# The Cooperation among Participants of Chinese Agricultural Machinery Insurance Based on Game Analysis

Authors: Ke Han; Yawei Wang
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

As the insurance for agricultural machinery features the attributes of quasi-public goods, its participants not only include farmers and insurance companies but also the government. The role played by the government will impose a direct impact on the development of the insurance for agricultural machinery. This paper sheds light on the behavior of the participants in agricultural machinery insurance from the aspects of economics and management. The game analysis is carried out on the participants to elaborate on the collaborative conditions for the long-term and stable development of agricultural machinery insurance. On this basis, the study provides policy recommendations on the role to be played by the government in agricultural machinery insurance.


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## Body

## 1. Behavioral Analysis on the Insurance Participants

The insured subjects of agricultural machinery insurance are farmers with agricultural machinery and farmers driving agricultural machinery. The agricultural machinery in agricultural machinery cooperatives (new agricultural businesses) is divided into two kinds. One kind is jointly funded by cooperatives, which are generally large machineries. The other is owned by farmers, which are uniformly dispatched by cooperatives during busy agricultural operations. So, the farmers in this paper not only refer to the owners and drivers of agricultural machinery but also include members in agricultural machinery cooperatives.In case that farmers are rational policy holders, there are two potential results after they take part in the insurance. First, they fail to obtain the insurance compensation without the occurrence of any accident. The other possible result is that they end up obtaining the compensation. Assuming that the input of farmers isI, the probability of failing to obtain compensation is p, and the income is R1; on the contrary, the probability is 1−p and the income is R2. In addition, δ refers to the premium, and q refers to the compensation [1].


### 1.1. Behavioral Analysis on whether Farmers Are Insured or Not

Based on the aforementioned assumption, the function of the expected profit for farmers isπ:
(1)π=pR1−δ+I+1−pR2−δ+I+q.The expected income is marked asR¯, R¯=pR1+1−pR2, the critical point of p is referred to as p∗, and the critical point of income is referred to as R∗. In case that p≤p∗, the result indicates that as the probability of obtaining the compensation grows higher, the farmers will be more likely to take out agricultural machinery insurance. Judging from the function of expected profit and that of expected income of farmers, it is found that the critical point to p∗ and R∗ must meet the following requirements:
(2)p∗R∗−δ+I+1−p∗R2−δ−I+q=0,p∗R∗+1−p∗R2=R¯.The result of solving Equation (1) is that p∗=R¯−δ−I+q/q. For farmers that take out the agricultural machinery insurance, their hope is to obtain the insurance compensation. In case they have a higher probability of getting insurance compensation, they are more likely to buy insurance. Correspondingly, when the probability of farmers of failing to obtain the compensation is p≤p∗=R¯−δ−I+q/q, farmers will sign up for the insurance. As can be seen from the equation p∗=R¯−δ−I+q/q, the factors that may impose an impact on farmers’ insurance are the premium δ and the insurance compensation q. If farmers are required to pay a lower level of premium but can obtain a higher level of insurance compensation, an increasing number of farmers will be more willing to buy insurance. The result consists with the farmers’ willingness to pay lower premium and obtain higher insurance compensation. Therefore, the condition for farmers to voluntarily get insured is p∗=R¯−δ−I+q/q≥p.


### 1.2. Behavioral Analysis on the Insurers

Against the backdrop of the market economy, profit-making is the primary goal of insurance companies. Whether the insurers develop insurance products for agricultural machinery depends on whether such products can lead to higher profits. Under circumstances of low profits and even losses, the insurers tend to be unwilling to provide corresponding insurance products. For instance, since the compulsory liability insurance for motor vehicle accidents features low premium and high risks, the insurance companies operating such business are encountered with severe losses and are unwilling to underwrite such insurance [2]. Therefore, the likelihood of obtaining profits holds key to insurance companies’ behavior of operating agricultural insurance.In case that all farmers are willing to take out agricultural machinery insurance and assuming that the operational and management cost of the insurers isk, then the function of the expected profit of the insurers is πc:
(3)πc=pδ−k+1−pδ−q−k.Assuming that the expected average profit of the insurers from the operation of agricultural machinery insurance isπ0, then the insurers will operate agricultural machinery insurance under the condition that πc≥π0. In addition, the insurers crave for higher premiums and lower insurance claims. Correspondingly, the condition for the insurers to be willing to operate agricultural machinery insurance is pδ−k+1−pδ−q−k≥π0.


### 1.3. Behavioral Analysis on whether the Government Provides Financial Subsidies to Agricultural Machinery Insurance

In case that the agricultural machinery insurance is not subject to the government subsidies, assume that the probability density function for farmers not receiving compensation isfp and that the distribution function is Fp. Accordingly, the average probability of farmers not receiving compensation subsequent to the application for agricultural machinery insurance is as follows:
(4)p¯=∫0p∗pfpdp∫0p∗fpdp=∫0p∗pfpdpFp∗.Partial derivative ofq is as follows:
(5)∂p¯∂q=p∗fp∗⋅∂p∗/∂q⋅FP∗−∂Fp∗/∂q⋅∫0p∗pfpdpFp∗2.Moreover,Fp∗ is a composite function of q, and thus,
(6)∂Fp∗∂q=∂Fp∗∂p∗⋅∂p∗∂q=fp∗∂p∗∂q,(7)∂p∗∂q=−R¯−Iq2.Substitute Equations (6) and (7) into Equation (5):
(8)∂p¯∂q=−R¯−Iq2⋅fp∗Fp∗2⋅p∗Fp∗−∫0p∗pfpdp.Sincefp∗/Fp∗2⋅p∗Fp∗−∫0p∗pfpdp>0, R¯>I, then ∂p¯/∂q<0. In other words, the average probability p¯ of farmers not receiving compensation is inversely proportional to the insurance compensation q.Judging from the research findings, as the insurance loss ratioq grows larger, the average probability p¯ of farmers not receiving compensation will grow smaller, whereas the probability 1−p¯ of receiving compensation will grow larger. In this case, farmers are more willing to sign up for the insurance. However, in the meantime, due to the heightened risks of agricultural machinery accidents, the complexity of operational environment for such insurance, and the particularity of accident risks, the loss rates of agricultural machinery insurance are surging towards an increasingly higher level. Under circumstances of such a high loss ratio and coupled with the moral hazard and adverse selection of agricultural machinery insurance, the insurers can only increase the premium so that they can operate agricultural machinery insurance without loss.Constrained by the capacity to pay, farmers lack effective purchasing power when encountered with higher premiums, thus leading to insufficient real demand. To convert the potential demand of farmers into real demand, reducing the premium would be the most straightforward method. However, such reduction of premium under high risks will lead to severe losses for the insurers, and consequently, they might even withdraw from the market of the agricultural machinery insurance [3]. To address this issue, the government needs to provide financial subsidies to the low-premium insurance products so as to attain an appropriate point of premium, which allows the premium to be affordable for farmers on the one hand, while ensuring the insurers to operate without losses on the other hand. This point is recognized as the point of equilibrium to meet the interests of both farmers and insurers.Assume that the total amount of subsidies by the government to agricultural machinery insurance isU0, in which the amount of subsidies for the premium to be paid by farmers is λU0 (0<λ<1), and the subsidy to the insurers is 1−λU0. In addition, the level of subsidies to agricultural machinery insurance is faced with the constraints of the financial capacity of the government. Set the maximum amount of government subsidies to agricultural machinery insurance as U; then, U0≤U.To sum up, to ensure effective tripartite cooperation in the market of agricultural machinery insurance, the following three conditions need to be met:(9)p∗=R¯−δ−λU0−I+qq≥p,πc=pδ−k+1−λU0+1−pδ−k+1−λU0−q≥π0,U0≤U.The equation indicates that under the premise of adequate financial capacity of the government, subsidies are provided to agricultural machinery insurance. On the one hand, such subsidies serve as incentives for farmers to voluntarily take out the agricultural machinery insurance. On the other hand, the insurers will be able to earn certain profits during the operation of agricultural machinery insurance. In this way, we are able to facilitate the sound development of agricultural machinery insurance in China and leverage its role of reducing the incurred risks and losses of farmers. Moreover, government subsidies are expected to guarantee the rights and interests of farmers, improve the market share of insurance companies in the rural insurance market, and manifest the role played by the government in terms of public function.


