How to charge doctors and price medicines in a two-sided online healthcare platform with network externalities?

Platform business models are upending the value proposition of the companies in traditional industries, such as healthcare, through competitively pricing value-added services and products. In this study, we consider a unique, but important, on-demand two-sided service platform: a monopolistic online healthcare platform, which not only provides medical services to patients by connecting them with doctors, but also sells medical products to patients. A crucial operational decision for the platform is to set up a pricing scheme to encourage enough doctors and patients to participate on the platform. We consider two common pricing schemes in this study: fixed and linear pricing, in an empirically-grounded optimisation model that considers medicine pricing decisions together with network externalities. Our results show that the fixed pricing scheme almost always dominates the linear pricing scheme by providing the platform with higher expected profit. Our results also reveal the strong interdependence between the optimal doctor service pricing decision and optimal medicine pricing decision. However, these decisions are often made independently by different functional units of a firm, which could cause suboptimal platform financial performance. These results indicate the importance of pricing products and services in an integrated manner for maximising the profit of a product/service platform.


Introduction
The online healthcare platforms are growing fast globally due to their tremendous roles in improving healthrelated quality and reducing the cost of health care systems, particularly in developing countries (Kyaw et al. 2019;Marques and Ferreira 2020;Kraus et al. 2021). Many of these platforms in China, i.e. Guahao.com, Chunyu-Doctor, Amwell, Doctor-on-Demand, etc. focused on providing diagnosis services from doctors to patients, with a few starting to evolve into online 'diagnosis + medicine purchase' such as 111.com. From our interviews, plenty of patients and doctors are willing to join in the 'diagnosis + medicine purchase' platform due to the advantages of online diagnosis. 1 The 2020 COVID-19 pandemic is further increasing the popularity of such integrated platforms because patients want to avoid visits to hospitals and pharmacies to minimise their exposure to the virus (Sun, Lu, and Rui 2020;Hong, Chan, and Chua 2020;Byrne and Watkinson 2020;Pauzi and Juhari 2020;Bijlani 2021 online medical users reached 215 million, accounting for 21.7% of the 989 million Internet users by December 2020. However, before creating values for platform stakeholders, an online healthcare platform needs to attract enough reputable and independent doctors to offer diagnosis services as well as enough patients to purchase medical services and products. A higher number of doctors attract more patients to enter the platform, which could further motivate the participation of more doctors. This feature is documented as the positive crossmarket network externality in many existing studies (Katz and Shapiro 1985;Lee, Lee, and Lee 2003;Chu and Manchanda 2016;Niculescu, Wu, and Xu 2018;Huuki and Svento 2020;Chi, Fan, and Wang 2021). Different from Qiu and Whinston (2017) or Qiu, Vakharia, and Chhikara (2021), who focus on consumers' observational learning from their friends' or strangers' decisions. From our interviews with patients, most of the time, they do not necessarily know how many other patients are joining or quitting the platform, but they could see how many doctors are offering service on the platform. A larger number of available doctors is a strong signal for them, indicating a high level of platform credibility and ultimately motivating them to join the platform. Our interviews with doctors also shows that, a major determining factor for them to join an online healthcare platform, is whether the financial return is sufficient to compensate for their online diagnosis efforts, which also largely depends on enough online platform patients.
In most cases, an online healthcare platform has two streams of revenue: (1) one from doctors by charging them for using the platform, and (2) the other from patients by selling the medicines. That is different from traditional two-sided platforms, which are only intermediaries connecting service providers and consumers without selling products simultaneously (Dou, He, and Xu 2016;Kung and Zhong 2017;Tan, Anderson Jr, and Parker 2020). Due to the interactions between participation of doctors and that of patients, the service and product pricing scheme for an online healthcare platform is more complicated than that of most online service platforms. To understand the pricing challenges faced by such two-sided healthcare platforms, we conducted a set of interviews in both 2020 and 2021 with platform managers, doctors and patients (both participating and non-participating on the platform). From the platform's view, we learned that 111.com, a major healthcare platform in China, has been struggling with determining 'how to charge doctors reasonably to motivate the participation of experienced doctors and set reasonable and competitive medicine price to build a big enough online patient market', as commented by a platform operations manager. Both doctors and patients also informed us of the significant influences of platform pricing decisions on their participating motivations. Therefore, we explore this unique pricing challenge faced by online healthcare platforms and study how the platform should make pricing decisions with network externality in affecting the behaviours of doctors and patients.
In regards to the pricing scheme for doctors' diagnostic services, the platform needs to choose a scheme that best motivates doctor participation without compromising the platform's financial return. From our extensive research about major healthcare online platforms, we learned that the two common pricing schemes are: fixed pricing and linear pricing. In a fixed pricing scheme, doctors pay a fixed settlement fee when entering the platform and then earn 100% of the diagnosis revenue thereafter. In a linear pricing scheme, the platform extracts a certain ratio of the diagnosis fee every time a doctor finishes an online diagnosis service. From our interviews with managers of 111.com, we learned that there is a lack of understanding about which pricing scheme is better from a platform profit maximisation's view, especially when they struggle with pricing medicines at the same time.
Building upon these qualitative insights and the online platform pricing literature, we set up a math model to examine how a monopoly online healthcare platform maximises its expected profit through optimising medicine price and considering network externality. To better understand findings from the math model, we conducted interviews again with managers of 111.com company in China. Overall, our results provide novel insights into the complex patterns of the optimal medicine pricing decisions under different service pricing schemes. When comparing the fixed and linear service pricing schemes, we show that the fixed pricing scheme almost always dominates the linear pricing scheme by providing the platform with a higher expected profit. In addition, the math model reveals the strong interdependence between the doctor service pricing decision and medicine pricing decision. However, our interviewees informed us that these decisions are often made independently by different functional units of a firm, which could cause suboptimal platform financial performance. Overall, these findings contribute to the general pricing literature by grounding the pricing decision in a complex and novel online platform context (e.g. Dietl, Lang, and Lin 2013;Kung and Zhong 2017).
The rest of this paper is organised as follows. Section 2 reviews the related literature. Section 3 describes the problem and assumptions of our model. In Section 4, we develop our basic model for analysing the optimal pricing strategies in the fixed pricing scheme, and investigate some characteristics of the optimal solutions. In Section 5, we extend our basic model to explore the optimisation strategies in the linear pricing scheme.We further make comparisons between the fixed and linear pricing schemes in Section 6. Finally, we discuss the reality-check of quantitative findings, summarise the results of our paper and give some suggestions on topics for future research in Section 7. All proofs are provided in Appendix.

