. “Green box” policies are policies in the World Trade Organization (WTO) Agreement on Agriculture that support agriculture and do not distort agricultural trade and production. It is based on public interest, non-trade-distorting, and covers subsidies for government services, food reserves, disaster relief, and so on. In the field of agricultural insurance, it uses government subsidies to reduce the cost of farmers’ participation in insurance, increase the participation rate, and help agricultural anti-risk and industrialized development. It has been adopted by countries worldwide to safeguard farmers’ well - being, stabilize agricultural operations, and drive agricultural industrialization (Goodwin et al., 2004). In China, since the initiation of agricultural insurance financial subsidies in 2004, the agricultural insurance market has witnessed remarkable growth. As per the data from the Banking and Insurance Regulatory Commission (BIRC), in 2022, the agricultural insurance sector achieved a premium income of 121.94 billion yuan, offering risk protection worth 5.46 trillion yuan to 170 million households. The coverage rate of the three major staple crops (rice, wheat, and corn) exceeded 70%, making it a cornerstone in the realization of the “rural revitalization” initiative.

However, while the scale, insurance density, and coverage of China’s agricultural insurance market continue to expand, several issues have emerged. Notably, the level of risk protection remains low, and the types of covered risks are relatively single (Xie and Luo, 2019; Zhou et al., 2023). Moreover, the supply of insurance products tailored to local - characteristic and advantageous agricultural products is insufficient, and it is challenging to cover the business activities of new agricultural operators. This has led to a structural imbalance in the agricultural insurance market, which is a significant concern (Tuo and Zhang, 2018). A large number of new agricultural operators, due to their large capital investment and scale of operation, relatively concentrated market- type risk structure, and thus have a higher demand for agricultural insurance participation and premium payment ability (Zhu and Chen, 2022), but in contrast, China’s agricultural insurance products for new agricultural and specialty agricultural products are seriously inadequate, and the few targeted products that do exist in the market have a low level of risk protection, and even fewer of them are able to realize the coverage of the fluctuating risks of the market price. The few targeted products that exist in the market also have a low level of risk coverage, and even fewer can realize coverage for market price fluctuation risks. This structural contradiction has amplified the contradiction between the supply and demand structure of the agricultural insurance market, and at the same time, it has also become a key issue that needs to be solved in order to promote the reform of the agricultural supply side and accelerate the establishment of a new type of agricultural modernization and development system.

The motivation behind this research is to address the pressing need to resolve the structural imbalance in the agricultural insurance market for new agricultural and specialty agricultural products. The current situation restricts the development of new agricultural businesses and undermines the overall resilience of the agricultural industry. Our aim is to identify effective strategies to bridge the gap between supply and demand in this specific insurance market segment and promote its healthy and sustainable development.

The structural imbalance between the supply and demand of agricultural insurance has been discussed in the literature, such as the fact that insurance for agricultural products with special characteristics is characterized by strong regionality, high risk, high payout rate and high cost. Therefore, although farmers have a strong demand for insurance, insurance companies lack sufficient supply willingness and supply capacity. And although local governments have a strong positive will to promote the industrialization of regional characteristic agricultural products with agricultural insurance, they can hardly afford the high premium subsidies. Eventually, the development bottleneck of “government shouting, farmers waiting, and companies relying” was formed (Hu and Li, 2022).

The ineffectiveness of catastrophe sharing mechanisms, the arduous nature of obtaining actuarial information for insurance, and the prevalence of high moral hazard are fundamental factors that prevent insurance from widely covering agricultural products (Tian, 2016). In order to correct the imbalances in the agricultural insurance market and promote its high quality development, several key measures have been proposed in existing studies. These measures include increasing financial subsidies and establishing a comprehensive framework of financial incentives and subsidies (Kramer et al., 2022); incentivizing insurance institutions to engage in product innovation; and improving the actuarial efficiency of agricultural insurance to facilitate the implementation of “provincial pricing” and “regional pricing” strategies (Cai et al., 2024). In addition, some studies have advocated improving the agricultural insurance system to match the requirements of new agricultural business entities in the demonstration zones, to enhance the level of risk protection and promote the construction of agricultural demonstration zones (Zhang et al., 2024).

Overall, existing studies provide valuable perspectives from a few perspectives, including risk-protection capacity, product-innovation trajectories and subsidy strategies. However, these studies also have limitations from the perspective of local advantageous agricultural development.

First, while recognizing the need to expand government financial subsidies from basic agricultural products to special agricultural undertakings, previous studies have overlooked the financial constraints faced by municipal and county governments and the relative importance of local agricultural development in the government’s strategic objectives. This oversight affects the accurate assessment of local governments’ ability to subsidize agricultural insurance. Even if the government expands the scope and intensity of premium subsidies, the sustainability of these initiatives may be compromised due to the limited financial resources of local governments. Therefore, it becomes challenging to address the issue of “government subsidy supplementation” for county-level specialty agricultural insurance.

