Sun Huan Wang Haiyan

(School of Economics and Management, Southeast University, Nanjing 211189, China)

Abstract：To improve the inefficient prevention caused by customers’ unwillingness to adopt prevention strategies in health management, an incentive feedback mechanism that is based on game theory and contract design theory is introduced.The conditions for making customers and health maintenance organizations(HMOs)willing to participate in the proposed mechanism are given.A dual nonlinear programming model is used to identify the optimal prevention effort of customers and the pricing strategy of HMOs.Results show that to generate increased benefits, HMOs need to consider cost sharing when customers are not familiar with the proposed health services.When health services are gradually accepted, the cost sharing factor can be gradually reduced.Simulation shows that under random circumstances in which the market reaches a certain size, the proposed method exhibits a positive network externality.Motivated by network externality, HMOs only need to make their customers understand that the larger the number of participants, the greater the utility of each person.Such customers may then spontaneously invite others to purchase insurance.

Key words：customers’ prevention efforts; incentive feedback mechanism; healthcare service; health pricing strategy; health service optimization

The growth of healthcare expenditures globally has become an issue that requires effective operation modes.Healthcare costs in the US in 2017 accounted for 17% of the country’s gross domestic product(GDP), and they are expected to rise to 19.9% in 2025[1].In the same year in China, healthcare costs accounted for 6.2% of the country’s GDP, thereby bringing tremendous pressure on health participants[2].

To reduce the rising health costs, health maintenance organizations(HMOs)play a significant role in health management.Health maintenance is able to reduce cost because it focuses not only on the past(previously diagnosed disease)but also on the future(disease prevention)[3].Prevention services can reduce the future demands for diagnostic services and treatment[4].Kaiser Permanente is a well-known HMO.Members of the Kaiser HMO include insurance companies and medical groups.The organization can provide customers with insurance that includes prevention and treatment services to control customer health[5].Its operating mode is to bring customers and the organization together by using a fixed fee format, thereby encouraging the HMO to manage customers’ health through prevention and other low-cost methods.The fixed fee payment method has become mainstream in the US[6].

In China, some HMOs have learned how to use similar operation methods to design health insurance.For example, Ping An Insurance Company of China, in cooperation with hospitals, provides child dental insurance, which includes inspections, preventive services, and accident reimbursement.

To reduce possible treatment costs, some studies[7-9]have added prevention to their investigation of health insurance options.Prevention services include three aspects.The first aspect covers disease screening, such as colonoscopies[10-11]and breast cancer screens[12].The second aspect covers vaccinations[13-14].The third aspect covers mechanism designs[15].However, few of the existing studies have directly addressed how to motivate customers to participate in prevention activities.The lack of incentives for customers’ prevention efforts leads to low efficiency of prevention.As customers do not engage in any prevention activity in their daily life, prevention efforts burden them.The effect of prevention depends on the efforts of HMOs and customers.The existing operation mode focuses solely on motivation for HMOs and largely ignores incentives for customers.Prevention without customer involvement leads to inefficiency[16].

In academia, two ideas have been raised to solve the aforementioned problem.On the one hand, the literature focuses on cooperation.Andritsos et al.[9]developed a health co-production model to establish joint control between patients and hospitals for readmission.Mendon?a et al.[17]developed a game theory to model liver transplantation consultations involving alcohol-driven liver disease to improve patients’ cooperation.Many collaborative studies have also focused on information sharing[18-19], asymmetric relationships[20], and trust[21].Other works have adopted unique approaches to establish cooperation with other service fields; such approaches include cost sharing contracts[22], social networks, and behavioral models[23].These existing methods provide theoretical support for the current work.On the other hand, the literature covers incentive mechanism design.Care et al.[24]built a computerized decision support system to document peer reviews and abnormal feedback on diagnostic results to ensure accurate diagnoses.Mehrotra et al.[25]developed multivariate regression models to study the effects of patient incentives on receiving preventive care.The preventive effects were found to remain low mainly because of inefficient incentives.The problem captured in the study requires a highly effective way to incentivize customers.To reduce hospital readmissions, Liu et al.[26]developed a delay time analysis model to identify effective checkup plans for monitoring patients.To improve prevention efficiency, the study considered customers’ free time and preferences without considering how to ensure customers’ cooperation.Mehta et al.[8]built a model of consumers’ annual medical insurance plan decisions and periodic consumption decisions to guarantee their health.However, these existing studies passively consider customers’ free time and preferences.Hence, the current study is motivated by the need to solve the aforementioned problem by strengthening customers’ participation.