Literature review
In this paper, we examine the joint optimisation of settlement fee and medicine price in an online healthcare platform with network externalities. Therefore our paper relates to three research streams: on-demand service platforms, pricing in online healthcare platforms, and network externalities.
On-demand service platforms are usually platforms that connect independent service providers with customers who are waiting-time sensitive (Taylor 2018). Relevant literature about the on-demand service platforms is conducted from three aspects: (1) how platforms should price their services/products and how much they should pay their workers (Cachon, Daniels, and Lobel 2017;Taylor 2018;Bai et al. 2019;Choi et al. 2020). (2) how to match heterogeneous service providers with customers (Guo et al. 2018;Feng, Kong, and Wang 2018;Ozkan and Ward 2019). (3) whether the growth in on-demand services harms or benefits workers, customers, and the environment (Burtch, Carnahan, and Greenwood 2018;Benjaafar et al. 2018;Ming et al. 2019). Closest to our work are papers that study the price and wage problem of the platform. For example, Cachon, Daniels, and Lobel (2017) focus on the agents' decisions about whether to work in settings with uncertain demand and heterogeneous agent opportunity costs. In Banerjee, Riquelme, and Johari (2015), customers are heterogeneous, and seek service if and only if their valuation exceeds the price. And Bai et al. (2019) show that the optimal price of a platform is not necessarily monotonic in the market size, when either the labour pool size or the waiting cost is high. However, our work does not consider patients' waiting-time cost, because according to the statistics, online-diagnosing takes an average of seven minutes 2 , and from our interview with patients of 111.com, we learned that patients do not mind waiting for a few minutes online compared to the time it takes to go to offline hospitals. In this study, the healthcare platform needs to decide the medicine price as well as the doctor's settlement fee to attract doctors and patients to participate. Besides, the healthcare platform is in line with the distinctive features of on-demand service platforms, where capacity affects demand, and vice versa; and capacity can be controlled only indirectly via wages and prices, with capacity and demand varying temporally and spatially (Benjaafar and Hu 2020). Therefore, we study the on-demand economy from the new perspective of the healthcare industry.
Although healthcare e-commerce or online medical consultation are emerging rapidly in the past years, research about online healthcare platforms is still rare. Most studies focus on the development, supervision and feasibility of them, such as analysing the current challenges encountered by platform and future development opportunities, investigating doctors' online-offline behaviour dynamics, exploring the difficulties in the application of blockchain and other advanced technologies in the healthcare industry(e.g. Anderson and Agarwal 2017;Ko et al. 2019;Lantzy and Anderson 2020;L. Wang et al. 2020;Sharma and Joshi 2021;Govindan et al. 2022). Some studies have explored pricing strategies of the healthcare platforms. For example, Li and Huang (2019) studied the subsidy strategy for pharmaceutical e-commerce platform. X. Wang et al. (2021) explored the price strategies of a telemedicine service system under the government subsidy policy. Yang, Wang, and Zheng (2021) studied pricing strategy for multichannel healthcare services by considering the knowledge level and misdiagnosis rate. Different from these studies, which did not consider diagnose and medicine selling at the same time, the online healthcare platform discussed in this paper collect revenue from not only selling diagnosis services of doctors but also selling medicine to patients.
In exploring the pricing schemes of platforms, studies usually just focused on one type of pricing scheme, such as a lump-sum fee (Lin et al. 2019), per-transaction fee (Rochet and Tirole 2003), or twopart tariff (Reisinger 2014), without comparing different schemes. Learned from the interviews, a linear pricing scheme, i.e. 40% of doctor platform revenue, is currently adopted. But the platform is not sure whether they need to lower that percentage or switch to a fixed price scheme to avoid discouraging doctor participation. So in this study, we will compare two different pricing schemes based on the practice of healthcare platforms. Though in the context of non-healthcare online platforms, some studies compare the effectiveness of different pricing schemes. For instance, Sundararajan (2004) analysed optimal pricing for information goods by considering the different impact of unlimited-usage (fixed-fee) pricing and usage-based pricing strategies and obtained a pure fixed-fee is optimal. Dietl, Lang, and Lin (2013) developed a model of asymmetric competition between pay and a free media platform, and showed that charging advertisers a lump-sum fee in a free media platform generates higher profits than a per-consumer fee. Balasubramanian, Bhattacharya, and Krishnan (2015) considered a monopolist who could use either selling or pay-per-use pricing mechanisms, and showed that as long as the psychological cost associated with pay-per-use is low, profits from that mechanism are higher than those from selling. Kung and Zhong (2017) formulated an optimisation problem of a two-sided delivery platform by focusing on three pricing strategies, i.e. membership-based pricing, transaction-based pricing, and cross-subsidisation, it was shown these three strategies are equivalent. Different from these studies, which focused on charging one side a fee, taking other price parameters as exogenously given, or maximising the platform's profit, this study will examine the optimal pricing schemes for doctors in an online healthcare platform while jointly optimising settlement fee and medicine price. Katz and Shapiro (1985) firstly defined network externality as a utility, which gets from the number of consumers who consume the same product. With the development of online platforms, most studies have explored the pricing strategy of online platforms based on the network externalities (Parker and Van Alstyne 2005;Wang and Wang 2017;Zhang, Tian, and Xiao 2019). Armstrong (2006) discussed the determinants of the price strategies in monopolistic and competitive platforms, and showed that the network externality is a main factor affecting the platform's pricing-decisions. Lin, Wu, and Zhou (2014) found that in response to higher market liquidity, the platform raises the entry fee of buyers and lowers that of sellers when considering positive cross-side network effect and negative same-side network effect. Dou, He, and Xu (2016) investigated the pricing strategies of a monopolistic two-sided platform when it invests value-added service for one side, and compared with the case without investment, the price for uninvested user side could either increase or decrease as well, depending on the relative strength of network externalities. Niculescu, Wu, and Xu (2018) proposed a game-theoretic model to analytically conceptualise the interplay among the degree of same-side platform openness, the absorptive capacity of the entrant, and the intensity of network effects. Li and Huang (2019) presented a model of healthcare e-commerce platform featuring network externality to investigate its subsidy strategies, and obtained that the network externalities not only affect the subsidy strategies, but also affect the maximum profit of the platform. Therefore, in our study of the online healthcare platform, where the utility of participants on one side (patients or doctors) are affected by the number of entrants on the other side (doctors or patients), we will also consider Table 1. Comparison of this paper with Hagiu and Spulber (2013).