Second, the imbalance between supply and demand in the agricultural insurance market is due to the fact that, given the special risk characteristics of new agricultural products, it is difficult for insurance companies to set stable profit expectations for them. As a result, insurance companies often launch innovative products with “low premiums and low payouts”, which cannot meet the demand for “high premiums, high paying capacity, and high payouts” of new agricultural business entities and the development of regional specialty agriculture. Encouraging insurance companies to overcome the imbalance through competition and innovation essentially avoids the central question of “what motivates insurance companies to innovate”. In addition, these studies do not adequately incorporate the heterogeneity of agricultural regions, especially county-level characteristics and risk structures, into the analytical framework. This shortcoming leads to a lack of in-depth exploration of the supply of agricultural insurance and government incentive strategies for regionally characterized agricultural products.

Therefore, this paper attempts to utilize the evolutionary - game - analysis tool to comprehensively integrate the agricultural - insurance demand and risk structure of new agricultural management subjects and specialty - agricultural farmers. Decisions of the government, farmers and insurance companies in the agricultural insurance market affect each other and change constantly. The assumption of complete rationality in traditional game theory is difficult to be established in reality because of the serious information asymmetry of each subject, which makes it impossible to make accurate decisions. Evolutionary game theory, on the other hand, considers the limited rationality of the participants and allows them to adjust their strategies through trial and error and learning, which is more in line with the reality of the agricultural insurance market. Therefore, evolutionary game theory is chosen in this study to analyze the behavior of each subject and solve the problem of market structure imbalance more accurately. By comparing the two distinct policy - agricultural - insurance subsidy models, namely “subsidizing farmers” and “subsidizing institutions”, we aim to analyze their impacts on the participation and underwriting strategies of specialty - agricultural insurance products. Specifically, we will explore how these subsidy models influence the behavior of farmers and insurance companies in the agricultural insurance market. We also aim to determine the optimal subsidy model and policy adjustments that can effectively stimulate the supply and demand of agricultural insurance for specialty agricultural products, break the market - structure imbalance, and ultimately achieve the “high - quality development of agricultural insurance.” Under the three - dimensional framework of “rural revitalization, farmers’ income increase, and agricultural development”, we strive to establish a “win - win” market structure where the government, agricultural insurance companies, and farmers can mutually benefit and promote each other, thereby accelerating the implementation of the strategy of “expanding coverage, increasing product variety, and improving service standards” in the agricultural insurance sector.

This study is an important contribution both in terms of theoretical and practical implications. It provides multidimensional value to the development of the agricultural insurance field.

From the perspective of theoretical implications, this study has made important breakthroughs in the construction of theoretical models and the expansion of research perspectives. In terms of theoretical modeling, the risk structure and agricultural insurance demand of characteristic agricultural products and new agricultural operators are innovatively integrated into the revenue function. An evolutionary game analysis model of government-farmer-insurance company is constructed. The influence mechanism of each factor on the game results is precisely revealed. A solid theoretical framework is provided for subsequent research. In terms of research perspectives, this study focuses on the structural imbalance of the insurance market for new and specialty agricultural products. It breaks through the limitations of traditional agricultural insurance research. It fills the gaps in the field, enriches agricultural insurance research, and provides scholars with brand new perspectives and ideas.

From the perspective of practical implications. The study provides key support for policymaking and decision-making by market players. For policy makers, the study considers the financial constraints of municipal and county governments and the relative weight of agriculture in the government’s objectives and provides a scientific basis for assessing the carrying capacity of local governments’ agricultural insurance subsidies. It also proposes a supply-side incentive path and a series of policy recommendations, such as expanding the boundary of government support, improving the policy of “award in lieu of subsidy”, initiating cost subsidies for insurance companies, and optimizing the reinsurance mechanism, which points out the direction for optimizing the agricultural insurance policy system. For market players, the study provides strong guidance for insurance companies and new agricultural operators. Insurance companies can plan their business strategies more rationally and realize sustainable development based on the profitability conditions and strategic analysis in the study. New agricultural operators, on the other hand, can make rational decisions and increase their willingness to participate in insurance by understanding the rights and benefits under the subsidized model, thus promoting the balance between supply and demand and the healthy development of the insurance market.

To better illustrate the research context and the roles of different participants, we present the following Table 1.

Table 1 Research context and the roles of different participants.
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Theoretical background

Establishment of limited rationality constraints for participating subjects

The imperfect information between game subjects would lead to the difficulty of subjects to accurately anticipate the possible action strategies of the game object (Zhang, 2024), and therefore, in the process of continuous game, the strategy choices of the game subjects are characterized by finite rationality (Hou et al., 2024), i.e., the loss of sight of the future. As a result, based on the limited rationality setting, the evolutionary game that fully considers the trial and error and correction process of the subject of the game becomes the main tool for behavioral analysis under the incomplete contract.