To mobilize customers toward prevention cooperation, this study designs a cooperative and incentive mechanism involving HMOs and customers.Different from the general health maintenance mechanism that sets a fixed price, the new mechanism adds a cost sharing fee to motivate customers.Moreover, the proposed mechanism is unlike medical insurance that provides fixed premiums and reimbursement ratios as it uses flexible pricing to target different customers.

This work presents an approach to the design of optimal incentive strategies to enhance customers’ enthusiasm for prevention.As far as we know, few studies have investigated the prevention effect from the perspective of customer incentives.Relative to the passivity of waiting for treatment, the proposed method is proactive in terms of prevention.

In summary, this study aims to address the operation of the new pricing strategy for incentivizing customers to enhance their prevention engagement.First, the dual nonlinear programming model, including customer utility and HMO utility, is established.Second, the optimal strategies are calculated after verifying the existence of the optimal solution.Third, the correlation sensitivity analysis is given.Finally, the numerical study is presented.

1 Problem Statement

This work constructs an incentive feedback model with two subjects: the HMO and customers.The HMO is the leader that designs a new pricing strategy containing a fixed pricePand a cost sharing ratercsto incentivize customers for their prevention effort.The customers are the followers who decide on their prevention effort strategyep.Our task is to find the optimal strategies for the HMO and the customers to obtain their maximum utility.

The decision sequence is as follows:In the beginning, the HMO offers an insurance plan with prevention services and possible treatment services for customers who become ill; the related effects areηpandηt.The price of the insurance consists of the fixed pricePand cost sharing ratercs.Then, the customers pay the fixed pricePto the HMO, and the organization manages the customers’ health.To save costs and add benefits, the HMO provides prevention services, which help reduce the probability of customers becoming ill.When receiving the prevention services, customers need to determine their prevention effort.After the prevention period, two results are expected: First, when customers are healthy until the end of the insurance period, no additional charges are incurred.Second, when customers are sick and need treatment, they should pay the cost sharing feercsCt, whereCtis the cost of treatment andrcsis the cost sharing rate given by the HMO to incentivize customers to exert prevention efforts further.Under the pressure of increased costs, customers might be more willing to cooperate through their prevention efforts.The incentive feedback mechanism is shown in Fig.1.

Fig.1 Incentive feedback mechanism and plan design

For HMOs, the trade-offs lie on the cost sharing ratercsand fixed priceP.On the one hand, an increase in the cost sharing ratercsincreases customer cost pressure, which in turn stimulates customers to exert prevention efforts and improve their health status, thereby reducing the treatment cost of the HMO.On the other hand, an increase inrcsmay cause it to exceed the total price limitation.The total price limitation is set to ensure the advantage of the new pricing strategy; it is given byP+rcsCt(1-ep)rd≤P0, where(1-ep)rdis the probability of illness after customer prevention efforts andrdrepresents the initial probability of illness.With the consequent rise in total price, customers may refuse to participate, therefore reducing the benefits received by the HMO.If the fixed pricePis too high, then customers become unwilling to pay for insurance; if the fixed price is too low, then the HMO cannot make ends meet.From the customer perspective, if customers exert negligible prevention efforts, then the cost sharing rate of their illness increases; if customers exert excessive prevention efforts, then the cost of prevention increases.

2 Model and Analysis

In this section, we construct a dual model of two subjects: the customers and the HMO.The customers determine their prevention strategies to reduce their costs and gain benefits.The HMO sets a new pricing strategy to maximize its utility.After proving the existence of the optimal solution, we obtain the relevant optimal strategy.Furthermore, we conduct sensitivity analysis and numerical simulation on the optimal strategy.

2.1 Modeling

To characterize the relationship between the degree of disease in terms of deteriorationηdand the effectiveness of preventionηtand treatmentηp, we assume that customers can recover through prevention and treatment under ideal conditions.