Model setup Differences Similarities
Hagiu and Spulber (2013) The platform charges participation fees p to buyers and w to sellers, and the platform can offer buyers an amount x of first-party content. The platform maximises its profit by setting the participation fees and first-party content.
(1) Hagiu and Spulber (2013) assume sellers are identical, but this paper assume doctors are heterogeneous for their opportunity cost φ are different.
(1) Both papers assume buyer (patient) incurs a heterogenous personal cost of adopting the platform equal to i.
Buyer i incurs a personal cost of adopting the platform equal to i, buyer i's net benefit from joining the platform is (2) Hagiu and Spulber (2013) only consider buyers participating the platform or not, but we consider two kinds of consumers (i.e. only seeking diagnosis, both seeking diagnosis and purchasing medicines), because the healthcare platform both attracts doctors to provide diagnosis services but sells medicines.
(2) Both papers maximise the profit of the platform.
Each seller's net benefit from joining the platform is V(π, x, w) = π(x)n − w − φ, where π(x) > 0 is the profit per buyer made by each seller and φ > 0 is a seller's fixed cost of 'porting' his product to the platform.
(3) Hagiu and Spulber (2013) consider the platform can offer buyers an amount x of first-party content, thus motivating the participation of buyers and sellers, to increase the profit; but in our model the platform stimulates the participation of patients by setting the medicine price, to improve the revenue.
(3) Both papers consider the network externality.
(4) Hagiu and Spulber (2013) explore strategy of the platform under fixed pricing scheme, but we explore the pricing strategy under two pricing schemes (fixed or linear) and compare them to obtain the better one. This paper The platform charges participation fee w to doctors, and sells medicines to patients with price p. The platform optimises its revenue by setting the participation fee and medicine price. Patients are heterogeneous with regard to their type i, and patient incurs a personal cost of adopting the platform, which is equal to i. Patient i's net utility after taking the diagnosis only is U i = u 0 + ms − π − i, and that after both taking the diagnosis and buying medicines is U i = u 0 + ms − π − i − p. Doctors' diagnostic service levels are the same, and differ only in their opportunity cost φ, the gross utility associated with a doctor on the platform is network externality, and explore its impact on the pricing strategies.
Our model mainly refers to the work of Hagiu and Spulber (2013) in terms of the utilities of two parts joining in the platform. The main difference is that our work involves two kinds of consumers (i.e. only seeking diagnosis, both seeking diagnosis and purchasing medicines), and motivates the participation of patients and doctors by properly setting participation fee and medicine price, while Hagiu and Spulber (2013) only consider consumers participating the platform or not, and the platform attracts buyers and sellers by offering first-party content and setting participation fees. Moreover, we list the comparison between Hagiu and Spulber (2013) and this paper in Table 1.
In short, our paper will consider the platform connecting patients and doctors online while selling medicines, and explore how charge modes, market size, and other factors influence the optimal strategies of platforms.

Model description
Combining qualitative findings, which come from interviews with 111.com and its users, with previous studies, we consider a monopolistic online healthcare platform that sells medicines. It connects two user groups: doctors and patients, as illustrated in Figure 1 (see the notation in Table 2). To join the platform, doctors need to pay a settlement fee. Then they can provide diagnostic services for patients online and earn revenue from diagnosis fee. Patients can choose whether to seek diagnostic services or both accept diagnostic services and buy medicines on the platform based on their utility. In the next section, we describe the characteristics of patients and doctors, and propose how the platform can maximise its profit. Moreover, the assumptions involved in this article are firmly grounded in the empirical context (see Table 3) and have strong literature support.
Patients are heterogeneous with regard to their type i, and patient i incurs a personal cost of adopting the platform, which is equal to i. We assume that i is uniformly distributed in [0, N]. Also, a patient enjoys utility s(s ∈ (0, 1)) per doctor on the platform. Since doctors charge where u 0 is the initial utility of a patient, and we assume that patient's initial utility is large enough (i.e. u 0 ≥ π), for the platform has provided many attracting value-added services to the market patients, ms is the value of doctors participation to patients, and π is per-diagnosis fee. Patients will join the platform if, and only if, their net utility is not negative. The expected demand of patients taking a diagnosis is In addition, patients could buy medicines directly on the platform after diagnosis. The platform sells many different kinds of medicines, but in this study, we assume that only one kind of medicine is sold. Patients who both seek diagnosis and purchase medicines on the platform would buy at most one therapeutic course of medicines at price p after diagnosis. Thus, patient j's net utility after both taking the diagnosis and buying medicines is U j = u 0 + ms − π − p − j. Patients will buy medicines only when their net utility is non-negative. Then, the expected demand for buying medicines is The online healthcare platform develops a new patient market for doctors. Therefore, doctors should pay for accessing this online patient market. Building on the online platform literature as well as our extensive research on healthcare platforms, our paper explore two different but common modes to charge doctors, which are fixed and linear pricing schemes. In the fixed pricing scheme, the platform charges a fixed settlement fee to the doctors. From our interviews with managers, doctors, and patients of major healthcare platforms, we learned that patients usually only seek basic and non-complicated diagnosis services online due to communication challenges. So we assume that doctors' diagnostic service levels are the same, and differ only in their opportunity cost φ, which means doctors have different concerns about online diagnosis and their offline profits differ (Dou, He, and Xu 2016;Bai et al. 2019, etc.). Then the gross utility associated with a doctor on the platform is V F (n 1 , w) = 'I worry about the lack of income from online diagnosis.' Doctors differ in their opportunity cost, which means doctors' offline profits are different. Do you have any concerns about joining the platform?
'Though it may ..., attract more patients, I am warried that taking diagnosis online will take too much of my time with family.' '...but it is not as accurate as face-to-face diagnosis, and it may take longer time to communicate, which will take up my normal work time.' To patients: 'The App will push relevant health knowledge to me every day ...' Patient's initial utility is large enough (i.e. u 0 ≥ π). Apart from convenience and good treatment effect, why do you want to choose online diagnosis/medicine?
'...The platform has free pharmacists to guide the purchase, when buying medicines or some small problems are not sure, I can find them to consult first...' 'If the platform provide medical insurance function, video consultation, etc. I will be more inclined to it.' To platform managers: 'Now we provide some value-added services, such as pushing health knowledge, sending regular text greetings, guiding people purchase medicines. And we are considering adding a community interactive section in the App to enhance the communication between doctors and patients, thereby increasing the users' activity and satisfaction. With the development of technology, we will also add video diagnosis function in the future.' Patient's initial utility is large enough (i.e. u 0 ≥ π).
What will you do to attract more people to choose your platform? To platform managers: 'Once patient enter the platform, she can see the current states of doctors (busy or idle), when the patient chooses a doctor to consult, the system will trigger for the doctor's telephone reminder, in order to ensure that the doctor can diagnose timely. And if the doctor is temporarily engaged, the system will match a suitable doctor according to the patient's consultation questions. Thus a doctor usually responds within a few minutes, which will greatly improve the patient's diagnostic experience.' Patient's waiting time is zero.
How to ensure that patients enter the platform can be diagnosed in a timely manner? To patients: 'Diagnosing online is very quicker than going to hospital, ...' Patient's waiting time is zero.
How do you think of the online-diagnosis in terms of waiting time?
'Every time I ask a question, a doctor calls me within minutes.' 'Although the doctor could not reply right away, I would rather consult the doctor on the platform than spend hours going to the hospital.' where n 1 is the demand of patients taking a diagnosis, m is the number of doctors who enter the platform, and φ is the opportunity cost of each doctor, which is uniformly distributed over the interval [0, M]. As such, the doctors will access the platform if and only if their expected utilities are non-negative, and the number of doctors on the platform in the fixed pricing scheme is In the linear pricing scheme, instead of charging doctors a fixed settlement fee, the platform charges doctors a share ratio (α) of the diagnosis fee after each diagnosis, that is, given revenue π per-diagnosis, one doctor must transfer απ to the platform and retain the rest (1 − α)π. It is natural to assume α ∈ (0, 1). Therefore, each doctor's net utility is Similarly, the participation of doctors in the linear pricing scheme is The online healthcare platform not only connects patients and doctors but also sells medicines. Without loss of generality, we assumed that the cost of medicines is zero. In the fixed pricing scheme, the platform charges the medicine fee to patients and collects a settlement fee from doctors. Then, the platform's profit is F P (p, w) = pn 2 + mw. In the linear pricing scheme, the platform charges the medicine fee to patients and extracts a ratio of revenue from doctors instead of charging doctors a settlement fee. Thus, the platform's profit is L P (p) = απn 1 + pn 2 .