Specifically, there is a typical information asymmetry problem among the participating subjects of featured agricultural insurance: First, the government subject, under the guidance of rural revitalization strategy of agricultural industrialization and local featured agricultural development, although the local government has the financial expenditure preference to stimulate the innovation of featured agricultural insurance products and drive the development of featured agricultural development through the financial subsidy method, it is difficult to accurately identify the actual featured agricultural insurance market capacity, i.e., it is impossible to accurately anticipate the final number of insured farmers and the total amount of financial subsidy to be paid under the established financial subsidy strategy. Accurately identify the actual characteristics of the agricultural insurance market capacity, that is, can not accurately expect in the established financial subsidy strategy under the final number of insured farmers and need to pay the total amount of financial subsidies, and do not understand the insurance company’s current characteristics of the agricultural insurance products and its possible future operating income and loss level. Second, the main body of the insurance company, compared with traditional agriculture, the risk type of specialty agriculture is mainly concentrated in price fluctuations, supply and demand imbalance and other market-type risks, which makes it difficult for insurance companies to realize the operational risk coverage of specialty agricultural insurance products, and at the same time cannot accurately observe the possible payout pressure and losses caused by farmers’ moral hazard. Thirdly, the main body of farmers, the large scale of investment in specialty agricultural business, the complexity of the type of risk, but also the lack of basic agricultural products of the government price intervention guarantee, so it is difficult to accurately anticipate the future potential risk loss, the potential future loss of the risk compensation may be obtained, and the cost of bargaining with the insurance company and so on is not completely clear. Information asymmetry among the main players in specialty agricultural insurance is extremely high. It is difficult for the government to grasp the market capacity, farmers’ demand and the operation status of the company; it is difficult for the insurance company to cope with the risk of specialty agriculture and monitor the moral risk of farmers; it is difficult for the farmers to predict the risk loss, compensation and bargaining cost. This makes the subject’s decision-making show limited rationality, and traditional game theory is difficult to apply. Evolutionary game theory assumes of finite rationality, which can portray the trial and error, learning and strategy adjustment of subjects in the game. In the agricultural insurance market, the decision-making of each subject is dynamically optimized, and evolutionary game theory helps to understand the logic of their behavior and market trends, providing an effective tool for research. Therefore, the limited rationality feature is significantly present in the three-principle strategy game of “government-agricultural insurance company-characterized agricultural management subjects” in the operation process of agricultural insurance products.

Research methodology

Game parameterization for insurance companies

Assume that an insurance company has an end-of-period asset present value of W. Under a government initiative, the company agrees to provide agricultural insurance services to a specialized agricultural business entity. The services are categorized by types of agricultural products. Let g denote the premium set for the agricultural insurance product and let φ represent the compensation coefficient stipulated in the insurance contract.

Considering that most agricultural insurance suppliers are comprehensive insurance companies. Specialty agricultural management subjects have higher income and stronger risk awareness, so insurance companies have more potential to expand life insurance, property insurance and other businesses through agricultural insurance services. We further assume that the insurance company can obtain additional revenue V by carrying out specialty agricultural insurance business. This revenue includes the insurer’s reputation appreciation, non-agricultural insurance business expansion income, rural market channel revenue, etc.

In order to further identify the impact of the two differentiated government subsidy modes of “supplementing farmers” and “insuring institutions” on the supply and demand of specialty agricultural insurance, it may be assumed that the government has implemented the “supplementing farmers” premium subsidy for specialty agricultural insurance. But considering the “supplementing farmers” premium subsidy for specialty agricultural insurance. It is not possible for insurers to realize additional revenue V through the development of specialty agricultural insurance business. It may be assumed that the government has implemented the “supplementing farmers” premium subsidy for specialty agricultural insurance, but considering that insurance companies provide specialty agricultural products insurance with strong risk payout pressure, and that it is difficult to control the operation risk by setting a higher premium level in the framework of policy-based agricultural insurance, therefore, in order to strengthen the incentives for insurance companies to supply products, the government agrees to provide insurance companies with a total amount of S. However, due to the complexity of the types of risks in specialty agriculture and the pressure on cost protection, the operating subsidy provided by the government to the insurance company is in the form of a total subsidy, i.e.,. The subsidy S does not vary with the number of insured farmers and the actual amount of claims paid out by the insurance company, but is entirely determined by the government’s negotiation with the insurance company beforehand, and is paid out annually.

Further, in a market with strong government intervention, insurers have the willingness to cater to the government’s behavior in order to seek government support in other insurance products, territorial openness, credit and land rationing, etc. Therefore, it is assumed that insurers will lose the potential dividend of responding to the government’s signals if they do not provide specialty agricultural insurance services, i.e., insurers have to pay a cost Q for not providing specialty agricultural insurance services.

Game parameterization for specialized agricultural business entities

Specialty agricultural products and specialty agricultural operations have the characteristics of large-scale investment, long return cycle, complex risk types, strong impact of market supply and demand and market prices. But at the same time, the expected returns of the operators are also higher, so specialty agricultural business entities have a strong ability to pay premiums and willingness to pay premiums. But at the same time, there is a “high premiums, high claims” and “market-based risk coverage” demand preference for agricultural insurance. But at the same time there are “high premiums, high claims” and “market-oriented risk coverage” of agricultural insurance demand preferences. Under these characteristics, let the maturity value of farmers’ special agricultural income at the end of the period be R, and the risk loss level be ηg, and then assume that the proportion of government subsidy in the determined premium g is (1-η), then the farmers’ out-of-pocket premium is ηg.

Because of the typical regional characteristics of featured agriculture, the featured agricultural insurance follows the localized characteristics of “one place (county), one negotiation”. And the final insurance contract is actually the result of full negotiation between the government, the insurance company, and the featured agricultural management subjects. In order to incentivize insurance companies to supply special agricultural insurance products and government policies, the main body of special agricultural business may need to make commitments in terms of business scale, inputs and risk control, i.e., it needs to pay the negotiation cost of T, so as to obtain the risk protection of post-disaster φΔi. However, when it does not participate in the insurance policy, due to the complexity of the business risk structure and the weak individual risk control ability, it may be assumed that it must assume a relatively high amount of other disaster mitigation inputs L when it lacks risk protection. L when it lacks risk protection.