ηd=ηt+ηp

(1)

Eq.(1)implies the following result: If the preventive effect increases, then the therapeutic effect can be reduced.Therefore, the cost of treatment can be reduced.

The utility of a customer who chooses the health insurance plan can be expressed as follows:

u(ep)=h+(1-ep)rd(-ηd+epηp+ηt)-

α1(P+(1-ep)rdrcsCt)-β1ep

(2)

The utility is composed of the customer’s initial health statush, health control utility(1-ep)rd(-ηd+epηp+ηt), spending utilityP+(1-ep)rdrcsCt, and health prevention costβ1ep, whereα1is the customer’s price sensitivity andβ1is the cost effect sensitivity factor for the prevention effort.The health control utility consists of the risk of illness and the health status after the intervention.Following Andritsos et al.[9], we obtain(1-ep)rd, which is the probability of illness after customer prevention efforts.Referring to the work of Mehta et al.[8], we obtain the health status after intervention-ηd+epηp+ηt.Ifepis small, then the probability of disease(1-ep)rdand the degree of disease severity-ηd+epηpare high.In this case, further treatmentηtis needed.

The customer’s optimization can be expressed as follows:

maxu(ep)=h+(1-ep)rd(-ηd+epηp+ηt)-

α1(P+(1-ep)rdrcsCt)-β1ep

(3)

s.t.

epηp+ηt≥β3ηd

(4)

0

(5)

Eq.(4)is set to guarantee the customer’s minimum health benefit.In this equation,β3is a constant that stands for the minimum guaranteed health rate,β3≤1.Eq.(5)provides the range for the variableep.In the model, the customer chooses his/her effort levelep(0

Different from the customer’s utility, the HMO’s utility needs to consider not only the utility cost difference but also the number of customers.The number of customers participatingNcpdepends on the fixed price and the customers’ prevention efforts; it is divided into the following two parts:

Ncps=(1-ep)rd(a-α2P)

(6a)

Ncph=(1-ep)(1-rd)(a-α2P)

(6b)

whereais the basic market size;α2is the price-sensitive parameter for market size; andPis the fixed price.Eq.(6a)represents the number of customers who have been ill during the given period.Eq.(6b)represents the number of customers who are healthy during the period.The HMO’s utilities from these different customers are represented as

Us=P-Cph-(1-rcs)Ct

(7a)

Uh=P-Cph

(7b)

Eq.(7a)presents the utility from a customer who has been ill during the period.It consists of the benefit from the fixed priceP, the prevention costCph, and the remaining treatment(1-rcs)Ct.Eq.(7b)presents the utility from the customer who is healthy during the period.It consists of the benefit from the fixed pricePand the prevention costCph.We assume thatCphis part of fixed priceP, that is,Cph=β4P.

We obtain the NLP that maximizes the HMO’s utility.The HMO’s optimization can be expressed as follows:

maxU(P,rcs)=UsNcps+UhNcph=

(P-Cph-(1-rcs)Ct)(1-ep)·

rd(a-α2P)+(P-Cph)(1-ep)·

(1-rd)(a-α2P)

(8)

s.t.

P+(1-ep)rdrcsCt≤P0

(9)

0

(10)

P>Cph+(1-rcs)Ct

(11)

Eq.(8)captures the HMO’s utility from customers who choose the insurance plan.Eq.(9)indicates that the comprehensive priceP+(1-ep)rdrcsCtshould not exceed the price of general insuranceP0.It is the total price limit.Eq.(10)provides the range of the cost sharing ratercs.Eq.(11)presents the range of fixed priceP, which means that pricePshould be greater than the HMO’s costsCph+(1-rcs)Ct.The utility of the HMO also depends on customers’ prevention effortsep.

2.2 Optimal solutions

In this section, we calculate and analyze the optimal strategies for customers and the HMO separately and then obtain the management result.

As a precondition of this study, we analyze the conditions leading to customers’ willingness to buy health insurance(the existence of customers’ utility).Then, we find the conditions in which the HMO would turn to the new pricing strategy and the conditions for the HMO’s optimal utility(the optimal solution for the existence of the HMO’s utility).On the basis of these conditions, we obtain the equilibrium decision between customers and the HMO.In addition, we analyze the impact of other indicators under the optimal strategies.