Fixed pricing scheme
In this section, the optimal strategies for medicine price and settlement fee in fixed pricing scheme are investigated. The sequence of events is as follows: First, the platform sets the settlement fee (w) for doctors, and the medicine price (p) for patients. Next, the individual doctor decides whether to participate in the platform to provide diagnosis services, and the patients decide whether to seek diagnosis only or both seek diagnosis and purchase medicines on the platform. The online healthcare platform chooses the optimal medicine price and settlement fee to maximise its expected profit with constraints, which can be expressed as follows: By optimising the platform's profit in (5) with the above constraints, we can derive the optimal solutions in the fixed pricing scheme, which are shown in Table 4, where p F * is the optimal medicine price, w F * is the optimal settlement fee, and F * is the maximum profit of the platform. And Theorem 4.1 shows the corresponding conditions of each potential optimal solution.
Theorem 4.1: The optimal solutions of online healthcare platform in the fixed pricing scheme are given by: Theorem 4.1 demonstrates a five-threshold optimal strategy for the platform as the patient's initial utility Table 4. Potential optimal solutions of the fixed pricing scheme.
increases, as well as respective number of participating patients (n F 1 , n F 2 ) and doctors (m F ). From the perspective of the patient's initial utility, as it increases, the platform should set different pricing strategies to attract more patients for diagnosis and medicines. When the patient's initial utility is low in (1) and (2) of Theorem 4.1, the optimal strategy as Case1 and Case2 is to set the entrance fee and medicine price with part of patients accepting diagnosis and buying medicines. When the patient's initial utility is medium, as in (3), (4) and (5) of Theorem 4.1, we will get the optimal joint pricing solutions for Case3 and Case4, or Case5, attracting all the patients to ask for diagnosis but part of them buying medicines. Last, when the patient's initial utility is relatively large as u 0 ≥ max{π + N − Ms, 2−s 2 2 π + 4−s 2 4 N}, then under the optimal pricing strategies Case6 and Case7, the number of patients seeking diagnosis reaches market saturation. Furthermore, in this case, whether the platform sets a joint pricing to make all the patients buy medicines relies on the market size of doctors, which is decided by the tradeoff between the revenue from doctors and that of selling medicines. The logic behind the variation of the optimal joint pricing is that the platform could improve the participation of patients by attracting more doctors (i.e. lowering doctors' entrance fee) and lowering medicine price. With the increase of patient's initial utility, the cost of bringing more patients by pricing decreases, then the platform prefers to adjust the joint pricing of the entrance fee and medicine to attract more patients for diagnosis and medicines, which increases its profit. Figure 2 clearly shows the optimal solutions in the fixed pricing scheme.
Based on the optimal solutions in Table 4 and Theorem 4.1, we know the potential optimal solutions rely on patient's initial utility and market size as well as marginal benefit of doctor participation. Thus, we firstly examine how the medicine price and fixed settlement fee change with these parameters and explore the special behaviours that will give the platform important implications. The features are summarised in Proposition 4.1 and Proposition 4.2. Then we explore how do network externalities generated by the healthcare platform affect users decisions and the platform profit, as shown in Proposition 4.3.

Proposition 4.1:
In the fixed pricing scheme, (1) As Proposition 4.1 shows, when the patient's utility is at a low level, which means patients do not trust its services, so the platform earns little from selling medicines, then the platform's main revenue comes from the doctor's entrance fee. In this state, with the increase of patient's utility, the platform could improve its marginal revenue from doctors by increasing the entrance fee, while little loss occurs from its negative effect on the doctors' participation. When the patient's utility is at a medium level, as it increases, the number of patients buying medicines online increases, then the revenue from selling medicines plays an important role. And the platform could decrease the doctor's entrance fee to attract more doctors, which increases the patient's utility from network externalities as well as the number of patients buying medicines. Moreover, the increase of revenue from selling medicines is higher than the loss of revenue from doctors by lowering the entrance fee. Last, when the patient's initial utility is large enough, that is, the platform has greater confidence in the entry of patients, the platform will not pay too much attention to the utility of patients when setting the settlement fee. Besides, the medicine price increases with the patient's initial utility. This is because the more confidence patients have on the platform, the more patients will enter the platform and buy medicine, which allows the platform to increase the medicine price to make higher profit. We know patient's initial utility depends more on the efforts made by the platform, including but not limited to the platform's promotion of its services. Proposition 4.1 helps the healthcare platform to understand the importance of grasping patient's initial utility and making appropriate pricing decisions.
Next, we will examine how market sizes of doctors and patients as well as marginal benefit of doctor participation affect the optimal settlement fee and medicine price.