## 1.1. Behavioral Analysis on whether Farmers Are Insured or Not

Based on the aforementioned assumption, the function of the expected profit for farmers isπ:
(1)π=pR1−δ+I+1−pR2−δ+I+q.The expected income is marked asR¯, R¯=pR1+1−pR2, the critical point of p is referred to as p∗, and the critical point of income is referred to as R∗. In case that p≤p∗, the result indicates that as the probability of obtaining the compensation grows higher, the farmers will be more likely to take out agricultural machinery insurance. Judging from the function of expected profit and that of expected income of farmers, it is found that the critical point to p∗ and R∗ must meet the following requirements:
(2)p∗R∗−δ+I+1−p∗R2−δ−I+q=0,p∗R∗+1−p∗R2=R¯.The result of solving Equation (1) is that p∗=R¯−δ−I+q/q. For farmers that take out the agricultural machinery insurance, their hope is to obtain the insurance compensation. In case they have a higher probability of getting insurance compensation, they are more likely to buy insurance. Correspondingly, when the probability of farmers of failing to obtain the compensation is p≤p∗=R¯−δ−I+q/q, farmers will sign up for the insurance. As can be seen from the equation p∗=R¯−δ−I+q/q, the factors that may impose an impact on farmers’ insurance are the premium δ and the insurance compensation q. If farmers are required to pay a lower level of premium but can obtain a higher level of insurance compensation, an increasing number of farmers will be more willing to buy insurance. The result consists with the farmers’ willingness to pay lower premium and obtain higher insurance compensation. Therefore, the condition for farmers to voluntarily get insured is p∗=R¯−δ−I+q/q≥p.


## 1.2. Behavioral Analysis on the Insurers

Against the backdrop of the market economy, profit-making is the primary goal of insurance companies. Whether the insurers develop insurance products for agricultural machinery depends on whether such products can lead to higher profits. Under circumstances of low profits and even losses, the insurers tend to be unwilling to provide corresponding insurance products. For instance, since the compulsory liability insurance for motor vehicle accidents features low premium and high risks, the insurance companies operating such business are encountered with severe losses and are unwilling to underwrite such insurance [2]. Therefore, the likelihood of obtaining profits holds key to insurance companies’ behavior of operating agricultural insurance.In case that all farmers are willing to take out agricultural machinery insurance and assuming that the operational and management cost of the insurers isk, then the function of the expected profit of the insurers is πc:
(3)πc=pδ−k+1−pδ−q−k.Assuming that the expected average profit of the insurers from the operation of agricultural machinery insurance isπ0, then the insurers will operate agricultural machinery insurance under the condition that πc≥π0. In addition, the insurers crave for higher premiums and lower insurance claims. Correspondingly, the condition for the insurers to be willing to operate agricultural machinery insurance is pδ−k+1−pδ−q−k≥π0.


## 1.3. Behavioral Analysis on whether the Government Provides Financial Subsidies to Agricultural Machinery Insurance

In case that the agricultural machinery insurance is not subject to the government subsidies, assume that the probability density function for farmers not receiving compensation isfp and that the distribution function is Fp. Accordingly, the average probability of farmers not receiving compensation subsequent to the application for agricultural machinery insurance is as follows:
(4)p¯=∫0p∗pfpdp∫0p∗fpdp=∫0p∗pfpdpFp∗.Partial derivative ofq is as follows:
(5)∂p¯∂q=p∗fp∗⋅∂p∗/∂q⋅FP∗−∂Fp∗/∂q⋅∫0p∗pfpdpFp∗2.Moreover,Fp∗ is a composite function of q, and thus,
(6)∂Fp∗∂q=∂Fp∗∂p∗⋅∂p∗∂q=fp∗∂p∗∂q,(7)∂p∗∂q=−R¯−Iq2.Substitute Equations (6) and (7) into Equation (5):
(8)∂p¯∂q=−R¯−Iq2⋅fp∗Fp∗2⋅p∗Fp∗−∫0p∗pfpdp.Sincefp∗/Fp∗2⋅p∗Fp∗−∫0p∗pfpdp>0, R¯>I, then ∂p¯/∂q<0. In other words, the average probability p¯ of farmers not receiving compensation is inversely proportional to the insurance compensation q.Judging from the research findings, as the insurance loss ratioq grows larger, the average probability p¯ of farmers not receiving compensation will grow smaller, whereas the probability 1−p¯ of receiving compensation will grow larger. In this case, farmers are more willing to sign up for the insurance. However, in the meantime, due to the heightened risks of agricultural machinery accidents, the complexity of operational environment for such insurance, and the particularity of accident risks, the loss rates of agricultural machinery insurance are surging towards an increasingly higher level. Under circumstances of such a high loss ratio and coupled with the moral hazard and adverse selection of agricultural machinery insurance, the insurers can only increase the premium so that they can operate agricultural machinery insurance without loss.Constrained by the capacity to pay, farmers lack effective purchasing power when encountered with higher premiums, thus leading to insufficient real demand. To convert the potential demand of farmers into real demand, reducing the premium would be the most straightforward method. However, such reduction of premium under high risks will lead to severe losses for the insurers, and consequently, they might even withdraw from the market of the agricultural machinery insurance [3]. To address this issue, the government needs to provide financial subsidies to the low-premium insurance products so as to attain an appropriate point of premium, which allows the premium to be affordable for farmers on the one hand, while ensuring the insurers to operate without losses on the other hand. This point is recognized as the point of equilibrium to meet the interests of both farmers and insurers.Assume that the total amount of subsidies by the government to agricultural machinery insurance isU0, in which the amount of subsidies for the premium to be paid by farmers is λU0 (0<λ<1), and the subsidy to the insurers is 1−λU0. In addition, the level of subsidies to agricultural machinery insurance is faced with the constraints of the financial capacity of the government. Set the maximum amount of government subsidies to agricultural machinery insurance as U; then, U0≤U.To sum up, to ensure effective tripartite cooperation in the market of agricultural machinery insurance, the following three conditions need to be met:(9)p∗=R¯−δ−λU0−I+qq≥p,πc=pδ−k+1−λU0+1−pδ−k+1−λU0−q≥π0,U0≤U.The equation indicates that under the premise of adequate financial capacity of the government, subsidies are provided to agricultural machinery insurance. On the one hand, such subsidies serve as incentives for farmers to voluntarily take out the agricultural machinery insurance. On the other hand, the insurers will be able to earn certain profits during the operation of agricultural machinery insurance. In this way, we are able to facilitate the sound development of agricultural machinery insurance in China and leverage its role of reducing the incurred risks and losses of farmers. Moreover, government subsidies are expected to guarantee the rights and interests of farmers, improve the market share of insurance companies in the rural insurance market, and manifest the role played by the government in terms of public function.


## 2. Game Analysis on Effective Tripartite Cooperation

Based on the analysis of subject behavior, this part will analyze the binary game of government, farmers, and insurers.


### 2.1. Game Analysis on the Government and Insurers

Assume the behavior of government is (provision of subsidies, absence of subsidies) and insurers is (operation, nonoperation). Based on the analysis of subject behavior, the gaming behavior between the government and insurers [4] is specified in Table 1.Table 1
Payment matrix of the government and insurers.
GovernmentInsurersOperationNonoperationProvision of subsidies−1−λU0, δ−1−pq−k+1−λU00, 0Absence of subsidies0,δ−1−pq−k0, 0Judging from the results of Table1, in case that the government supports the insurers of agricultural machinery insurance by subsidizing them, these insurers are expected to obtain subsidies of 1−λU0. When the losses of insurers are inferior to the government subsidies, they tend to be more willing to operate agricultural machinery insurance. Under such circumstances, the expected profit of insurers is πc=pδ−k+1−λu0+1−pδ−k+1−λU0−q. When πc>π0, it is the most effective method to encourage insurers to operate agricultural machinery insurance. At this time, the income of insurers is δ−1−pq−k+1−λU0, whereas the national financial expenditure is 1−λU0. If the government supports the operation of agricultural machinery insurance and yet the insurers determine not to underwrite such insurance, neither parties will encounter loss nor profit. If the government provides no subsidies at all, the insurers risk facing losses while operating agricultural machinery insurance. If the government provides no subsidies and the insurers do not carry out related business, neither party will encounter any gain nor loss.Based on the above analysis, the insurers are able to make profits only with the support from the government, which provides them with the incentives for more proactively operating agricultural machinery insurance. For the government, the optimal level of profits is 0, which suggests that it is more cost effective not to provide any support to the agricultural machinery insurance. However, since the agricultural machinery insurance is a quasi-public goods with positive externalities, its economic and social benefits shall be fully taken into account during assessment of such insurance policy. In the near term, the government’s financial strength is likely to be weakened to some extent due to subsidies to the agricultural machinery insurance. Nevertheless, in the long run, the reduced financial revenue will be converted into public welfare, enabling the farmers to obtain benefits thanks to the preferential agricultural policy implemented by the state. If the government does not provide any subsidies and the insurers do not carry out any agricultural machinery insurance, once an accident related to agricultural machinery occurs, it will incur losses of life and property for farmers, dragging them back to poverty. This scenario is likely to heighten the uncertainties of rural social life and is harmful to the harmonious development of rural society [5]. Hence, from the aspect of social welfare, the optimal game combination between the government and insurers is that the government provides subsidies, whereas the insurers operate the agricultural machinery insurance.