To identify which condition leads to customers’ willingness to purchase insurance, we consider two conditions: 1)The condition leading to customers’ benefit(u>0)when choosing the new health insurance; 2)The condition leading to the benefit of the new health insurance being greater than that of the general one(Δu>0).

Lemma 1 shows the conditions that makeu> 0.

Lemma 1 shows that the customers’ choice of health insurance depends on the fixed pricePand cost sharing ratercsdesigned by the HMO.The aforementioned constraints relate toα1; the biggerα1is, the smaller the set of constraints.Lemma 1 reveals that the more sensitive the customer is to price, the smaller the range of choices for the health organization’s pricing.Therefore, the HMO should choose to cooperate with customers who are willing to exert prevention efforts.

For condition 2), we establish a comparison groupu′=u(ep= 0)to analyze the condition in which customers choose the new health insurance.

(12)

Lemma2Ifβ1<(2-ep)rdηp, then Δu>0.

Lemma 2 shows that the new pricing strategy appeals to customers once the cost sharing factorβ1is less than the customers’ prevention cost constraints(2-ep)rdηp.

After assessing the conditions in which customers purchase insurance, we study the maximum customer utility under prevention effort strategies.Lemma 3 shows the results.

Lemma3Given the HMO’s initial pricingPandrcs, the customer’s optimal prevention effort is

(13)

under which the customer’s optimal utility is

(14)

To satisfy the inequality 0

(15)

Before calculating the optimal utility for the HMO, two constraints need to be considered: the constraint that makes the HMO prefer the new operation mode and the existence of the optimal strategy of the HMO.

To obtain the first constraint, we establish a comparison groupU′=U(ep=0,Cph=0,rcs=0).

U′=(P-Ct)rd(a-α2P)+P(1-rd)(a-α2P)

(16)

whereU′ is the HMO’s utility without any prevention effort.We set ΔU=U-U′.

ΔU=(a-Pα2)((-2P+rcsCt)rd+ep((1-rcs)rdCt+

P(-1+β4))-P(-2+β4))

(17)

Lemma4If(1+β4)P<(rcs+1)Ctrd, then ΔU>0.

To answer the second question, we need to prove that the HSU model has an optimal solution.As the HSU model is an NLP with nonlinear constraints, we can verify which condition leads to the existence of the HSU model by proving that it is convex programming.To prove that the HSU model is convex programming, we need to verify that the maximum objective function and the larger inequality symbol constraints are concave and that the smaller inequality symbol constraints are convex.

Lemma5IfPandrcsmeet the following condition: then the HSU model is convex programming and can achieve an optimal solution.

4α1α2(a-Pα2)(α1rdrcsCt-β1)(1-β4)-(α2β1+

α1(-α2((2rcs+1)rdCt+2P(-1+β4))+

a(-1+β4)))2≥0

To solve the HSU model, we provide the optimal solutions without constraints and then introduce them to the constraints from Lemmas 1 to 5.

Tab.1 Optimal fixed price and cost sharing rate

Then, we introduce the above solutions to the constraints in Lemmas 1 to 5 and the participation constraint(a>α2P)and ensure that the fourth solution meets all the constraints.We substitute the fourth solution into the above inequality group and obtain the following results:

Theorem1If the market size meets the inequalities(1-β3)ηd<ηp, then the optimal solutions of the HSU model and customer utility(CU)model are

(18)

(19)

(20)

and the maximum utilities for the customer and the HMO are

(21)

(22)

Theorem 1 shows that if market sizeameets the above constraints, thenrcsandPare optimal solutions.We should note that because(1-β3)ηd<ηp,Ctrdα1-β1+6rdηp>Ctrdα1-β1+3rd((1-β3)ηd+ηp).That is, the constraint aboutain Theorem 1 is not empty.

2.3 Sensitivity analysis

The managerial insight is that in the initial period, the HMO should set a relatively high cost sharing rate to reduce the negative effect of the high market price sensitivity.When the health service is sufficiently understood, the cost sharing factor can be reduced as the market price sensitivity decreases.

Therefore, to obtain increased customer participation, the HMO needs to decrease the cost sharing rate when the probability of the customer being ill increases.