Different from traditional agricultural operations, the income of specialty agricultural business entities highly depends on business category/technology choices and product sales channels. Such entities are more vulnerable to market price fluctuations, making channel/market information particularly valuable. Under the “small farmers, big market” supply-demand structure, assume insurance companies offer insured farmers customized disaster prevention technical services and scaling/industrialization technical-info support to reduce future claim probabilities in specialty agricultural insurance. With information spillover, this grants “free-rider” benefits to uninsured homogeneous farmers. Since insurance companies provide technical-info services to mitigate payout risks φΔi, their market-oriented service scale relies entirely on specialty agricultural entities’ potential losses. Here, define uninsured farmers’ free-rider gain as δΔi, where δ denotes the free-rider gain coefficient.

Game parameterization for municipal and county local governments

Under the guidance of the strategic idea of “expanding the surface, increasing the products and raising the standard” for the high-quality development of agricultural insurance. It has become the core idea of China’s agricultural insurance development to realize the expansion of policy subsidy boundaries to featured agriculture and featured agricultural products with the main body of the local government. And it also rapidly enters into the framework of the local government’s performance appraisal objectives. Therefore, the local governments of cities and counties have the policy task of promoting the industrialization and branding of featured agriculture through the implementation of policy agricultural insurance, and provide premium subsidies for farmers to participate in featured agricultural insurance, and at the same time provide subsidies for insurance companies to subsidize their operating costs S. Assuming that they are constrained by their limited financial payment capacity, the premium subsidy coefficient given to farmers is (1 - η), and there are 0 < (1 − η) < 1.

Further, in the current local government sectional governance structure, two key tasks exist: executing higher-level government mandates and developing the territorial economy. Promoting regional agricultural specialties—core performance content for local governments—can both boost the regional economy and enhance governance performance, thereby generating promotion capital. Therefore, this paper assumes that local government bodies adopt only a [Subsidize] strategy, ignoring the [no subsidize] option. Their subjective discretion is reflected solely in adjusting premium subsidies for operators and cost subsidies for insurance companies.

In the above three subjects’ game, the local government as an implicit participant exists a single strategy characteristic, while the strategy sets of farmers and insurance companies are [insured, uninsured] and [underwritten, not contracted], respectively.

Evolutionary Game analysis result: specialized agricultural business entities and insurance companies

Under the government’s pre-agreed premium subsidy (1-η)g for specialty agriculture and operating expense subsidy S to insurance companies, specialty agricultural operators exhibit strong insurance demand. However, due to insurance companies’ profitability constraints, high premiums set within the policy agricultural insurance framework may deviate from “policy” and “public welfare” attributes. Consequently, insurance companies tend to continue the traditional agricultural insurance model of “low insurance coverage and low compensation”. This creates a deviation between farmers’ risk protection needs and the actual level of agricultural insurance protection, making it difficult to effectively stimulate the demand of specialty agricultural operators. As a result, operators form a “low insurance” model. Therefore, the operators will form the “agricultural insurance can only cover part of the risk” of the expectations, which will lead to the characteristics of agricultural operators to participate in the insurance decision-making differentiation, it may be remembered that x for the characteristics of agricultural business subjects to choose the probability of [insured] strategy, then (1-x) for the probability of choosing the strategy [uninsured].

As the insurance company provides special agricultural insurance products under the break-even constraint. It needs to realize the product coverage of some market-type risks in order to pry the demand of special agricultural operators to participate in the insurance. But it also implies a rise in the pressure of risk payout, and the difficulty of the product actuarial risk beforehand, and there is a resistance to the pricing of the high premiums, so the insurance company supplying the special agricultural insurance products has a strong constraint of the operating costs. But at the same time, the insurance company to provide special agricultural insurance products, that is, has a broad market demand, but also can realize the integration of life insurance, property insurance business channels, but also by actively responding to the government’s actions can obtain potential benefits, so the insurance company exists in the [underwriting] and [non-coverage] strategy choice, may wish to record y for the insurance company to choose the probability of the [underwriting] strategy, (1-y) for the insurance company to choose the probability of the [non-coverage] strategy. strategy.

Accordingly, the initial return matrix of the static game between the specialty agricultural management subjects and insurance companies under different strategies can be calculated, and the corresponding actual returns under different strategies are detailed in Table 2:

Table 2 Benefits Matrix of specialized agricultural business entities and insurance companies.
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Specialized Agricultural Business Entities:

The expected utility when [insured] is:

$${U}_{{\rm{insured}}}={\rm{y}}[{\rm{R}}-\eta {\rm{g}}+(\varphi -1){\rm{E}}(\Delta {\rm{i}})]+(1-{\rm{y}})[{\rm{R}}-{\rm{E}}(\Delta {\rm{i}})-{\rm{L}}-{\rm{T}}]$$
(1)

The expected utility when [not insured] is:

$${U}_{{\rm{not\; insured}}}={\rm{y}}[{\rm{R}}+(\delta -1){\rm{E}}(\Delta {\rm{i}})-{\rm{L}}]+(1-{\rm{y}})[{\rm{R}}-{\rm{E}}(\Delta {\rm{i}})-{\rm{L}}]$$
(2)