### 2.2. Game Analysis between Farmers and the Government

On the premise that the insurers operate agricultural machinery insurance, the behavior of government is (provision of subsidies, absence of subsidies), and farmer is (insured, uninsured); the game behavior between farmers and the government is specified in Table2.Table 2
Payment matrix of farmers and the government.
FarmerGovernmentProvision of subsidiesAbsence of subsidiesInsured1−pq−δ−l+λU0, −λU01−pq−δ−l, 0Uninsured0, 00, 0If the government provides subsidies to the premium and that farmers are willing to insure, the income of farmers subsequent to the occurrence of accidents is1−pq−δ−l+λU0. When the condition of p∗=R¯−δ−λ0U−I+q/q≥p is met, farmers will reach the maximum positive income, whereas the government’s income is negative. However, the positive externality of the agricultural machinery insurance allows the entire society to enjoy the social benefits due to the spillover effects. In case that the government does not provide subsidies to the premium, the income of farmers is 1−pq−δ−l, whereas the income of the government is 0. In case that farmers are uninsured and the government does not provide subsidies, neither parties will be subject to gains nor losses. Therefore, in the case of underwriting by the insurers, the optimal game combination between farmers and the government is that farmers get insured and that the government provides subsidies to the premium.


### 2.3. Game Analysis of Farmers and the Insurers

Game analysis of farmers and the insurers includes government provides no subsidies at all and government provides subsidies. Whether the government subsidizes, the behavior of insurers is (operation, nonoperation), and farmer is (insured, uninsured).


#### 2.3.1. Government Provides No Subsidies at All

If the premium income exceeds the amount of the insurance compensation, the insurers are able to obtain higher profits, whereas the utility of farmers will diminish and vice versa [6]. Assuming that the losses incurred by agricultural machinery accidents are l, then the payment matrix of farmers and insurers is specified in Table 3.Table 3
Payment matrix of the government and farmers.
FarmersInsurersOperationNonoperationInsured1−pq−δ−l, δ−1−pq−k−l, 0Uninsured−l, 0−l, 0Judging from Table3, if farmers apply for the agricultural machinery insurance and the insurers are willing to underwrite such insurance, when the accident occurs, farmers are able to obtain income 1−pq−δ−l from the insurance, whereas the insurers shall compensate farmers for the loss 1−pq, and their income is δ−1−pq−k. Assuming that farmers are willing to take out the agricultural machinery insurance and yet that insurers are unwilling to underwrite such insurance, then when the risk occurs, farmers have to bear the risks themselves because they cannot afford the insurance. Their loss is l, and insurers can gain no profits at all. Suppose that the insurers operate the agricultural machinery insurance and the farmers do not take out the insurance. When an accident occurs, the loss of farmers is l and the income of the insurers is 0. Assuming that farmers do not get insured and that insurers do not operate agricultural machinery insurance, then when an accident occurs, the loss of farmers is l, whereas the insurers will have no income.During the gaming procedure, the primary goal of decision-making by each participant is to maximize their own income. However, the benefits of participants are not only merely related to their respective behavior of decision-making but also dependent on the behavioral strategies adopted by other players. Therefore, each participant must take into account the decisions made by other participants when deciding over their own behaviors. For farmers, when the insurers operate agricultural machinery insurance, no matter whether1−pq−δ−l exceeds zero, the insurance compensation can cover their losses to a certain extent. Therefore, the optimal strategy for farmers is to “get insured.” On the contrary, if the insurers do not operate the agricultural machinery insurance, farmers will be faced with the same losses whether they are insured or not. For the insurers, when farmers apply for the agricultural machinery insurance, due to the low premium and high risk of the agricultural machinery insurance, losses may occur while operating the agricultural machinery insurance. In other words, when δ−1−pq−k<0, the optimal strategy is “nonoperation.” If farmers do not take out the agricultural machinery insurance, the insurers will be faced with no losses at all no matter whether it underwrites the insurance or not. Therefore, the result of the game between farmers and insurers is that farmers are able to apply for the agricultural machinery insurance and that the insurers do not underwrite such insurance. However, in fact, the insurers should not only merely aim at maximizing their own profits but also assume social responsibilities and achieve social benefits. Therefore, certain insurers are unwilling to operate the compulsory liability insurance for motor vehicle accidents, but they dare not refuse to underwrite such insurance and often adopt the form of bundled sales.


#### 2.3.2. Government Provides Subsidies

Under circumstances when the government provides subsidies, the payment matrix between farmers and insurers is specified in Table4.Table 4
Payment matrix of the government and farmers.
FarmersInsurersOperationNonoperationInsured1−pq−δ−l+λU0, δ−1−pq−k+1−λU0−l, 0Uninsured−l, 0−l, 0Judging from Table4, when farmers take out the agricultural machinery insurance and that insurers underwrite such insurance, and in case that the agricultural machinery accident occurs, then the income of farmers is 1−pq−δ−l+λU0, whereas the income of insurers is δ−1−pq−k+1−λU0, and both of which exceed zero. In case that farmers are willing to sign up for the agricultural machinery insurance and that insurers operate this business, when an agricultural machinery accident occurs, then the loss of farmers is l. In case that farmers and insurers do not take out the agricultural machinery insurance, subsequent to the occurrence of the agricultural machinery accident, then the loss of farmers is l and the income of insurers is 0. It can be concluded that the optimal combination would be that the government provides subsidies, farmers are willing to take out the insurance, and that insurers are willing to underwrite such insurance [7]. At this time, the benefits of both sides would reach the maximum level and that the tripartite cooperation is effective.To sum up, the optimal strategy of tripartite cooperation is as follows: farmers sign up for the agricultural machinery insurance, insurers operate such insurance, and that the government provides support to the insurance [8].


## 2.1. Game Analysis on the Government and Insurers

Assume the behavior of government is (provision of subsidies, absence of subsidies) and insurers is (operation, nonoperation). Based on the analysis of subject behavior, the gaming behavior between the government and insurers [4] is specified in Table 1.Table 1
Payment matrix of the government and insurers.
GovernmentInsurersOperationNonoperationProvision of subsidies−1−λU0, δ−1−pq−k+1−λU00, 0Absence of subsidies0,δ−1−pq−k0, 0Judging from the results of Table1, in case that the government supports the insurers of agricultural machinery insurance by subsidizing them, these insurers are expected to obtain subsidies of 1−λU0. When the losses of insurers are inferior to the government subsidies, they tend to be more willing to operate agricultural machinery insurance. Under such circumstances, the expected profit of insurers is πc=pδ−k+1−λu0+1−pδ−k+1−λU0−q. When πc>π0, it is the most effective method to encourage insurers to operate agricultural machinery insurance. At this time, the income of insurers is δ−1−pq−k+1−λU0, whereas the national financial expenditure is 1−λU0. If the government supports the operation of agricultural machinery insurance and yet the insurers determine not to underwrite such insurance, neither parties will encounter loss nor profit. If the government provides no subsidies at all, the insurers risk facing losses while operating agricultural machinery insurance. If the government provides no subsidies and the insurers do not carry out related business, neither party will encounter any gain nor loss.Based on the above analysis, the insurers are able to make profits only with the support from the government, which provides them with the incentives for more proactively operating agricultural machinery insurance. For the government, the optimal level of profits is 0, which suggests that it is more cost effective not to provide any support to the agricultural machinery insurance. However, since the agricultural machinery insurance is a quasi-public goods with positive externalities, its economic and social benefits shall be fully taken into account during assessment of such insurance policy. In the near term, the government’s financial strength is likely to be weakened to some extent due to subsidies to the agricultural machinery insurance. Nevertheless, in the long run, the reduced financial revenue will be converted into public welfare, enabling the farmers to obtain benefits thanks to the preferential agricultural policy implemented by the state. If the government does not provide any subsidies and the insurers do not carry out any agricultural machinery insurance, once an accident related to agricultural machinery occurs, it will incur losses of life and property for farmers, dragging them back to poverty. This scenario is likely to heighten the uncertainties of rural social life and is harmful to the harmonious development of rural society [5]. Hence, from the aspect of social welfare, the optimal game combination between the government and insurers is that the government provides subsidies, whereas the insurers operate the agricultural machinery insurance.


## 2.2. Game Analysis between Farmers and the Government

On the premise that the insurers operate agricultural machinery insurance, the behavior of government is (provision of subsidies, absence of subsidies), and farmer is (insured, uninsured); the game behavior between farmers and the government is specified in Table2.Table 2
Payment matrix of farmers and the government.
FarmerGovernmentProvision of subsidiesAbsence of subsidiesInsured1−pq−δ−l+λU0, −λU01−pq−δ−l, 0Uninsured0, 00, 0If the government provides subsidies to the premium and that farmers are willing to insure, the income of farmers subsequent to the occurrence of accidents is1−pq−δ−l+λU0. When the condition of p∗=R¯−δ−λ0U−I+q/q≥p is met, farmers will reach the maximum positive income, whereas the government’s income is negative. However, the positive externality of the agricultural machinery insurance allows the entire society to enjoy the social benefits due to the spillover effects. In case that the government does not provide subsidies to the premium, the income of farmers is 1−pq−δ−l, whereas the income of the government is 0. In case that farmers are uninsured and the government does not provide subsidies, neither parties will be subject to gains nor losses. Therefore, in the case of underwriting by the insurers, the optimal game combination between farmers and the government is that farmers get insured and that the government provides subsidies to the premium.