Proposition 4.2:
In the fixed pricing scheme, the sensitivity analysis on the market size of doctors M, the market size of patients N, and the marginal benefit of doctor participation s can be obtained as: (2) There are four cases about the impacts of the market size of patients N; There are three cases about the marginal benefit of doctor participation s; Proposition 4.2 presents the impact of different parameters on the potential optimal solutions in the fixed pricing scheme. The intuition behind Proposition 4.2(1) is that increases in the market size of doctors lead to the platform lowering the threshold to attract more doctors to participate in the platform, because the increasing number of doctors diagnosing in the platform has a larger network externality to the patients, then motivates more patients to seek diagnosis online. In addition, the optimal medicine price increases with doctors' market size because more potential doctors in the platform will increase the patient's utility, which indirectly prompts the platform to raise medicine retailing prices to earn more profit.
From Proposition 4.2(2), we can know that there are three thresholds for patients' market size (N), and when the market size of patients is relatively small, if the market size of doctors is small, the optimal settlement fee (w F ) increases with patients' market size, but if doctors' market size (M) is large, the optimal settlement fee decreases with patients' market size. Next, with the increase of patients' market size, the optimal settlement fee increases with patients' market size, then decreases with it, and finally irrelevant to it. This is because more potential patients seek for online healthcare services will indirectly promote higher demand for doctors, so the platform needs to lower settlement fee to let more doctors participate. In addition, we can conclude that the medicine price decreases first and then increases as patients' market size increases, which means when the potential demand for online healthcare services is small, lowering the medicine price to attract more patients can bring higher profit, but a higher potential demand for online healthcare services promotes the platform to raise the medicine price, then a higher profit can be achieved.
In addition, when it comes to the impact of the marginal benefit of doctor participation on the optimal solutions, as shown in Proposition 4.2(3), the settlement fee generally decreases as the strength of the network externalities increases. This implies that the stronger the network externalities of doctors to patients, the lower settlement fee should the platform set for the doctors. Because as the increase of the strength of the network externalities, more patients would take diagnosis online and buy medicines with a large number of doctors on the platform. Thus, the platform has incentive to lower the settlement fee to attract more doctors participating, while the revenue brought by larger number of patients taking diagnosis and buying medicines dominates the loss of revenue from doctors. Besides, we can find that in a small region, i.e. when the strength of the network externalities falls at a medium level, the settlement fee increases with the strength of the network externalities. The reason is that under certain condition, the decrease of the doctors' participation would not lower the number of patients taking diagnosis and buying medicines, as the strength of the network externalities increases. Then the platform could slightly increase the settlement fee for doctors to obtain more profit. Last, the medicine price increases with the strength of the network externalities, because the stronger network externality promotes more patients joining in the platform, then a higher profit can be achieved by raising the medicine price without worrying about the lack of patients on the platform.
From Proposition 4.2, we know doctors' market size has a monotonous effect on the settlement fee and medicine price. When doctors' market size becomes larger, the platform could lower the settlement fee but raise medicine price to get more profit. However, the impact of patients' market size on pricing is much complicated, where the adjustment of the optimal prices relies on the range of patients' market size. In addition, when the platform perceives a relatively higher level of network externalities, it can set a lower settlement fee, but a higher medicine price. In order to further explore the effect of network externalities, then we investigate how network externalities generated by the healthcare platform affect the users' participation and the platform's profit.

Proposition 4.3:
In the fixed pricing scheme, when the marginal benefit of doctors participation increases, both the demands of patients for diagnosis (n 1 ) and medicines (n 2 ) increase until they reach saturation, and the maximum profit of the platform ( F ) also varies with the strength of the network externality, i.e. ∂ F * ∂s > 0.
Through Proposition 4.3, it can be intuitively observed that if the scale of network externalities is greater, more patients will seek diagnosis online, and thus the demand for medicines increases. The reason is that, with the strength of network externality increases, unit participation of doctors on the platform has a greater impact on the patient's utility. Then the expansion of the doctors will attract more patients to participate, and the volume of diagnosis and buying medicines increases. In addition, with the increase of the strength of network externalities, the profit of the platform becomes greater. Because with the increases of network externalities, not only the demands for diagnosis and medicines increase, but also the medicine price increases, which both bring higher profits for the platform. In short, the network externalities have a positive effect on the platform's profit, and in practice the platform could provide value-added services (i.e. services that patients can enjoy free of charge) to increase the strength of network externality, such as establishing doctor-patient communication community to enhance the doctor-patient interaction.