The average expected utility is:

$${\bar{U}}_{{\rm{specialized}}\; {\rm{agricultural}}\; {\rm{business}}\; {\rm{entities}}}={\rm{x}}\{{\rm{R}}-{\rm{E}}(\Delta {\rm{i}})-{\rm{L}}-\Delta {\rm{G}}+{\rm{y}}[-\eta g+\varphi E(\Delta {\rm{i}})+{\rm{L}}+\Delta {\rm{G}}]\}+(1-{\rm{x}})[{\rm{R}}-1{\rm{E}}(\Delta {\rm{i}})-{\rm{L}}+{\rm{y}}\delta E(\Delta {\rm{i}})]$$
(3)

The expected utility of the insurance company’s [underwriting] strategy is:

$$\begin{array}{lll}{U}_{{\rm{underwriting}}} &=& x[W+g+S-\varphi E(\Delta {\rm{A}})+{\rm{V}}]\\&&+\,(1-x)[W-\delta E(\Delta i)+{\rm{S}}]\end{array}$$
(4)

The revenue function of the insurance company when choosing the [non-coverage] strategy is:

$${U}_{{\rm{non}}-{\rm{coverage}}}=x(W-{\rm{Q}})+(1-x)W$$
(5)

Furthermore, assuming that the insurance company provides insurance products for specialized agricultural business entities, it can leverage existing agricultural insurance sales channels, outlets, and professional service personnel. As a result, no additional operating costs are incurred. The insurance company’s potential operational losses may only arise from significant risk compensation pressures and potential actuarial losses. Therefore, the average expected utility of the insurance company can be expressed as:

$${\bar{U}}_{{\rm{Insurance\; companies}}}={\rm{y}}\{x[g+V-(\varphi -\delta )E(\Delta i)]+W+S-\delta E(\Delta i)\}+(1-{\rm{x}})(W-x{\rm{Q}})$$
(6)

In the framework of evolutionary game theory with discrete strategies, the replicator dynamic equation for specialized agricultural business entities (Li and Wang, 2022) can be expressed as follows:

$$F(x)=x(1-x)\{y[L+T-\eta g+(\varphi -\delta )E(\Delta i)]-T\}$$
(7)

And the replicator dynamic equation for the insurance company entities is:

$$G(x)=y(1-y)\{x[g+V+Q-(\varphi -\delta )E(\Delta i)]+S-\delta E(\Delta i)\}$$
(8)

Given the dynamic equations of stable evolution, the five evolutionary game equilibrium points are: (0,0), (0,1), (1,0), (1,1), and E5 (x*, y*), where provide details or definitions of \({x}^{* }=\frac{\delta E(\Delta i)-S}{g+V+Q-(\varphi -\delta )E(\Delta i)}\), \({y}^{* }=\frac{T}{L+T-\eta g+(\varphi -\delta )E(\Delta i)}\) if available.

In practical terms, the dynamic perspective of the evolutionary game adopted in this study implies that the actors involved in the agricultural insurance market (government, farmers and insurance companies) do not make one-time decisions. Instead, they continuously adjust their strategies in a changing market environment.

For example, an insurance company does not offer brand new contracts in every period. It will adjust its underwriting strategy and the specific terms of the insurance contract according to changes in market feedback, payouts, government subsidy policies, and farmers’ insurance participation behavior. If the damage to specialty agricultural products in a certain period exceeds expectations and the amount of payout increases, the insurance company may raise the premiums of some insurance products or adjust the indemnity coefficients in the subsequent period in order to balance the income and expenditure and control the risks.

For the government, although it will not change its subsidy policy frequently and drastically, it will adjust the level and manner of the subsidy policy at the right time in accordance with the overall development of the agricultural insurance market, the financial situation and the extent to which the policy objectives have been achieved. For example, if it is found that the underwriting incentive of insurance companies is not high or the participation rate of farmers does not meet expectations, the Government may appropriately raise the operating cost subsidy for insurance companies or increase the proportion of premium subsidy for farmers to stimulate market supply and demand.

This dynamic adjustment process is reflected in the return matrix (e.g., Table 3). The value of each return in the matrix will change with the change of each subject’s strategy. In each round of the game, each subject will adjust its own strategy according to the current return situation and the expectation of the future. This adjustment in turn affects the outcome of the next round of the game. The whole market evolves in such a dynamic process until it reaches a stable equilibrium state.

Table 3 Simplified Payoff Matrix of the Game.
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In the dynamic strategy space of continuous games, the stability of equilibrium points actually depends on the strategy convergence stability of the system’s Jacobian matrix. The following is a stability analysis of the Jacobian matrix. For convenience of expression, letters A to G are used to represent the corresponding parts of the payoff matrix. The correspondence is shown in Table 3.