## 2.3. Game Analysis of Farmers and the Insurers

Game analysis of farmers and the insurers includes government provides no subsidies at all and government provides subsidies. Whether the government subsidizes, the behavior of insurers is (operation, nonoperation), and farmer is (insured, uninsured).


### 2.3.1. Government Provides No Subsidies at All

If the premium income exceeds the amount of the insurance compensation, the insurers are able to obtain higher profits, whereas the utility of farmers will diminish and vice versa [6]. Assuming that the losses incurred by agricultural machinery accidents are l, then the payment matrix of farmers and insurers is specified in Table 3.Table 3
Payment matrix of the government and farmers.
FarmersInsurersOperationNonoperationInsured1−pq−δ−l, δ−1−pq−k−l, 0Uninsured−l, 0−l, 0Judging from Table3, if farmers apply for the agricultural machinery insurance and the insurers are willing to underwrite such insurance, when the accident occurs, farmers are able to obtain income 1−pq−δ−l from the insurance, whereas the insurers shall compensate farmers for the loss 1−pq, and their income is δ−1−pq−k. Assuming that farmers are willing to take out the agricultural machinery insurance and yet that insurers are unwilling to underwrite such insurance, then when the risk occurs, farmers have to bear the risks themselves because they cannot afford the insurance. Their loss is l, and insurers can gain no profits at all. Suppose that the insurers operate the agricultural machinery insurance and the farmers do not take out the insurance. When an accident occurs, the loss of farmers is l and the income of the insurers is 0. Assuming that farmers do not get insured and that insurers do not operate agricultural machinery insurance, then when an accident occurs, the loss of farmers is l, whereas the insurers will have no income.During the gaming procedure, the primary goal of decision-making by each participant is to maximize their own income. However, the benefits of participants are not only merely related to their respective behavior of decision-making but also dependent on the behavioral strategies adopted by other players. Therefore, each participant must take into account the decisions made by other participants when deciding over their own behaviors. For farmers, when the insurers operate agricultural machinery insurance, no matter whether1−pq−δ−l exceeds zero, the insurance compensation can cover their losses to a certain extent. Therefore, the optimal strategy for farmers is to “get insured.” On the contrary, if the insurers do not operate the agricultural machinery insurance, farmers will be faced with the same losses whether they are insured or not. For the insurers, when farmers apply for the agricultural machinery insurance, due to the low premium and high risk of the agricultural machinery insurance, losses may occur while operating the agricultural machinery insurance. In other words, when δ−1−pq−k<0, the optimal strategy is “nonoperation.” If farmers do not take out the agricultural machinery insurance, the insurers will be faced with no losses at all no matter whether it underwrites the insurance or not. Therefore, the result of the game between farmers and insurers is that farmers are able to apply for the agricultural machinery insurance and that the insurers do not underwrite such insurance. However, in fact, the insurers should not only merely aim at maximizing their own profits but also assume social responsibilities and achieve social benefits. Therefore, certain insurers are unwilling to operate the compulsory liability insurance for motor vehicle accidents, but they dare not refuse to underwrite such insurance and often adopt the form of bundled sales.


### 2.3.2. Government Provides Subsidies

Under circumstances when the government provides subsidies, the payment matrix between farmers and insurers is specified in Table4.Table 4
Payment matrix of the government and farmers.
FarmersInsurersOperationNonoperationInsured1−pq−δ−l+λU0, δ−1−pq−k+1−λU0−l, 0Uninsured−l, 0−l, 0Judging from Table4, when farmers take out the agricultural machinery insurance and that insurers underwrite such insurance, and in case that the agricultural machinery accident occurs, then the income of farmers is 1−pq−δ−l+λU0, whereas the income of insurers is δ−1−pq−k+1−λU0, and both of which exceed zero. In case that farmers are willing to sign up for the agricultural machinery insurance and that insurers operate this business, when an agricultural machinery accident occurs, then the loss of farmers is l. In case that farmers and insurers do not take out the agricultural machinery insurance, subsequent to the occurrence of the agricultural machinery accident, then the loss of farmers is l and the income of insurers is 0. It can be concluded that the optimal combination would be that the government provides subsidies, farmers are willing to take out the insurance, and that insurers are willing to underwrite such insurance [7]. At this time, the benefits of both sides would reach the maximum level and that the tripartite cooperation is effective.To sum up, the optimal strategy of tripartite cooperation is as follows: farmers sign up for the agricultural machinery insurance, insurers operate such insurance, and that the government provides support to the insurance [8].


## 2.3.1. Government Provides No Subsidies at All

If the premium income exceeds the amount of the insurance compensation, the insurers are able to obtain higher profits, whereas the utility of farmers will diminish and vice versa [6]. Assuming that the losses incurred by agricultural machinery accidents are l, then the payment matrix of farmers and insurers is specified in Table 3.Table 3
Payment matrix of the government and farmers.
FarmersInsurersOperationNonoperationInsured1−pq−δ−l, δ−1−pq−k−l, 0Uninsured−l, 0−l, 0Judging from Table3, if farmers apply for the agricultural machinery insurance and the insurers are willing to underwrite such insurance, when the accident occurs, farmers are able to obtain income 1−pq−δ−l from the insurance, whereas the insurers shall compensate farmers for the loss 1−pq, and their income is δ−1−pq−k. Assuming that farmers are willing to take out the agricultural machinery insurance and yet that insurers are unwilling to underwrite such insurance, then when the risk occurs, farmers have to bear the risks themselves because they cannot afford the insurance. Their loss is l, and insurers can gain no profits at all. Suppose that the insurers operate the agricultural machinery insurance and the farmers do not take out the insurance. When an accident occurs, the loss of farmers is l and the income of the insurers is 0. Assuming that farmers do not get insured and that insurers do not operate agricultural machinery insurance, then when an accident occurs, the loss of farmers is l, whereas the insurers will have no income.During the gaming procedure, the primary goal of decision-making by each participant is to maximize their own income. However, the benefits of participants are not only merely related to their respective behavior of decision-making but also dependent on the behavioral strategies adopted by other players. Therefore, each participant must take into account the decisions made by other participants when deciding over their own behaviors. For farmers, when the insurers operate agricultural machinery insurance, no matter whether1−pq−δ−l exceeds zero, the insurance compensation can cover their losses to a certain extent. Therefore, the optimal strategy for farmers is to “get insured.” On the contrary, if the insurers do not operate the agricultural machinery insurance, farmers will be faced with the same losses whether they are insured or not. For the insurers, when farmers apply for the agricultural machinery insurance, due to the low premium and high risk of the agricultural machinery insurance, losses may occur while operating the agricultural machinery insurance. In other words, when δ−1−pq−k<0, the optimal strategy is “nonoperation.” If farmers do not take out the agricultural machinery insurance, the insurers will be faced with no losses at all no matter whether it underwrites the insurance or not. Therefore, the result of the game between farmers and insurers is that farmers are able to apply for the agricultural machinery insurance and that the insurers do not underwrite such insurance. However, in fact, the insurers should not only merely aim at maximizing their own profits but also assume social responsibilities and achieve social benefits. Therefore, certain insurers are unwilling to operate the compulsory liability insurance for motor vehicle accidents, but they dare not refuse to underwrite such insurance and often adopt the form of bundled sales.


## 2.3.2. Government Provides Subsidies

Under circumstances when the government provides subsidies, the payment matrix between farmers and insurers is specified in Table4.Table 4
Payment matrix of the government and farmers.
FarmersInsurersOperationNonoperationInsured1−pq−δ−l+λU0, δ−1−pq−k+1−λU0−l, 0Uninsured−l, 0−l, 0Judging from Table4, when farmers take out the agricultural machinery insurance and that insurers underwrite such insurance, and in case that the agricultural machinery accident occurs, then the income of farmers is 1−pq−δ−l+λU0, whereas the income of insurers is δ−1−pq−k+1−λU0, and both of which exceed zero. In case that farmers are willing to sign up for the agricultural machinery insurance and that insurers operate this business, when an agricultural machinery accident occurs, then the loss of farmers is l. In case that farmers and insurers do not take out the agricultural machinery insurance, subsequent to the occurrence of the agricultural machinery accident, then the loss of farmers is l and the income of insurers is 0. It can be concluded that the optimal combination would be that the government provides subsidies, farmers are willing to take out the insurance, and that insurers are willing to underwrite such insurance [7]. At this time, the benefits of both sides would reach the maximum level and that the tripartite cooperation is effective.To sum up, the optimal strategy of tripartite cooperation is as follows: farmers sign up for the agricultural machinery insurance, insurers operate such insurance, and that the government provides support to the insurance [8].


## 3. Construction of the Mechanism of Tripartite Cooperation among Participants

The game among subjects to the agricultural machinery insurance is not a one-time game, but a repeated one in the long run. The optimal scenario is to realize the sound operation of agricultural machinery insurance; namely, the government properly provides subsidies, and insurers are willing to operate such insurance with certain profits, whereas farmers pay a certain premium and are willing to take out the insurance. By adopting the theory of repeated game, this paper elaborates on the multistage gaming procedures among varying subjects to the agricultural machinery insurance, so as to identify the conditions for building a long-term and stable cooperative relationship among different subjects [9].