Concurrently, the HMO looks favorably on customers recommending it to others as this behavior increases the organization’s utility.

2.4 Numerical study

To further study the relationship between CUsuand market sizea, we use the control variable method for our simulation under random circumstances.

(5α2β1+α1(Ctrdα2+a(-1+β4)))·

To rule out the impact of other factors on market size and utility, we use the control variable method in our research.Fig.2(a)shows that the customer’s utility changes withα1anda, whereα1∈(0,3].Fig.2(b)shows that the customer’s utility changes withα2anda, whereα2∈(0,3].Fig.2(c)shows that the customer’s utility changes withβ1anda, whereβ1∈[0,40].Fig.2(d)shows that the customer’s utility changes withβ4anda, whereβ4∈(0,1).Fig.2(e)shows that the customer’s utility changes withCtanda, whereCt∈[0,200].Fig.3(f)shows that the customer’s utility changes withrdanda, whererd∈(0,1].Fig.2(g)shows that the customer’s utility changes withηpanda, whereηp∈(0,40].Excluding the range of market size a in Fig.2 of[0,50], the range ofain Figs.2(a)to(g)is[0,500].

Fig.2(a)shows that the largerα1is, the more obvious the tendency of the curve between CUsuand market sizeawill be.The curve tendency is that as a increases,udecreases first and then increases.Fig.2(a)indicates that whenα1is low, customers may not urge their friends to join the HMO; whenα1is large, the recommendation strategy is wise.

Fig.2(b)shows that as the market sizeaincreases, CUsudecreases first and then increases whenα2is low.Asα2grows, the increase in CUsuslows down.Whenα2is large enough, CUsudecreases as the market size a grows.Although the degree of bending weakens, the CUs always decreases first and then increases as the market size increases regardless of howα2changes.Fig.2(b)indicates that the HMO should focus on the initial period to expand the market size because of the lowα2.

Figs.2(c)to(f)show that regardless of the changes inβ1,β4,Ct,andrd, as the market sizeaincreases, the CUsudecreases first and then increases.The differences among the figures are as follows: Asβ1andCtincrease, CUsudecreases.Meanwhile, the increase ofβ4leads to an increase in CUsu.Asβ1andCtincrease, the growth of CUsuaccelerates.Figs.2(b),(d), and(f)also show that asα2,β4,andrdincrease, the curve between market sizeaand customer utilityuweakens.Different from that in the other figures, the utility of the customer in Fig.2(g)is above zero.Similarly,ηpdoes not change the tendency of theu-acurve.Fig.2(d)indicates that when the market size is large, the HMO should improve the prevention ratio in fixed priceβ4.Fig.2(f)shows that the incentive mechanism is relatively effective in cases when the probability of illness is low.Fig.2(g)indicates that only when the market size is large enough can the prevention effect given by the HMO be effective.This result is explained as follows: When the sample size is small, the effect of prevention is not obvious.Only when the sample size is large enough, can the impact of effective prevention be easily observed.

(a)

From Fig.2, we conclude that customers will not recommend insurance to their friends when the market size is too small.When the market size is above the inflection point, recommending insurance to others not only benefits the customers’ utility while reducing their prevention efforts but also increases the HMO’s utility.These results indicate that our method is affected by positive network effects[28]when the market size reaches a certain degree.

3 Conclusions

1)To proactively defend against the rising costs of healthcare due to customers’ inefficient prevention efforts, we design a mechanism for increasing the utility of customers and health organizations by stimulating customers’ prevention efforts.This mechanism combines customer efforts and health advice to make prevention increasingly effective.In the proposed model, the effectiveness of prevention is improved by reducing the risk and severity of customers’ illnesses.The given operation model improves not only customers’ health status but also their participation.

2)This study presents the conditions under which the new mechanism is superior to the traditional health management strategy.The proposed mechanism is applicable to different customers.

3)Numerical experiments prove that the proposed method has positive network effects.That is, increased participation improves the benefits gained by customers and HMOs.

4)To cope with the increased probability of illness, HMOs need to increase their cost sharing rates.However, under the conditions in which customers are price insensitive and unwilling to pursue prevention while the prevention costs are high, HMOs should reduce cost sharing.