The Jacobian matrix is given by:

$$J=\left|\begin{array}{cc}{(1-2y)}^{* }\left[{({\rm{E}}-{\rm{F}}+{\rm{H}}-{\rm{G}})}^{* }x-({\rm{H}}-{\rm{G}})\right] & {({\rm{E}}-{\rm{F}}+{\rm{H}}-{\rm{G}})}^{* }{y}^{* }(1-{\rm{y}})\\ -{({\rm{B}}-{\rm{D}}+{\rm{C}}-{\rm{A}})}^{* }{x}^{* }(1-{\rm{x}}) & {(1-2x)}^{* }\left[({\rm{B}}-{\rm{D}})-{({\rm{B}}-{\rm{D}}+{\rm{C}}-{\rm{A}})}^{* }y\right]\end{array}\right|$$
(9)

From this, the determinant of matrix J is:

$$\begin{array}{l}\det J={(1-2y)}^{* }{\left[{({\rm{E}}-{\rm{F}}+{\rm{H}}-{\rm{G}})}^{* }{\rm{x}}-({\rm{H}}-{\rm{G}})\right]}^{* }{({\rm{l}}-2x)}^{* }\left[({\rm{B}}-{\rm{D}})-{({\rm{B}}-{\rm{D}}+{\rm{C}}-{\rm{A}})}^{* }y\right]\\ +{({\rm{E}}-{\rm{F}}+{\rm{H}}-{\rm{G}})}^{* }{y}^{* }{(1-{\rm{y}})}^{* }{(A-C+D-B)}^{* }{x}^{* }(1-x)\end{array}$$
(10)

The trace of the matrix is:

$${\rm{tr}}\,J=(1-2y)* \left[{({\rm{E}}-{\rm{F}}+{\rm{H}}-{\rm{G}})}^{* }x-({\rm{H}}-{\rm{G}})\right]+\,{(1-2x)}^{* }\left[({\rm{B}}-{\rm{D}})-{({\rm{B}}-{\rm{D}}+{\rm{C}}-{\rm{A}})}^{* }y\right]$$
(11)

Substituting the payoff matrix into these expressions yields:

$$\begin{array}{l}A-C=(\varphi -\delta )E(\Delta i)+L-\eta g\\ B-D=-T\\ E-F=g+S-\varphi E(\Delta i)+V+Q\\ G-H=S-\delta E(\Delta i)\end{array}$$
(12)

The equilibrium point analysis is summarized in Table 4 as follows:

Table 4 Simplified equilibrium point analysis of the game.
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For specialized agricultural business entities, unlike staple crop products, the lack of a price protection mechanism results in a higher proportion of market volatility risk within their risk structure. Under the constraints of significant upfront capital investment, farmers typically lack alternative mechanisms for risk diversification. Therefore, they exhibit a stronger willingness to smooth risk expectations through agricultural insurance participation and display a more pronounced risk-averse preference. In fact, the new agricultural operators, due to their reliance on a single crop variety and high-risk concentration, as well as a lack of non-agricultural income compensation, demonstrate a stronger preference for agricultural insurance participation (Ye and Zhu, 2018). In this case, A − C > 0 and B − D < 0, indicating that specialized agricultural business entities have a strong willingness to purchase insurance given the effective supply of agricultural insurance products. Therefore, the distribution of equilibrium points in the strategy game in Table 4 mainly depends on the signs of E − F and G − H, as discussed below:

1. E-F > 0, G-H > 0

At this point, the replicated dynamic system has four strategy equilibrium points, namely E1(0,0), E2(0,1), E3(1,0), E4(1,1). Their stability properties are shown in Table 5 and Fig. 1.

Table 5 Stability analysis results of equilibrium point when E-F > 0 and G-H > 0.
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Fig. 1
figure 1

Stability analysis results of equilibrium point when E-F > 0 and G-H > 0.

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At this point, all other convergence points are either saddle points or unstable points, with only the point (1,1) being a stable game convergence strategy. This means that the game ultimately converges to the [insured, underwriting] strategy.

In reality, this situation shows that when the insurance company can obtain positive returns from operating specialty agricultural insurance. When government subsidies and other factors make the relevant conditions satisfied, farmers and insurance companies will eventually reach a strategic combination of [insured, underwriting]. This means that the agricultural insurance market can realize a good match between supply and demand. Farmers are willing to participate in insurance to protect their own risks, insurance companies have incentives to provide insurance services, and the market tends to develop stably.

2. E-F > 0, G-H < 0

At this point, in addition to the original equilibrium points E1(0,0), E2(0,1), E3(1,0), E4(1,1), the replicated dynamic system presents a new equilibrium point, E5(x*, y*), where

$$\begin{array}{c}{y}^{* }=\frac{B-D}{B-D+C-A}=\frac{-{\rm{T}}}{-{\rm{T}}+\eta G-L-\varphi E(\Delta i)+\delta E(\Delta i)}\\ {x}^{* }=\frac{H-G}{E-F+H-G}=\frac{\delta E(\Delta i)-S}{g+S-\varphi E(\Delta i)+V+Q-S+\delta E(\Delta i)}\end{array}$$
(13)

The results of the stability analysis of each equilibrium point are shown in Table 6 and Fig. 2.

Table 6 Stability analysis results of equilibrium point when E-F > 0 and G-H < 0.
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Fig. 2
figure 2

Stability analysis results of equilibrium point when E-F > 0 and G-H < 0.

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At this point, the only stable game strategy convergence point remains (1,1), with the multi-agent game ultimately converging to the [insured, underwriting] strategy.

From a realistic point of view, despite the emergence of a new equilibrium point, the final multi-subject game still converges to the [insured, underwritten] strategy. This suggests that even if certain combinations of government subsidies and market factors change. As long as the insurance company operates the specialty agricultural insurance with positive profit, the market can still reach the ideal stable state, and the interests of farmers and insurance companies can be coordinated in this situation.