### 3.1. Repeated Game

Repeated game refers to the multistage dynamic game with the same structure, and each game is referred to as the stage game [10]. This paper introduces the game model of the repeated “prisoner’s dilemma” to analyze the gaming procedure of the agricultural machinery insurance. Assume that the players mired in the prisoner’s dilemma are faced with two options, namely, cooperation or deception, and each player makes his respective choice separately. If aiji,j=1,2 represents the income of participant one under varying combinations of strategies, biji,j=1,2 represents the income of participant two under varying combinations of strategies. The income of participants consists with the following relationship:
(10)a21>a11>a22>a12,b12>b11>b22>b21.Judging from Table5, the game of the “prisoner’s dilemma” features a unique Nash equilibrium of pure strategy (deception, deception), and the benefits of both sides are (a22, b22). However, the Nash equilibrium is Pareto invalid given that the return of strategy combination (cooperation, cooperation) is (a11, b11), which exceeds (a22, b22). However, the strategy of (cooperation, cooperation) is not the Nash equilibrium of the game and will not be adopted in a single-stage game.Table 5
Game matrix of “prisoner’s dilemma”.
Participant twoCooperationDeceptionParticipant oneCooperationa11, b11a12, b12Deceptiona21, b21a22, b22The game among the varying participants to the agricultural machinery insurance conforms to the game model on the “prisoner’s dilemma.” In case that both participants adopt the strategy of “cooperation” and carefully operate the agricultural machinery insurance, then the market of such insurance is expected to grow healthily, and all participants are able to obtain higher benefits. However, if one participant opts for the behavior of “deception,” then his income will be relatively high in the short term, whereas the income of the other party that honestly operates the agricultural machinery insurance will encounter greater losses. If both parties distrust each other and consequently adopt the behaviors of “deception,” then the market conditions are likely to deteriorate for the agricultural machinery insurance, and neither side will earn much income. Since the Nash equilibrium of the model is (deception, deception), which is consistent with the overall optimal result, adverse selection takes place in the market of operation. Therefore, it is necessary to analyze the procedure of repeated game among participants to the agricultural machinery insurance, so as to identify the cooperation and incentive mechanism for the long-term and stable development of the agricultural machinery insurance [11].Assuming that in the infinitely repeated game among subjects to the agricultural machinery insurance, the results of thet−1 stage game can be observed from the outset of each t stage. In other words, all participants are aware of their opponent’s strategy over the past stages of the game. In the meantime, the total income of participants is the sum of the discounted values of income obtained from the game throughout all stages. Let the discount factor be δ0<δ<1, which is the discount coefficient of the income generated from the game of agricultural machinery insurance at each stage to the income generated from the game of the previous stage. Suppose the discount factor δ of both sides to the game is equivalent to each other.


### 3.2. Strategy for the Repeated Game

During the repeated game of the agricultural machinery insurance, participants can decide upon their respective choice at this stage based on the previous game behavior of both parties. Participants usually adopt the trigger strategy; namely, if the other party constantly opts for the strategy of “cooperation,” the participant will continue to opt for the “cooperation” as well. Once the other party chooses to “cheat” at a certain stage, then the player will be triggered to cheat for a later period of time, or even forever [12]. Two typical trigger strategies are the grim strategy and the tit for tat strategy.


#### 3.2.1. Grim Strategy

The grim strategy indicates that both sides of the game opt for the “cooperation” during the first stage, and in staget>1, if the results of all previous t−1 stages are (cooperation, cooperation), then the participant would choose “cooperation”; otherwise, the behavior of “deception” would be selected.Assuming that participant one first takes the opportunistic approach during thet stage of the game and opts for “deception” instead of “cooperation,” then he is able to obtain 5 units of benefits during this stage. In the meantime, given that participant one opts for “deception,” participant two will then constantly choose to “cheat” so as to punish participant one. Therefore, the income of participant one during each subsequent stage would be one unit. In this case, the total income obtained by participant one is as follows:
(11)a21+a22δ+a22δ2+⋯a22δn+⋯=a21+a22δ1−δ.Assuming that participant one constantly chooses “cooperation” during the game, his income during stage T would be 4 units. Moreover, participant two will constantly opt for “cooperation,” and thus, the total income of participant one is as follows:(12)a11+a11δ+a11δ2+⋯+a11δn+⋯=a111−δ.Under circumstances of the grim strategy, once participant one chooses to “cheat,” participant two will choose to punish participant one permanently. At this time, the total income obtained by participant one will be inferior to the benefits brought along by the cooperation, namely,(13)a21+a22δ1−δ≤a111−δ.By solving this inequality, the following result is obtained:(14)δ≥a21−a11a21−a22.In other words, when the discount factor isδ≥a21−a11/a21−a22, the grim strategy is the subgame Nash equilibrium of the repeated game of the agricultural machinery insurance.


#### 3.2.2. Tit for Tat Strategy

The tit for tat strategy indicates that both parties opt for “cooperation” during the first stage, whereas the strategy selected in any stage oft>1 is the same as the strategy adopted by the other party in t−1 stage. The primary attribute of such strategy is that if participant one deceives participant two, then participant two will resume cooperation only after he also manages to deceive participant one once [11].Assume that during thet stage of the game of agricultural machinery insurance, participant two opts for “cooperation,” whereas participant one opts for “deception.” During the t+1 stage, in case that participant one decides upon the strategy of “cooperation,” then participant two will take “deceptive” measures to retaliate. If participant one alternately opts for “cooperation” and “deception,” the total income is as follows:
(15)a21+a12δ+a21δ2+a12δ3+a21δ4+⋯+a21δ2n−2+a12δ2n−1+⋯=a21+a12δ1−δ2.If participant one constantly chooses “cooperation” during the game, then participant two will always choose “cooperation.” Hence, the total income of participant one is as follows:(16)a11+a11δ+a11δ2+⋯+a11δn+⋯=a111−δ.In the case of the tit for tat strategy, if participant one chooses to “cheat,” then participant two will impose certain punitive measures on participant one. At this time, the total income of participant one will also be less than that of both parties during cooperation, namely,(17)a21+a12δ1−δ2≤a111−δ.By solving Equation (17), the following result is obtained: δ≥a21−a11/a11−a12.In other words, when the discount factor isδ≥a21−a11/a11−a12, then the tit for tat strategy is the subgame Nash equilibrium of the repeated game of agricultural machinery insurance.Judging from the holistic analysis on the above two triggering strategies, within a certain range of discount factor, the related parties to the agricultural machinery insurance will gradually tend to cooperate with each other after repeated game. This is the result of each subject’s comprehensive assessment of both short-term and long-term benefits. When the long-term benefits obtained by each participant that has opted for “cooperation” are greater than the maximum benefits obtained by the short-term “deception,” each subject will naturally opt for the strategy of long-term cooperation.


#### 3.2.3. Stability of Long-Term Cooperation among Subjects of the Agricultural Machinery Insurance

The long-term cooperation among varying subjects of the agricultural machinery insurance mainly depends on the comparison of benefits derived from “cooperation” with those from “deception.” When the expected benefits obtained by each participant from cooperation are greater than those from deception, each participant will opt for cooperation in the long run, and the agricultural machinery insurance will experience health growth for a long time [13].Assuming thatu represents the difference between benefits from cooperation and those from deception, in case that both parties implement the “tit for tat strategy”, then the u of each party is as follows:
(18)u1=a111−δ−a21+a22δ1−δ=a21−a22δ−a21−a111−δ,(19)u2=a111−δ−a21+a12δ1−δ2=a11−a12δ−a21−a111−δ2.The two equations were differentiated with respect to the discount factorδ0<δ<1, and the following result is obtained; namely, u1′>0 and u2′>0. The result indicates that the difference between the benefits of “cooperation” and “deception” obtained by the participants is monotonically increasing with the discount factor δ. In other words, as the value of δ increases, there will be larger discrepancy between the long-term benefits from “cooperation” and those from “deception” as well as greater opportunity loss incurred by “deception.” Therefore, increasing δ is a vital measure of encouraging the cooperation among varying subjects of the agricultural machinery insurance.To establish a long-term mechanism of cooperation among subjects of the agricultural machinery insurance, the following measures are recommended. (1) We shall establish a punitive mechanism. A certain punitive system is advised to be set up under the leadership of the government, so as to increase the cost borne by participants not to cooperate, so that all subjects will find it hard to engage in illegal operation. We shall facilitate each subject to constantly opt for cooperation during the game of decision-making among cooperation, violation, and punishment, so as to avoid the risks incurred by opportunistic behaviors in the form of violation and adverse selection [14]. (2) We shall establish an information sharing mechanism. The government shall establish a platform of information sharing among varying subjects to increase information transparency and to promote timely exchange of information. The improvement of information symmetry among subjects will make it harder to commit violation. In the meantime, due to the higher expectation of future income, the discount factor will grow larger, and all subjects will pay closer attention to long-term benefits [15–17].