Linear pricing scheme
The linear pricing scheme is similar to the fixed scheme, except that there is a share ratio for the diagnosis fee that the platform receives from doctors, but the platform will not charge doctors a fixed settlement fee. That is, there are no barriers preventing doctors accessing the platform, but the platform will get an απ per-diagnosis. Next, we present the equilibrium solutions in the linear pricing scheme and explore some characteristics of the optimal solutions. The model can be reformulated as s.t. m L = min{(1 − α)πn 1 /m, M}, n 1 = min{u 0 + ms − π , N}, n 2 = min{max{u 0 + ms − π − p, 0}, N}, 0 Solving the objective function (6) as well as the above constraints allows the optimal solutions to be obtained, which can be seen in Table 5, where p L * is the optimal medicine price, and L * is the maximum profit of the platform. We also show the conditions under which the potential solutions will be optimal in Theorem 5.1. Denote that Theorem 5.1: In the linear pricing scheme, the platform's optimal strategies are shown as follows: (1−α)π + π − Ms}, then Case 1 is optimal, and n L 1 = u 0 + m 1 s − π, n L 2 = u 0 +m 1 s−π 2 , m = m 1 ; (2) When max{π , π + N − s As Theorem 5.1 shows, there are three-thresholds for patient's initial utility, which divide the optimal strategies into six cases as shown in Table 5. Figure 3 shows the optimal pricing solutions of medicines in different regions in terms of patients' market size (N) and patient's initial utility (u 0 ). Theorem 5.1 also presents the demands of patients taking a diagnosis, buying medicines, and the number of doctors providing online services under the optimal pricing. We can find that when patient's initial utility is relatively large, i.e. u 0 ≥ max{π , π + N − s √ (1 − α)πN, π + N − Ms}, then the platform should set an optimal medicine price to attract all patients buying medicines after diagnosis. Besides, when the market size of doctors is higher than a threshold, i.e. M ≥ √ (1 − α)πN, the optimal medicine price is related to the share ratio, which shows the necessity of joint pricing for doctors and medicines.
Next, we will examine how patient's initial utility, market sizes of doctors and patients as well as marginal benefit of doctor participation affect the optimal medicine price.  (1) Proposition 5.1 presents the impacts of different parameters on the optimal medicine price in the linear pricing scheme. First, the optimal medicine price increases with the patient's initial utility. Because more patients choose online diagnosis with the increase of the initial utility, then the platform will raise medicine price to maximise its profit. Second, we observe that the relationship between the optimal medicine price and patient's market size is not monotonous. There are seven thresholds for the patient's market size. If the patient's market size is relatively small (i.e. N ≤ max{ M 2 (1−α)π , u 0 +Ms−π 2 }), the optimal medicine price will decrease after two peaks as the patient's market size increases; and if the patient's market size is relatively large, the medicine price is irrelevant to the patient's market size. This is because when there is less demand for online diagnosis, the platform needs to balance its profit with the number of patients entering the platform according to the actual situation, so as to raise or lower the medicine price. And when the demand for online diagnosis reaches a higher level, the platform will set medicine price without considering its impact on patients. Third, the optimal medicine price increases with the doctor's market size, because the larger the doctor's market size, the greater the patient's utility from network externality, which would compensate for patient loss due to the higher price. Finally, with the increase of the strength of network externalities, the optimal medicine price increases. We know the network externalities are decided by ms, so the strength of network externalities (s) and the doctor's market size (M) have the same effect on the optimal solutions, the stronger the network externalities, the more patients are willing to seek services and buy medicines on the platform, then the platform can raise the price of medicine to maximise its profit.
From this proposition, we know in the linear pricing scheme, the platform should set the medicine price by considering several variables. Specifically, the optimal medicine price increases with the patient's initial utility, doctor's market size, and the strength of network externalities. Besides, the platform needs to adjust the medicine price in accordance to changes in patient's market size, by balancing the revenue brought by higher demand for online medicines and a larger marginal profit of selling medicines.
Last, in the linear pricing scheme, the demand of patients for medicines is also affected by the number of doctors participating in the platform, which is in turn decided by the share ratio for doctors, so we next explore the impact of doctor's share ratio on medicine pricing. From Proposition 5.2, we can see that doctor takes the entrance cost (i.e. the share ratio α) into account, when deciding whether to participate in the platform. Obviously, when the ratio is too large, the platform will take more of the doctors' income, which reduces the doctors' willingness to enter the platform, then the number of doctors enters the platform turns to be non-increasing with the increase of share ratio. In addition, the decrease of doctors number will indirectly affect the efficiency of the platform's diagnosis, reducing patient's onlinediagnosing experience. Therefore, it is necessary for the platform to make up for patients by reducing medicine price. Moreover, when the online-diagnosing services become more popular, all the doctors in the market are willing to enter the platform to provide services, and the number of doctors in the platform reaches saturation. At this time, the platform will consider more from the perspective of patients and its own profit when pricing medicines, so the price of medicine will not be affected by the share ratio. In practice, when only a part of the total doctors population join in the platform, the platform should consider lowering the medicine price if a higher share ratio is already set to balance service and medicine revenue.
In addition, to further explore the effect of network externalities on the platform, we investigate how patients' participation and the platform's profit vary with the network externalities.

Proposition 5.3:
In the linear pricing scheme, when the strength of network externality brought by online doctors to patients increases, both the demands of patients for diagnosis (n 1 ) and medicines (n 2 ) increase until they reach saturation, and the maximum profit of the platform ( L ) also increases with the strength of the network, i.e. ∂ L * ∂s > 0.
Proposition 5.3 shows that in the linear pricing scheme, the greater the network externalities of the doctors, the more patients will seek diagnosis online, and thus there would be more patients buying medicines online. With the development of the healthcare platform, the strength of network externality increases, then unit participation of doctors on the platform has a greater positive impact on patient's utility. Therefore, the same number of online doctors will attract more patients to participate, and the demands for online diagnosis and medicines increase, which in turn increases the profit of the platform. Thus, in practice, the platform needs to take some measures to strengthen the strength of network externality, such as providing value-added services and strengthening doctor-patient interaction.

Comparison
As of now, we have examined the optimal pricing decisions within each service pricing scheme. From a profitmaximising point-of-view, however, an online healthcare platform also needs to decide the pricing scheme, either fixed or linear, that needs to be implemented. In this section, we explore which pricing scheme is more advantageous for the healthcare platform, and the result is given in Proposition 5.3.

Proposition 6.1:
Proposition 6.1 reveals that when patient's initial utility is relatively large, which can be enhanced by providing more value-added services, the platform is better to choose the fixed pricing scheme to get more profits. The reason why fixed scheme is better than linear scheme is that the online healthcare industry is immature, which means that there are greater risks. Thus, the settlement fees paid by the participating doctors will ensure that the platform earns more fixed profit than it would by using the linear pricing scheme where the platform needs to undertake the risk of less participating patients. This result is consistent with the results of some previous studies (Dietl, Lang, and Lin 2013;Kung and Zhong 2017). In addition, if the platform wants to apply the linear pricing scheme, then the share ratio should be set at a certain value (i.e. α = max{1 − M 2 πN , 1 − s 2 N 4π }); otherwise, the profit obtained by the platform will be less than that in the fixed pricing scheme. Besides, we see that when the patient's initial utility is relatively small, it is hard to tell which pricing scheme makes the platform more profitable, because with small patient's initial utility, the number of participating patients is low, which makes the platform less attractive to doctors. In this case, liner pricing scheme has an advantage of attracting more doctors for no entrance barrier; while fixed pricing scheme takes an advantage on ensuring the platform fixed revenue from doctors. Which advantage of two pricing schemes dominates depends on many factors, such as the doctor's market size, the patient's market size and the strength of network externalities on the platform. In summary, when the healthcare platform makes a charging mechanism for doctors, mostly a fixed pricing scheme is more profitable.