3. E-F < 0, G-H > 0

At this point, the equilibrium point distribution of the replicated dynamic system includes E1(0,0), E2(0,1), E3(1,0), E4(1,1), E5(x*, y*), where

$$\begin{array}{c}{y}^{* }=\frac{B-D}{B-D+C-A}=\frac{-{\rm{T}}}{-{\rm{T}}+\eta G-L-\varphi E(\Delta i)+\delta E(\Delta i)}\\ {x}^{* }=\frac{H-G}{E-F+H-G}=\frac{\delta E(\Delta i)-S}{g+S-\varphi E(\Delta i)+V+Q-S+\delta E(\Delta i)}\end{array}$$
(14)

The results of the equilibrium point stability analysis can be found in Table 7 and Fig. 3.

Table 7 Stability analysis results of equilibrium point when E-F < 0, G-H > 0.
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Fig. 3
figure 3

Stability analysis results of equilibrium point when E-F < 0, G-H > 0.

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As shown in Fig. 3, there is no pure strategy equilibrium at this point, with the equilibrium point being E5. In this case, the system will struggle to achieve a static equilibrium in the game, and the interaction between farmers and the insurance company will exhibit repeated dynamic strategy adjustments, making it difficult to form a final stable convergence strategy.

In the actual agricultural insurance market, this situation reflects the fact that when the insurance company operates the featured agricultural insurance with negative profitability, the system is difficult to reach static equilibrium, and the strategies between farmers and insurance companies will be constantly adjusted dynamically. This may lead to the market in an unstable state, and it is difficult for both parties to form a stable cooperative relationship, which is not conducive to the sustainability of agricultural insurance business.

4. E-F < 0, G-H < 0

At this point, the equilibrium points of the replicated dynamic system are reduced to four: E1(0,0), E2(0,1), E3(1,0), E4(1,1). The stability results of each equilibrium point are shown in Table 8 and Fig. 4.

Table 8 Stability analysis results of equilibrium point when E-F < 0, G-H < 0.
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Fig. 4
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Stability analysis results of equilibrium point when E-F < 0, G-H < 0.

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At this point, the dominant strategy for both farmers and the insurance company is [uninsured, non-coverage], leading both parties to exit the agricultural insurance market.

This situation means that both farmers and insurance companies tend to choose the strategy of [uninsured, non-coverage], and both parties withdraw from the agricultural insurance market. This suggests that in the case of poor profitability and other unfavorable conditions for insurance companies, the agricultural insurance market will shrink. It will not be able to provide risk protection for farmers, which will hinder the stable development of agriculture.

In summary, considering that only stable equilibrium points can ultimately become the preferred strategies for game participants in continuous games, while unstable equilibrium points will continuously adjust and fluctuate, this indicates that only when specialized agricultural insurance companies have positive profits (E - F > 0) will underwriting become the dominant strategy. This ensures the achievement of a stable equilibrium at (E4(1,1). However, when the profits of specialized agricultural insurance companies are negative (E - F < 0), even if the company receives government subsidies S for operational costs and ensures that G - H < 0, the dominant strategy for the insurance company will still be [no underwriting]. In this scenario, the government’s operational cost subsidy ensures that the insurance company does not incur losses and serves as the sole condition supporting the provision of insurance products.

Comparing the two differentiated subsidy models of the government, under the assumption of bounded rationality, local governments are unable to accurately predict the actual market size after providing insurance subsidies. Therefore, in practice, financial subsidies for specialized agriculture in China have mostly adopted a pre-determined model, where the government commits to providing premium subsidies for agricultural insurance at a fixed rate for farmers. However, since insurance companies face significant moral hazard losses when offering underwriting services for specialized agricultural products and lack sufficient actuarial data to achieve “low-margin” operations, to ensure E − F > 0, the insurance company’s strategic options are either to increase the premium g that farmers must pay for participation or to limit the scale of insurance services, thus restricting the number of insured farmers. The ultimate result is either an increase in fiscal subsidy pressure, which dampens farmers’ willingness to participate in insurance, or agricultural insurance services being offered on a small scale, leading to the phenomenon of “one loss and it stops, one trial and it ends”.

In contrast, if the government adjusts the subsidy model from “Protecting farmer” to “Protecting institutions,” while keeping the premium g constant, the direct subsidies to farmers are eliminated. This indeed reduces the incentives for farmers to participate in insurance. However, given that new specialized agricultural operators have higher profitability, a concentrated risk structure, and strong demand for risk protection, the weakened incentive impact can be effectively offset. At the same time, direct subsidies to insurance companies ensure that they can provide insurance services to specialized agricultural operators on a larger scale. Under the direct subsidy approach, the impact of user scale is effectively controlled, allowing insurance companies to stabilize and observe the break-even point from E − F > 0 to E − F < 0. Transitioning from “Protecting farmer” to “Protecting institutions” still involves business scale limitations but can effectively stimulate the supply of specialized agricultural insurance, ensuring that insurance services are provided on a larger scale.

