3.2. Model Construction
A signaling game is a dynamic game with incomplete information. A typical signaling game consists of two players: the sender and the receiver. As the family doctor and the resident are two parties with asymmetric information, there is a sequence of actions between them in the process of implementing contracted services, so a signaling game is suitable for solving the research problem in this paper. Assuming that the game revolves around contracted services between family doctors and residents of the area, it has been shown that, in the healthcare service market, patients are unable to assess the service quality of doctors due to a lack of professional knowledge, and there is a large information asymmetry between the two [
27], which in this paper signifies that residents only know the a priori probability of family doctor competency and the number of contracted personalized service packages, and it is difficult to confirm the actual competency of family doctors. Therefore, referring to existing studies that took the number of comments in the online consultation service as a basis for judging the doctors’ service levels [
28], this paper took the number of signed personalized service packages as a key signal for residents to judge the competency levels of family doctors. When the number of contracts exceeds the average level, family doctors will be judged as having a high competency level; on the contrary, if the number of contracts falls below the average level, family doctors’ competency level will be judged as low.
The specific game process is as follows: Firstly, nature (N) determines the family doctor type with a prior probability , the type space , and and represent high-competency and low-competency family doctors, respectively, with a prior probability , . Secondly, family doctors observe their own type and then select the signal from the signal set to transmit; in the signal set , and represent higher and lower contracted quantities, respectively. Finally, according to the competency signal transmitted by the family doctor, the residents use the Bayesian rule to modify the prior probability to obtain to which type of posterior probability the family doctor belongs, and then select their own strategies , where and represent the signing of a personalized service package and basic service package, respectively.
Regarding the parameter design for the signal recipient—the resident: The total cost for the resident to sign up for the personalized service package is , where is the cost of investigation, that is, the cost paid by the resident to find the actual competency level of the family doctor, such as making inquiries with people who are familiar with the situation, and is the actual cost paid by the resident for signing up for the personalized service package. The total cost for residents to register for the basic service package is zero, and access is free upon registration. This paper assumes that residents are rational economic individuals, and considering that signing up for the basic service package is free, they all choose to register under the strong advocacy of the local government. Regarding the utility, for residents who sign up for the personalized service package, if they correspond to a high-competency family doctor, the utility they obtain includes the basic utility brought by the basic public health service and the value-added utility brought by satisfaction with personalized services; on the contrary, if they correspond to a low-competency family doctor, the value-added utility obtained by the residents is (), under the condition that the family doctor does not disguise their own competency. represents the service gap between a high-competency family doctor and a low-competency family doctor, where the gap decreases with . Under the condition that the family doctor disguises his or her competency, a successful disguise results in the resident obtaining value-added utility , but residents will not be able to obtain value-added utility if the disguise fails, which also means that the gap between reality and the ideal based on individual cognitive bias will bring some losses to residents.
Regarding the parameter design of the signal sender—the family doctor: In terms of cost, the marginal cost of basic public healthcare service is
, and the marginal cost of personalized healthcare service for family doctors with a high or low competency level is, respectively,
and
, where
; the higher the competency, the lower the marginal cost of service. In addition, a family doctor with a low competency level may act speculatively to disguise his or her competency in order to obtain more benefits, thereby sending a false signal of high competency to the resident. The camouflage cost of transmitting this false signal is
. This speculative behavior generally means that family doctors do not focus on improving their own competency but use “being signed up,” “signing up for,” and “false signing up” based on China’s unique social structure of acquaintances such as “favor and face” to increase the number of contracts for their own personalized service packages [
21]. With regard to benefits, family doctors will receive funding
for basic public healthcare services regardless of the service package for which they register, which is mainly funded by the basic medical insurance fund and government subsidies. In addition, family doctors who sign up for the personalized service package will also receive payment
for the personalized service package, of which
, which means that the self-paid portion of the residents who register for the personalized service package will be included in the family doctor’s salary in proportion. This will motivate family doctors’ efforts to a certain extent, but at the same time, it also breeds opportunistic behavior, a key point of this study. Finally, if the resident who registers for the personalized service package is satisfied with the family doctor, the family doctor will receive an additional benefit
, which mainly represents the improvement of the family doctor’s reputation. More residents may sign up for the personalized service package with them in the future. For family doctors with high competency, the probability that residents are dissatisfied with them is very small and may be ignored. For a family doctor with low competency, the probability of success in disguise is
,
. If the disguise succeeds, the family doctor will receive the same additional benefits as a family doctor with high competency; if the disguise fails, the residents will not be satisfied with them. At this time, they will not only be unable to obtain additional benefits but will also be subject to certain penalties
.
The benefits to the family doctor and the resident under different conditions are shown in
Figure 1. The benefits to the family doctor’s income are specifically expressed as
,
,
,
, and
. The benefits to the resident are specifically expressed as
,
,
, and
.
3.3. Equilibrium Analysis
Assuming that the probability of a false signal of competency from a family doctor is (), there are , , , and . According to the Bayesian rule, the posterior probability of the residents is , , , and .
3.3.1. Separating Equilibrium
Since competent family doctors will not release a signal for a low number of contracts, there is only one condition left for a possible separating equilibrium strategy for family doctors, in which they send a signal for a high number of contracts, and family doctors with low competency send a signal for a low number of contracts. Under this condition, there are and .