## 3.1. Repeated Game

Repeated game refers to the multistage dynamic game with the same structure, and each game is referred to as the stage game [10]. This paper introduces the game model of the repeated “prisoner’s dilemma” to analyze the gaming procedure of the agricultural machinery insurance. Assume that the players mired in the prisoner’s dilemma are faced with two options, namely, cooperation or deception, and each player makes his respective choice separately. If aiji,j=1,2 represents the income of participant one under varying combinations of strategies, biji,j=1,2 represents the income of participant two under varying combinations of strategies. The income of participants consists with the following relationship:
(10)a21>a11>a22>a12,b12>b11>b22>b21.Judging from Table5, the game of the “prisoner’s dilemma” features a unique Nash equilibrium of pure strategy (deception, deception), and the benefits of both sides are (a22, b22). However, the Nash equilibrium is Pareto invalid given that the return of strategy combination (cooperation, cooperation) is (a11, b11), which exceeds (a22, b22). However, the strategy of (cooperation, cooperation) is not the Nash equilibrium of the game and will not be adopted in a single-stage game.Table 5
Game matrix of “prisoner’s dilemma”.
Participant twoCooperationDeceptionParticipant oneCooperationa11, b11a12, b12Deceptiona21, b21a22, b22The game among the varying participants to the agricultural machinery insurance conforms to the game model on the “prisoner’s dilemma.” In case that both participants adopt the strategy of “cooperation” and carefully operate the agricultural machinery insurance, then the market of such insurance is expected to grow healthily, and all participants are able to obtain higher benefits. However, if one participant opts for the behavior of “deception,” then his income will be relatively high in the short term, whereas the income of the other party that honestly operates the agricultural machinery insurance will encounter greater losses. If both parties distrust each other and consequently adopt the behaviors of “deception,” then the market conditions are likely to deteriorate for the agricultural machinery insurance, and neither side will earn much income. Since the Nash equilibrium of the model is (deception, deception), which is consistent with the overall optimal result, adverse selection takes place in the market of operation. Therefore, it is necessary to analyze the procedure of repeated game among participants to the agricultural machinery insurance, so as to identify the cooperation and incentive mechanism for the long-term and stable development of the agricultural machinery insurance [11].Assuming that in the infinitely repeated game among subjects to the agricultural machinery insurance, the results of thet−1 stage game can be observed from the outset of each t stage. In other words, all participants are aware of their opponent’s strategy over the past stages of the game. In the meantime, the total income of participants is the sum of the discounted values of income obtained from the game throughout all stages. Let the discount factor be δ0<δ<1, which is the discount coefficient of the income generated from the game of agricultural machinery insurance at each stage to the income generated from the game of the previous stage. Suppose the discount factor δ of both sides to the game is equivalent to each other.


## 3.2. Strategy for the Repeated Game

During the repeated game of the agricultural machinery insurance, participants can decide upon their respective choice at this stage based on the previous game behavior of both parties. Participants usually adopt the trigger strategy; namely, if the other party constantly opts for the strategy of “cooperation,” the participant will continue to opt for the “cooperation” as well. Once the other party chooses to “cheat” at a certain stage, then the player will be triggered to cheat for a later period of time, or even forever [12]. Two typical trigger strategies are the grim strategy and the tit for tat strategy.


### 3.2.1. Grim Strategy

The grim strategy indicates that both sides of the game opt for the “cooperation” during the first stage, and in staget>1, if the results of all previous t−1 stages are (cooperation, cooperation), then the participant would choose “cooperation”; otherwise, the behavior of “deception” would be selected.Assuming that participant one first takes the opportunistic approach during thet stage of the game and opts for “deception” instead of “cooperation,” then he is able to obtain 5 units of benefits during this stage. In the meantime, given that participant one opts for “deception,” participant two will then constantly choose to “cheat” so as to punish participant one. Therefore, the income of participant one during each subsequent stage would be one unit. In this case, the total income obtained by participant one is as follows:
(11)a21+a22δ+a22δ2+⋯a22δn+⋯=a21+a22δ1−δ.Assuming that participant one constantly chooses “cooperation” during the game, his income during stage T would be 4 units. Moreover, participant two will constantly opt for “cooperation,” and thus, the total income of participant one is as follows:(12)a11+a11δ+a11δ2+⋯+a11δn+⋯=a111−δ.Under circumstances of the grim strategy, once participant one chooses to “cheat,” participant two will choose to punish participant one permanently. At this time, the total income obtained by participant one will be inferior to the benefits brought along by the cooperation, namely,(13)a21+a22δ1−δ≤a111−δ.By solving this inequality, the following result is obtained:(14)δ≥a21−a11a21−a22.In other words, when the discount factor isδ≥a21−a11/a21−a22, the grim strategy is the subgame Nash equilibrium of the repeated game of the agricultural machinery insurance.


### 3.2.2. Tit for Tat Strategy

The tit for tat strategy indicates that both parties opt for “cooperation” during the first stage, whereas the strategy selected in any stage oft>1 is the same as the strategy adopted by the other party in t−1 stage. The primary attribute of such strategy is that if participant one deceives participant two, then participant two will resume cooperation only after he also manages to deceive participant one once [11].Assume that during thet stage of the game of agricultural machinery insurance, participant two opts for “cooperation,” whereas participant one opts for “deception.” During the t+1 stage, in case that participant one decides upon the strategy of “cooperation,” then participant two will take “deceptive” measures to retaliate. If participant one alternately opts for “cooperation” and “deception,” the total income is as follows:
(15)a21+a12δ+a21δ2+a12δ3+a21δ4+⋯+a21δ2n−2+a12δ2n−1+⋯=a21+a12δ1−δ2.If participant one constantly chooses “cooperation” during the game, then participant two will always choose “cooperation.” Hence, the total income of participant one is as follows:(16)a11+a11δ+a11δ2+⋯+a11δn+⋯=a111−δ.In the case of the tit for tat strategy, if participant one chooses to “cheat,” then participant two will impose certain punitive measures on participant one. At this time, the total income of participant one will also be less than that of both parties during cooperation, namely,(17)a21+a12δ1−δ2≤a111−δ.By solving Equation (17), the following result is obtained: δ≥a21−a11/a11−a12.In other words, when the discount factor isδ≥a21−a11/a11−a12, then the tit for tat strategy is the subgame Nash equilibrium of the repeated game of agricultural machinery insurance.Judging from the holistic analysis on the above two triggering strategies, within a certain range of discount factor, the related parties to the agricultural machinery insurance will gradually tend to cooperate with each other after repeated game. This is the result of each subject’s comprehensive assessment of both short-term and long-term benefits. When the long-term benefits obtained by each participant that has opted for “cooperation” are greater than the maximum benefits obtained by the short-term “deception,” each subject will naturally opt for the strategy of long-term cooperation.


### 3.2.3. Stability of Long-Term Cooperation among Subjects of the Agricultural Machinery Insurance

The long-term cooperation among varying subjects of the agricultural machinery insurance mainly depends on the comparison of benefits derived from “cooperation” with those from “deception.” When the expected benefits obtained by each participant from cooperation are greater than those from deception, each participant will opt for cooperation in the long run, and the agricultural machinery insurance will experience health growth for a long time [13].Assuming thatu represents the difference between benefits from cooperation and those from deception, in case that both parties implement the “tit for tat strategy”, then the u of each party is as follows:
(18)u1=a111−δ−a21+a22δ1−δ=a21−a22δ−a21−a111−δ,(19)u2=a111−δ−a21+a12δ1−δ2=a11−a12δ−a21−a111−δ2.The two equations were differentiated with respect to the discount factorδ0<δ<1, and the following result is obtained; namely, u1′>0 and u2′>0. The result indicates that the difference between the benefits of “cooperation” and “deception” obtained by the participants is monotonically increasing with the discount factor δ. In other words, as the value of δ increases, there will be larger discrepancy between the long-term benefits from “cooperation” and those from “deception” as well as greater opportunity loss incurred by “deception.” Therefore, increasing δ is a vital measure of encouraging the cooperation among varying subjects of the agricultural machinery insurance.To establish a long-term mechanism of cooperation among subjects of the agricultural machinery insurance, the following measures are recommended. (1) We shall establish a punitive mechanism. A certain punitive system is advised to be set up under the leadership of the government, so as to increase the cost borne by participants not to cooperate, so that all subjects will find it hard to engage in illegal operation. We shall facilitate each subject to constantly opt for cooperation during the game of decision-making among cooperation, violation, and punishment, so as to avoid the risks incurred by opportunistic behaviors in the form of violation and adverse selection [14]. (2) We shall establish an information sharing mechanism. The government shall establish a platform of information sharing among varying subjects to increase information transparency and to promote timely exchange of information. The improvement of information symmetry among subjects will make it harder to commit violation. In the meantime, due to the higher expectation of future income, the discount factor will grow larger, and all subjects will pay closer attention to long-term benefits [15–17].