The evolutionary game analysis confirms that government policy subsidies (S) to specialized agricultural insurance companies can effectively drive product innovation and supply, ensuring “insurance availability for willing participants” within an ideal market scale. The theoretical basis for government premium subsidies at coefficient (1−η) stems from agriculture’s public good nature, aligning with China’s “policy + market” agricultural insurance framework. This framework provides comprehensive coverage for basic agriculture and selective coverage for specialized sectors. While directly stimulating farmer demand, the “subsidizing farmers” model imposes significant operational cost pressures on insurance companies when services shift to new agricultural operators. This discourages innovation, expands services, and accurately identifying break-even points, fostering a “fear of difficulty/loss” mentality. Adjusting risk allocation among farmers, insurers, and the government—clarifying new operators’ risk-bearing capacity and needs—along with moderating the rigid “low-premium” model to “moderate premiums with high compensation” can establish a reasonable cost-sharing structure. Stimulating insurers to supply specialized products via a “market-oriented, policy-supported” approach offers a feasible path to expand the “policy + market” model to local specialized agriculture and accelerate agricultural insurance’s high-quality development.

Conclusion

Discussion

Based on the findings, we summarize three policy recommendations. The specific recommendations are summarized in Table 9.

Table 9 Policy recommendations.
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Expand the boundaries of Government support and improve the details of the “Award Instead of Subsidy” Policy

In the current agricultural insurance market, the dynamic adjustment mechanism is crucial to maintaining the stability and development of the market. The government’s subsidy policy adjustments directly affect the decisions of insurance companies and farmers (Lu et al., 2024). When the government adjusts the level of subsidies, insurance companies will reassess the benefits and risks of providing specialty agricultural insurance products. If the subsidy increases, insurance companies may expand their business more aggressively by expanding the scope of coverage or lowering the premium threshold to attract more farmers to participate in the insurance; conversely, if the subsidy decreases, insurance companies may contract their business and raise premiums to ensure profitability (Feng and Bai, 2024).

Initiating Subsidies for Insurance Companies’ operational costs to stimulate innovation in niche agricultural insurance products

In countries such as Japan, Israel, and Brazil, governments subsidize the operating costs of insurance companies to incentivize them to lower premiums and increase risk coverage. In China, however, subsidies for agricultural insurance are mainly directed to farmers, due to weak social credit awareness and opaque costs for insurers. While this ensures the efficiency of subsidies, it lacks a mechanism to incentivize insurers to innovate, resulting in insurers lacking incentives to develop targeted products in the specialty agricultural insurance market, or even adopting a “test the waters and give up” or “exit at a loss” strategy, leading to market failure (Li et al., 2023). Research shows that shifting subsidies to insurance companies can effectively promote product innovation. Therefore, we should launch a pilot program of cost subsidies for insurance companies in the field of specialty crops as soon as possible. For products with a strong industry foundation, “index insurance” and “insurance + futures” market-oriented models should be promoted. For specialty crops that are more affected by price fluctuations, operating cost subsidies should be implemented and adjusted annually according to local conditions. At the same time, the Government should promote differentiated innovation among different insurance companies, integrate resources, and establish centralized agricultural insurance departments at the county and village levels, so as to reduce the promotion costs of insurance companies, improve the accuracy of services, reduce the pressure on government subsidies, and support the development of specialty agricultural operators.

Further improvement of the reinsurance mechanism to provide a safety net for insurance companies’ innovations

Distinct from traditional agriculture, the risk structure of specialty agricultural products is more complex, with diverse risk types and high levels of uncertainty. Additionally, region-specific and brand-oriented specialty agricultural products typically involve numerous participating farmers and exhibit converging risk structures. This complexity exposes insurance companies to significant payout pressures due to external market shocks and major disasters. To address this, it is crucial to refine the operational details of agricultural insurance reinsurance mechanisms (Yu et al., 2024). Beyond the current reinsurance company operations, it is advisable to expand the scale of disaster relief and reinsurance funds for specialty agricultural insurance at the provincial and municipal levels. This could involve a government-funded initial investment, with a gradual exit over subsequent year. Such measures would establish a buffer and risk-sharing mechanism for concentrated risks faced by insurance companies, thus reducing the risk accumulation in specialty agricultural insurance operations. This approach would provide a safety net for insurance company innovations in agricultural insurance, thereby mitigating issues of insufficient supply willingness characterized by “trial and failure” scenarios.

Concluding remarks

In the agricultural sector product upgrading and structural optimization, accelerate the new agricultural industrialization, regional characteristics of agricultural products leading to become an important content of the promotion of high-quality development of agriculture, which puts forward a new task for the agricultural insurance services in the development of new agriculture. At this stage, there is a structural imbalance in China’s agricultural insurance market in the field of new agricultural products, and there is a lack of relevant agricultural insurance products, and a large number of new agricultural operators are “willing to be insured but not insured”. Therefore, this paper tries to fully integrate the risk characteristics, income structure and willingness to participate in insurance of new agricultural households in an evolutionary game analysis framework, comparing the impacts of the government’s two differentiated financial subsidy modes of “supplementing farmers” and “supplementing institutions” on the game strategies of farmers and insurance companies, with a view to exploring the impacts on the game strategies of farmers and insurance companies, with a view to exploring the impacts on the game strategies of new agricultural households and insurance companies. This study compares the impacts of the two differentiated financial subsidy modes of “supplementing farmers” and “supplementing institutions” on the game strategies of farmers and insurance companies, with a view to exploring the policy paths of prying the innovation of new agricultural insurance products and accelerating the “increase of products, improvement of standards, and expansion of agricultural insurance”.