Discussions
Online healthcare platforms are quickly changing how patients seek diagnosis services and purchase medicines. In this article, we establish a mathematical model to explore how online healthcare platforms should jointly decide medicine price and settlement fee to maximise the platform's profit. Major assumptions of the math model are firmly grounded in the platform economy literature as well as our extensive research about 111.com, a major online healthcare platform in China that not only connects doctors and patients but also sells medicines online. We further used interviews with platform managers, doctors, and patients to triangulate the normative findings derived from the math model to reveal real-world online platform pricing and participating behaviours. We will firstly discuss the reality-check of quantitative findings, and then discuss the theoretical and managerial implications of our main findings.

Reality-check of quantitative findings: the platform's pricing decision
To better understand how the platform intuitively makes pricing decision with multiple influencing factors, we conducted a 2 nd round of interviews with two product managers of the healthcare platform in April of 2021. And we compare their pricing strategies with the normative mathematical results in our study (Table 6), which provide a reality-check of the quantitative results (i.e. whether results make sense in the real world, and if not, why) (Breitmayer 1991;Farmer et al. 2006). And from Table 6, we learn that there are similarities and differences between the decisions made by product managers of the platform and our math model results.

Theoretical implications
A unique feature of the online healthcare platform examined in this study is the dual revenue streams of the platform: the platform not only obtains revenue from doctors, or the service providers, but also from the service buyers (i.e. patients), by selling medicines. Therefore, the revenue stream of such a two-sided platform is more complex than traditional two-sided online platforms, which are only intermediaries connecting service providers and consumers without selling products (Dou, He, and Xu 2016;Liu et al. 2019). By mathematically modeling the pricing decisions while considering social factors, such as network externality, this study extends the online platform behavioural pricing literature in several ways.
First, our study shows the importance of jointly optimising product/service pricing decisions for maximising the platform's profit, with the existence of network externalities between doctors and patients on the two-sided platform. For instance, our normative results indicate that, when the linear price scheme for charging doctors is adopted, then if the number of doctors on the platform fails to reach market saturation, the optimal medicine price should decrease as the share ratio increases. The reason is that with the increase of share ratio, the number of doctors offering diagnosis on the platform decreases, then reducing the patient's utility. So the platform needs to lower the medicine price to attract more patients to buy medicines online. Moreover, when all the doctors in the market are willing to join the platform to provide services, the platform could optimise the medicine price regardless of the share ratio with doctors, because the share ratio could change the number of participating doctors, and then affect the demand of patients buying medicines. Once all the doctors join the platform, the number of participating doctors is constant as M, resulting in the impact of the share ratio on patients' utility disappears. Our finding is quite different from the existing studies in terms of the effects of network externalities on the platform's pricing strategy. This difference is due to the fact that the literature only considers one revenue stream of the platform (Kung and Zhong 2017), Specifically, Kung and Zhong (2017) explores the interaction of fees setting for customers and shippers (i.e. service providers) through network externalities, and find that the optimal charging fee from the customer is positively associated with the fee paid to shippers. However, 'Sure, the settlement fee depends on the market size. If the number of patients is high enough, our platform becomes more attractive to doctors, which allows us to set a higher settlement fee. If the number of doctors on the platform is high, we will reduce the fee because we can earn more revenue from more doctors, each paying us a little bit lower settlement fee.' This practice is generally consistent with recommendations from Proposition 4.2(1).
Proposition 4.2(2) shows that the market size of patients has a complex effect on the optimal settlement fee, such as when the number of patients is at a high level, the optimal settlement fee could decrease with, and ultimately be independent of, the number of patients.
When pricing medicine, do you consider the potential number of patients or doctors? If yes, how?
'Of course, when the number of patients buying medicines increases, we will definitely increase the prices of medicines to increase our income. But for now, because the platform is still in the early stage of expansion, we may consider setting a lower price than offline pharmacies to attract patients. And we didn't think too much about the number of doctors when making medicine pricing decisions.' The medicine pricing practice is partially consistent with Proposition 5.1(2) of the linear pricing scheme, which states about a positive relationship between the number of patients and medicine price.
However, Proposition 5.1(2) shows that the medicine price does not always increase with the number of patients, when the potential number of patients reaches a certain peak, medicine price remains the same.
In addition, we obtain ∂p/∂M ≥ 0 in our research, but in practice, the platform haven't considered their relationship.
When deciding medicine price, do you consider the share ratio of diagnosis fee?
'At present, we set a 40% share rate for doctors independently from our medicine pricing decision, because medicine sales and doctor management are separate departments, these decisions are made by different units at different time, instead of by one unit in an integrated manner. Because our platform is in the initial expansion period, it is more important for us to make decisions that can increase platform traffic.' The interview shows inconsistence between practice and normative math results. Specifically, from Proposition 5.2, we learn when not all docotrs are willing to enter the platform to provide services, the medicine price should decrease as the share ratio increases. Due to a lack of integrated decisionmaking system, the platform is not considering the share ratio when pricing medicine.
When deciding the charging mode for doctors, do you consider the medicine pricing decisions?
'Currently, we make these two types of decisions independently. Now we charge doctors a fee based on the proportion of revenue, about forty percent. In terms of medicine pricing, we mainly set medicine prices by comparing prices with our competitors and the cost of selling medicines. Maybe in the future, we will consider more comprehensive to optimise our decisions and enhance our competitiveness.' The current platform practice does not follow the recommendations from our results, which shows that the optimal medicine prices should differ between the two charging modes for doctors (i.e. fixed and linear). However, in practice, these decisions are made independently, thus showing no influence of the charging mode for doctors on medicine prices.
Comparing the fixed VS linear pricing scheme for doctors, which one is more beneficial to the initial and long-term development of the platform?
'I think in the initial development period, linear transaction-based scheme is favourable, which has no entry barrier for doctors, and they pay for the platform based on their revenue. But from the perspective of long-term development, I think the fixed scheme may benefit the platform more. Because with a fixed settlement fee, doctors can keep their whole revenue, which will motivate them to improve their service for patients.' From Proposition 6.1, we see that when patient's initial utility is relatively small, either scheme could be better, so the platform currently charges doctors with a linear pricing scheme may be reasonable.
With patient's initial utility increases, the fixed pricing scheme is better than the linear pricing scheme, which is consistent with the platform managers' insights, they believe that a fixed settlement fee may benefit the platform in the long term.
a In order to better understanding the development of online healthcare platform, and with the support from 111.com, we were able to interview both participating and non-participating doctors and patients as well as platform managers to gain a diverse set of insights about online healthcare platform. In total, we interviewed five participating doctors and five participating patients, four non-participating doctors and four non-participating patients who never participate the platform, and two platform managers who are responsible for the development and maintenance of platform. All the interviews were conducted by phone calls with two researchers from September to October in 2020, with the duration of interviews ranges from 11 to 36 minutes. When interviewing doctors, a key focus is to understand what drive their participation on 111.com, and which pricing scheme is more reasonable in their mind: fixed vs linear? When interviewing patients, we focused on understanding their opinions about the online diagnosis and value-added services. And when interviewing the platform managers, we focused on the decisions around service/medicine pricing.
considering the dual revenue streams of the platform, our study shows that in the linear scheme, the charging fee from doctors is negatively associated with the medicine price paid by patients through network externalities. Our theoretical results indicate that the number of revenue streams on the platform could change the impact of network externalities on the relationships between different pricing decisions. Second, our study reveals in the platform economy, the optimal pricing decisions of online healthcare platforms depend on the initial utility of patients, the market size of doctors and the strength of network externalities. Our math model shows that when the healthcare platform prices the medicines, the optimal price of medicines increases with the patient's initial utility, the market size of doctors, and the strength of network externalities. And this is different from the literature about on-demand service platforms (i.e. Bai et al. 2019;Benjaafar and Hu 2020), which show the optimal price is not necessarily monotonic when the market size of service providers, or the number of doctors in our study, increases. This difference shows an important difference between onesided and two-sided online platforms in terms of how the market sizes affect pricing decisions. These results set up a stage for future theoretical endeavours to explore the relationship between market sizes of different populations and pricing decisions in online platforms which have multiple streams of revenue.
Finally, our study contributes to the platform pricing literature by comparing the performance implications of two types of service pricing schemes. The math model shows that in most cases the fixed pricing scheme is optimal from the platform's view, in terms of maximising the platform's profit. This result contributes to the online platform pricing literature, which has started to compare performance implications of different pricing schemes for service providers (Sundararajan 2004;Dietl, Lang, and Lin 2013;Kung and Zhong 2017). However, none of the three above literature considered more than one revenue stream of the online platform but instead focus only on the transactions between the platform and service providers. Specifically, our model considers the platform has two revenue streams, and when the initial utility of patients is greater than a certain threshold, the fixed pricing scheme performs better than the linear pricing scheme. However, when the initial utility of patients is small, the size relationship of the platform profit under the two schemes cannot be obtained precisely, which is different from the result of Sundararajan (2004), Dietl, Lang, and Lin (2013) and Kung and Zhong (2017) that the fixed pricing strategy always dominates the linear pricing strategy. Therefore, with more revenue streams of the platform considered, the performance implications of pricing schemes (fixed vs linear) for the platform are distinctively different, since these pricing schemes vary on how they motivate participation of service providers, which would further affect the participation of service users as well as product demand for the platform.