## 3.2.1. Grim Strategy

The grim strategy indicates that both sides of the game opt for the “cooperation” during the first stage, and in staget>1, if the results of all previous t−1 stages are (cooperation, cooperation), then the participant would choose “cooperation”; otherwise, the behavior of “deception” would be selected.Assuming that participant one first takes the opportunistic approach during thet stage of the game and opts for “deception” instead of “cooperation,” then he is able to obtain 5 units of benefits during this stage. In the meantime, given that participant one opts for “deception,” participant two will then constantly choose to “cheat” so as to punish participant one. Therefore, the income of participant one during each subsequent stage would be one unit. In this case, the total income obtained by participant one is as follows:
(11)a21+a22δ+a22δ2+⋯a22δn+⋯=a21+a22δ1−δ.Assuming that participant one constantly chooses “cooperation” during the game, his income during stage T would be 4 units. Moreover, participant two will constantly opt for “cooperation,” and thus, the total income of participant one is as follows:(12)a11+a11δ+a11δ2+⋯+a11δn+⋯=a111−δ.Under circumstances of the grim strategy, once participant one chooses to “cheat,” participant two will choose to punish participant one permanently. At this time, the total income obtained by participant one will be inferior to the benefits brought along by the cooperation, namely,(13)a21+a22δ1−δ≤a111−δ.By solving this inequality, the following result is obtained:(14)δ≥a21−a11a21−a22.In other words, when the discount factor isδ≥a21−a11/a21−a22, the grim strategy is the subgame Nash equilibrium of the repeated game of the agricultural machinery insurance.


## 3.2.2. Tit for Tat Strategy

The tit for tat strategy indicates that both parties opt for “cooperation” during the first stage, whereas the strategy selected in any stage oft>1 is the same as the strategy adopted by the other party in t−1 stage. The primary attribute of such strategy is that if participant one deceives participant two, then participant two will resume cooperation only after he also manages to deceive participant one once [11].Assume that during thet stage of the game of agricultural machinery insurance, participant two opts for “cooperation,” whereas participant one opts for “deception.” During the t+1 stage, in case that participant one decides upon the strategy of “cooperation,” then participant two will take “deceptive” measures to retaliate. If participant one alternately opts for “cooperation” and “deception,” the total income is as follows:
(15)a21+a12δ+a21δ2+a12δ3+a21δ4+⋯+a21δ2n−2+a12δ2n−1+⋯=a21+a12δ1−δ2.If participant one constantly chooses “cooperation” during the game, then participant two will always choose “cooperation.” Hence, the total income of participant one is as follows:(16)a11+a11δ+a11δ2+⋯+a11δn+⋯=a111−δ.In the case of the tit for tat strategy, if participant one chooses to “cheat,” then participant two will impose certain punitive measures on participant one. At this time, the total income of participant one will also be less than that of both parties during cooperation, namely,(17)a21+a12δ1−δ2≤a111−δ.By solving Equation (17), the following result is obtained: δ≥a21−a11/a11−a12.In other words, when the discount factor isδ≥a21−a11/a11−a12, then the tit for tat strategy is the subgame Nash equilibrium of the repeated game of agricultural machinery insurance.Judging from the holistic analysis on the above two triggering strategies, within a certain range of discount factor, the related parties to the agricultural machinery insurance will gradually tend to cooperate with each other after repeated game. This is the result of each subject’s comprehensive assessment of both short-term and long-term benefits. When the long-term benefits obtained by each participant that has opted for “cooperation” are greater than the maximum benefits obtained by the short-term “deception,” each subject will naturally opt for the strategy of long-term cooperation.


## 3.2.3. Stability of Long-Term Cooperation among Subjects of the Agricultural Machinery Insurance

The long-term cooperation among varying subjects of the agricultural machinery insurance mainly depends on the comparison of benefits derived from “cooperation” with those from “deception.” When the expected benefits obtained by each participant from cooperation are greater than those from deception, each participant will opt for cooperation in the long run, and the agricultural machinery insurance will experience health growth for a long time [13].Assuming thatu represents the difference between benefits from cooperation and those from deception, in case that both parties implement the “tit for tat strategy”, then the u of each party is as follows:
(18)u1=a111−δ−a21+a22δ1−δ=a21−a22δ−a21−a111−δ,(19)u2=a111−δ−a21+a12δ1−δ2=a11−a12δ−a21−a111−δ2.The two equations were differentiated with respect to the discount factorδ0<δ<1, and the following result is obtained; namely, u1′>0 and u2′>0. The result indicates that the difference between the benefits of “cooperation” and “deception” obtained by the participants is monotonically increasing with the discount factor δ. In other words, as the value of δ increases, there will be larger discrepancy between the long-term benefits from “cooperation” and those from “deception” as well as greater opportunity loss incurred by “deception.” Therefore, increasing δ is a vital measure of encouraging the cooperation among varying subjects of the agricultural machinery insurance.To establish a long-term mechanism of cooperation among subjects of the agricultural machinery insurance, the following measures are recommended. (1) We shall establish a punitive mechanism. A certain punitive system is advised to be set up under the leadership of the government, so as to increase the cost borne by participants not to cooperate, so that all subjects will find it hard to engage in illegal operation. We shall facilitate each subject to constantly opt for cooperation during the game of decision-making among cooperation, violation, and punishment, so as to avoid the risks incurred by opportunistic behaviors in the form of violation and adverse selection [14]. (2) We shall establish an information sharing mechanism. The government shall establish a platform of information sharing among varying subjects to increase information transparency and to promote timely exchange of information. The improvement of information symmetry among subjects will make it harder to commit violation. In the meantime, due to the higher expectation of future income, the discount factor will grow larger, and all subjects will pay closer attention to long-term benefits [15–17].


## 4. Role of the Government during the Cooperation of Participants

Due to the nature of agricultural machinery insurance as quasi-public goods, the government has a vital role to play in agricultural machinery insurance [18]. Based on the analysis of the behavior of participants, it is found that the smooth promotion of agricultural machinery insurance in China is closely related to the support from the government. From the aspect of farmers, their willingness to get insured is not strong due to rather low educational level and limited risk awareness. In addition, since their household income level is low, most of the income of farmers is set aside for living consumption, education, and medical expenditure. As a result, their purchasing power is limited to take out the agricultural machinery insurance. Without government support, it is hard for farmers to convert potential demand into real demand under high premium. For the insurers, they are faced with the issues such as nonstandardized insurance market, low premium but high risk, inadequate insurance capacity, and moral hazard in agricultural machinery insurance market, and thus, the government needs to take measures to regulate the market. By leveraging the role of the government, we are able to better guarantee the actual demand of farmers and to enhance the willingness and capacity of insurers for the provision of insurance. Furthermore, we can cope with numerous issues arisen from the implementation of related system and ensure the development of the agricultural machinery insurance in China. During the effective cooperation of participants in the agricultural machinery insurance market, the government should roll out supportive policies on the one hand and set up the system and policy environment for the development of agricultural machinery insurance on the other hand. In addition, it shall enhance the publicity of agricultural machinery insurance [19].


### 4.1. Providing Legal Support

Law lays the foundation for all activities. With the support of law, we can ensure that “there are laws to abide by” during the implementation, regulate and standardize the behavior of participants with law, and ensure the sound development of market economy. The operation of the agricultural machinery insurance involves numerous departments, and the subjects involved in the market also have a complex nature [20]. Accordingly, it is necessary to clarify corresponding norms and constraints from the legal and regulatory perspective, so as to ensure the smooth implementation of policies. Although the “agricultural insurance regulations” clarify the business subject, financial subsidy, tax preference, and other issues of agricultural machinery insurance, there is a lack of regulations. Therefore, the government ought to standardize the scope and level of insurance, the system of premium subsidy, and other matters from the perspective of law and clarify the liabilities of the authorities in charge of agricultural machinery insurance management as well as the rights and obligations of participants, so as to ensure the healthy and orderly development of the agricultural machinery insurance market.


### 4.2. Providing Institutional Support

The purpose of operating the agricultural machinery insurance is to alleviate the risks and losses incurred by farmers’ agricultural production and to ensure the stability of farmers’ life. Therefore, the agricultural machinery insurance should be deemed as a system for building a harmonious society and promoting the development of new countryside. The government is ought to organize investigation and research, collect basic data, and formulate specific schemes of operation. In addition, the authority shall integrate related regulations with the specific national conditions in China and set up a management system of national agricultural machinery insurance and a unified platform, and establish a mechanism of reward and punishment as well as an information sharing system [21]. Last but not least, China shall form a management system of agricultural machinery insurance with the participation of all subjects under the unified coordination and leadership of the government.


### 4.3. Providing Financial Support

Through the game analysis of the participants, it can be seen that the optimal strategy can only be achieved in the tripartite game between farmers, insurance companies, and the government under the financial subsidy of the government; that is, the benefits of all parties can be maximized. Therefore, the government needs to improve the embarrassing situation of high premium, which farmers cannot pay, and low premium, which insurance companies lose money of agricultural machinery insurance through financial subsidy policy, so as to find a “balance point” between farmers’ payment ability and insurance price. According to the economic level of each region and the development of agricultural machinery insurance, the government should scientifically formulate the premium subsidy method and subsidy proportion and implement appropriate after-tax preferential measures [22].