(a) When residents receive a signal from family doctors that the number of contracts is high (), the expected benefit of their choice to sign up for the personalized service package is ; the expected benefit from choosing to sign up for the basic service package is . When , the residents’ optimal strategy is to sign up for the personalized service package. We can solve . In this case, competent family doctors always choose to send signals with a high number of contracts and will not deviate from the equilibrium path. As far as the reality is concerned, the proportion of residents who sign up for personalized service packages at their own expense is low, and the value of the services covered far exceeds the signing cost of residents. However, the premise for residents to obtain full access is to fully understand and actively use the contents of the personalized service package; at the same time, personalized service packages should be designed for different groups of people, to improve residents’ willingness to use them. Conversely, if residents do not understand the personalized service package or the personalized service package cannot meet their needs, then , where the corresponding signing cost is higher than the value-added utility obtained. In this scenario, the residents’ optimal strategy is to sign up for the free basic service package.
(b) When residents receive a signal from family doctors that the number of contracts is low (), the expected benefit of signing up for a personalized service package is ; the expected benefit of signing up for a basic service package is .
When , the residents’ optimal strategy is to sign up for personalized service packages. We can solve . Residents will choose to sign up for personalized service packages only when the service gap between various types of family doctors is small. When residents choose to sign up for personalized service packages, the family doctor’s benefit on the equilibrium path is , and the benefits on the disequilibrium path are and . If the benefits are satisfied by and , the family doctor does not deviate from the equilibrium path. We can solve . At this point, the family doctor and the resident reach a separating equilibrium, which is classified in this paper as Equilibrium Ⅰ, and the equilibrium solution is . In this equilibrium, all types of family doctors choose to release true signals and residents sign up for personalized service packages.
When , the optimal strategy for residents is to sign up for a basic service package, at which point , indicating that the residents are reluctant to sign up for a personalized service package when the service gap is large, that is, when the value-added utility received by the residents cannot cover the cost paid. At this point, the benefit of family doctors on the equilibrium path is , and the benefits on the disequilibrium path are and . When and , a separating equilibrium is achieved, and the solution is . Since , when , the family doctor and the resident reach a state of separating equilibrium, which is classified as Equilibrium Ⅱ, and the equilibrium solution is . In this equilibrium, family doctors with either high or low competency levels all release true signals, corresponding to contracted personalized service packages and basic service packages, respectively.
3.3.2. Pooling Equilibrium
The possible pooling equilibrium is one in which all types of family doctors release the signal of a high number of contracts, where there are and . When the residents receive the signal , they choose to sign up for the personalized service package with the expected benefit of , or the basic service package with the expected benefit of .
When , the solution is , which is also . In this case, if , the above formula is impossible. In other words, when the contracted cost is greater than the value-added utility, the residents do not choose to sign up for the personalized service package; in the case where the cost of residents signing up for a personalized service package is less than the value-added utility, when the probability of the family doctor’s success in disguise decreases, the proportion of doctors with actual high competency in the total family doctor team subsequently increases, and the optimal strategy of residents for this scenario is to sign up for the personalized service package. When residents choose to sign up for personalized service packages, high-competency family doctors will not deviate from the equilibrium path; they will always be willing to send signals of a high number of contracts. The benefit of low-competency family doctors on the equilibrium path is , and the benefits on the disequilibrium path are and . Only when and will the family doctor not deviate from the equilibrium path, and the solution is . In this scenario, the family doctor and the resident reach a pooling equilibrium, which is classified as Equilibrium Ⅲ, and the solution is or . The implication of this equilibrium is that all family doctors release the signal and residents sign up for the personalized service package. In this case, the low-competency family doctors trick residents into signing up for a personalized service package by inventing a false number of contracts to receive a handsome salary. This is a “free-riding” behavior, which means that rather than gaining the trust of residents by improving competency, family doctors instead take shortcuts to obtain a pay increase.
When , , there is . In this case, if , then the above formula must be established. In other words, when the cost of signing up for the personalized service package is greater than the value-added utility, the optimal strategy for the residents is to sign up for the basic service package; if , the proportion of high-competency doctors in a family doctor team decreases if a family doctor has a greater probability of success in disguise. More family doctors mask themselves instead of improving themselves, and the optimal strategy for residents is to sign up for the basic service package. When residents choose to sign up for basic service packages, high-competency family doctors always choose the equilibrium path of a high number of contracts. For low-competency family doctors, the benefit of the equilibrium path of a high number of contracts is , and the benefits of the disequilibrium path of a low number of contracts are and . When a low-competency family doctor chooses to release the signal , if the resident chooses to sign up for the basic service package, the family doctor’s income is . Since is obviously established, the family doctor will definitely deviate from the equilibrium path, and there will be no pooling equilibrium in this scenario. Therefore, there are two necessary conditions for achieving a pooling equilibrium: first, when the family doctor deviates from the equilibrium path to choose to release the signal , the resident chooses to sign up for the personalized service package; secondly, when the resident chooses to sign up for the personalized service package, the benefit of the family doctor on the disequilibrium path is less than that on the equilibrium path. In this case, the family doctor has no motive to deviate from the equilibrium path, and the two sides reach a pooling equilibrium. Satisfying , the solution yields , at which point the two sides achieve a pooling equilibrium, classified as Equilibrium Ⅳ, and the equilibrium solution is . In this equilibrium, all types of family doctors choose to release the signal , while residents choose to sign up for the basic service package; when residents receive the signal , they instead choose to sign up for the personalized service package, which is a phenomenon of adverse selection.
In summary, in the contracted service focused on family doctor competency, four equilibria will develop between family doctors and residents, as shown in
Table 1. Combined with the above analysis, it can be found that the cost of disguise
, the payment for the personalized service package acquired by family doctors, the value-added utility of the personalized service
acquired by residents, the service gap between high-competency family doctors and low-competency family doctors, and the proportion of high-competency family doctors in the family doctor team are the most important factors that affect the decision-making behavior of the subject. In order to facilitate the discussion, the parameters were abstracted into two states of large (high) and small (low). Then, a specific analysis of the real-life dilemmas that hinder the implementation of the personalized contracted service model was conducted in relation to the parameters in the model.