### 4.4. Providing Organizational Support

The organizational supply provided by the government during the implementation and promotion of agricultural machinery insurance is mainly reflected in the supervision of institutional supply. Government supervision serves as vital guarantee for the normal operation of agricultural machinery insurance. Since the agricultural machinery insurance is still in the pilot period, the relevant business of insurers is supervised by China Insurance Regulatory Commission, whereas the Agricultural Machinery Safety Association lacks corresponding regulatory measures to operate such insurance. In addition, in terms of the issuance and utilization of premium subsidy funds set aside for agricultural machinery insurance in each pilot area, there is a lack of corresponding regulatory system, which has imposed a direct impact on the business of agricultural machinery insurance and enthusiasm of insurers. Therefore, the government needs to set up and improve the regulatory system of agricultural machinery insurance [23] and to monitor the implementation of policies related to the agricultural machinery insurance as well as the management of subsidy funds, so as to enhance the regulatory efficiency and truly put in place the government’s supporting policies and preferential measures. In addition, the authority is ought to supervise the behaviors of all parties involved in agricultural machinery insurance, foster a healthy market environment for such insurance, and ensure that the insurance can give full play to its role of risk protection [24].


### 4.5. Enhancing Publicity

Since the agricultural machinery insurance is still in its infancy in China, the general public lacks the understanding of agricultural machinery insurance, and certain government departments are not taking proactive measures in promoting the agricultural machinery insurance given that the compulsory liability insurance for motor vehicle accidents is already in place, which hinders the extensive development of agricultural machinery insurance. To address these issues, we need to enhance the publicity of agricultural machinery insurance and improve the understanding of relevant authorities on such insurance. In addition, it is necessary to inform the insured farmers of the procedures and practice of agricultural machinery insurance business, so that farmers are fully aware of the insurance procedures, claim settlement methods, and expected claim amount. Moreover, farmers will be able to effectively enjoy the benefits thanks to the guarantee role of agricultural machinery insurance, increase their trust in the insurers, improve farmers’ enthusiasm to take out the agricultural machinery insurance, and enhance voluntary participation of agricultural machinery insurance. Last but not the least, it is recommended to publicize the risk aversion function of the agricultural machinery insurance and nurture the demand in related insurance market.


### 4.6. Establish the Reinsurance System of Agricultural Machinery Insurance

With the rapid development of agricultural mechanization and large-scale agricultural production, more and more large agricultural machinery will be applied to agriculture. The risk of machinery accidents is larger and larger, and the compensation is very high, so the insurers will face a severe test. How to resolve the operational risks of agricultural machinery insurance is an important problem. Therefore, it must establish reinsurance system to accelerate the development of agricultural machinery insurance. For the reinsurance, on the one hand, the government can establish an agricultural reinsurance company to undertake agricultural and agricultural machinery reinsurance. On the other hand, the government can provide reinsurance to China Reinsurance Company.


## 4.1. Providing Legal Support

Law lays the foundation for all activities. With the support of law, we can ensure that “there are laws to abide by” during the implementation, regulate and standardize the behavior of participants with law, and ensure the sound development of market economy. The operation of the agricultural machinery insurance involves numerous departments, and the subjects involved in the market also have a complex nature [20]. Accordingly, it is necessary to clarify corresponding norms and constraints from the legal and regulatory perspective, so as to ensure the smooth implementation of policies. Although the “agricultural insurance regulations” clarify the business subject, financial subsidy, tax preference, and other issues of agricultural machinery insurance, there is a lack of regulations. Therefore, the government ought to standardize the scope and level of insurance, the system of premium subsidy, and other matters from the perspective of law and clarify the liabilities of the authorities in charge of agricultural machinery insurance management as well as the rights and obligations of participants, so as to ensure the healthy and orderly development of the agricultural machinery insurance market.


## 4.2. Providing Institutional Support

The purpose of operating the agricultural machinery insurance is to alleviate the risks and losses incurred by farmers’ agricultural production and to ensure the stability of farmers’ life. Therefore, the agricultural machinery insurance should be deemed as a system for building a harmonious society and promoting the development of new countryside. The government is ought to organize investigation and research, collect basic data, and formulate specific schemes of operation. In addition, the authority shall integrate related regulations with the specific national conditions in China and set up a management system of national agricultural machinery insurance and a unified platform, and establish a mechanism of reward and punishment as well as an information sharing system [21]. Last but not least, China shall form a management system of agricultural machinery insurance with the participation of all subjects under the unified coordination and leadership of the government.


## 4.3. Providing Financial Support

Through the game analysis of the participants, it can be seen that the optimal strategy can only be achieved in the tripartite game between farmers, insurance companies, and the government under the financial subsidy of the government; that is, the benefits of all parties can be maximized. Therefore, the government needs to improve the embarrassing situation of high premium, which farmers cannot pay, and low premium, which insurance companies lose money of agricultural machinery insurance through financial subsidy policy, so as to find a “balance point” between farmers’ payment ability and insurance price. According to the economic level of each region and the development of agricultural machinery insurance, the government should scientifically formulate the premium subsidy method and subsidy proportion and implement appropriate after-tax preferential measures [22].


## 4.4. Providing Organizational Support

The organizational supply provided by the government during the implementation and promotion of agricultural machinery insurance is mainly reflected in the supervision of institutional supply. Government supervision serves as vital guarantee for the normal operation of agricultural machinery insurance. Since the agricultural machinery insurance is still in the pilot period, the relevant business of insurers is supervised by China Insurance Regulatory Commission, whereas the Agricultural Machinery Safety Association lacks corresponding regulatory measures to operate such insurance. In addition, in terms of the issuance and utilization of premium subsidy funds set aside for agricultural machinery insurance in each pilot area, there is a lack of corresponding regulatory system, which has imposed a direct impact on the business of agricultural machinery insurance and enthusiasm of insurers. Therefore, the government needs to set up and improve the regulatory system of agricultural machinery insurance [23] and to monitor the implementation of policies related to the agricultural machinery insurance as well as the management of subsidy funds, so as to enhance the regulatory efficiency and truly put in place the government’s supporting policies and preferential measures. In addition, the authority is ought to supervise the behaviors of all parties involved in agricultural machinery insurance, foster a healthy market environment for such insurance, and ensure that the insurance can give full play to its role of risk protection [24].


## 4.5. Enhancing Publicity

Since the agricultural machinery insurance is still in its infancy in China, the general public lacks the understanding of agricultural machinery insurance, and certain government departments are not taking proactive measures in promoting the agricultural machinery insurance given that the compulsory liability insurance for motor vehicle accidents is already in place, which hinders the extensive development of agricultural machinery insurance. To address these issues, we need to enhance the publicity of agricultural machinery insurance and improve the understanding of relevant authorities on such insurance. In addition, it is necessary to inform the insured farmers of the procedures and practice of agricultural machinery insurance business, so that farmers are fully aware of the insurance procedures, claim settlement methods, and expected claim amount. Moreover, farmers will be able to effectively enjoy the benefits thanks to the guarantee role of agricultural machinery insurance, increase their trust in the insurers, improve farmers’ enthusiasm to take out the agricultural machinery insurance, and enhance voluntary participation of agricultural machinery insurance. Last but not the least, it is recommended to publicize the risk aversion function of the agricultural machinery insurance and nurture the demand in related insurance market.


## 4.6. Establish the Reinsurance System of Agricultural Machinery Insurance

With the rapid development of agricultural mechanization and large-scale agricultural production, more and more large agricultural machinery will be applied to agriculture. The risk of machinery accidents is larger and larger, and the compensation is very high, so the insurers will face a severe test. How to resolve the operational risks of agricultural machinery insurance is an important problem. Therefore, it must establish reinsurance system to accelerate the development of agricultural machinery insurance. For the reinsurance, on the one hand, the government can establish an agricultural reinsurance company to undertake agricultural and agricultural machinery reinsurance. On the other hand, the government can provide reinsurance to China Reinsurance Company.


## 5. Conclusions

Based on the behaviors of farmers, insurers, and the government, this paper elaborates on the conditions for the effective cooperation among the three subjects of agricultural machinery insurance and carries out a game analysis on the behaviors of these three subjects. The research findings indicate that the optimal strategy of tripartite cooperation is as follows: farmers sign up for the agricultural machinery insurance, insurers underwrite such insurance, and the government supports the insurance. The basis for the long-term and stable cooperation of all parties involved in the game is the discount factor. Within a certain range, in case that the long-term benefits obtained by each participant from “cooperation” exceed the maximum benefits obtained through short-term “deception,” all parties will opt for the strategy of long-term cooperation, which can facilitate the healthy development of agricultural machinery insurance. The government plays a critical role in agricultural machinery insurance due to its nature of quasi-public goods. The smooth implementation of such insurance in China depends on the support from the government. During the effective cooperation among participants of the agricultural machinery insurance, the government should enhance the supporting policies on the one hand and set up the system and policy environment for the development of agricultural machinery insurance on the other hand. In addition, the authority shall enhance the publicity of agricultural machinery insurance.



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*Source: 3915672-2022-03-11.xml*