Practical implications
Although most mathematical propositions are consistent with the platform's pricing behaviours, we do identify some inconsistent areas. For instance, the online healthcare platform 111.com is using the linear pricing scheme when collecting revenue from participating doctors. This practice is different from the normative mathematical results, which suggests a fixed fee scheme as the better option. When interviewing with the platform managers, we learn about the rationale behind adopting the linear pricing scheme, which sets a lower threshold of entry for doctors and is supposed to reduce the perceived risk of participation by doctors (i.e. you only pay the platform when you get revenue from patients). Therefore, given that the platform is still in its early stage of market expansion, the linear pricing scheme still has its virtue, though the platform admits that, in the long term, it makes sense to switch to the fixed pricing scheme for doctors. And from the interviews with doctors, we learned that a fixed pricing scheme increases the perceived fairness by doctors 3 , since they can keep 100% of the diagnosis revenue after paying a fixed settlement fee. Thus, our results indicate a need for the platform to consider adopting a fixed pricing scheme in the future to ensure a sustainable revenue stream from doctors. Then if the platform adopts the fixed pricing scheme in the future, both the participation of patients and doctors should be considered to set the settlement fee. When there is a small size of patients joining the platform, then a low settlement fee or even a subsidy for doctors should be adopted. The reason is that a lower settlement fee will let more doctors to enter the platform, then through the network externality to attract more patients seeking services from platform, which forms a virtuous cycle. As the flow of patients and doctors increases, then the platform could strategically raise the settlement fee, without loss of participants. That is, strategic pricing for doctors is more reasonable in practice. Although in the early stage the platform may get less from doctors or subsidise doctors, in the long run, it will benefit from the strategic pricing.
In addition, when pricing medicines, managers of 111.com usually assume that patients decide whether to buy medicines online based on the utilities they realise directly from the platform. Therefore, the platform can improve the patients' utilities by providing additional value-added services (such as the establishment of doctor-patient communication community, pharmacist shopping guide, caring message, etc.) to increase the probability of buying medicines. However, the utility brought by the network externalities of participating doctors seems not recognised by platform managers. From our study, the optimal medicine price is positively associated with the strength of network externalities, in addition to the effect of patient's direct utility from the platform. Although the platform cannot regulate the network externalities, it can estimate the network externalities by observing the impact of more doctors' access on the number of diagnosing or buying medicines. Because the network externalities work as follows: the increase in the number of doctors will improve the utility of patients receiving diagnosis or buying medicines on the platform, thus increasing the number of patients receiving diagnosis and buying medicines. At the same time, the increase in the number of patients on the platform will also increase the doctors' revenue of providing services. So the platform should not only provide better value-added services to strengthen doctors'/patients' preferences, thus guiding more users join the platform, but also lower the settlement fee for doctors based on the scale of network externalities so as to expand the platform market more efficiently. In this way, more patients will join the platform due to the indirect value they obtain from accessing a bigger online doctor market.

Limitations and future directions
Our study investigates how the online 'diagnosis + medicine purchase' healthcare platform determines the medicine price and settlement fee jointly with considering network externality. Although we have identified some interesting results from a theoretically and empirically grounded math model, there are still gaps between our model setting and the real-world practice. For instance, we assume the platform only provides one type of medicine to keep the model tractable so that we could focus on the interactions among the platform, patients and doctors. In practice, a platform could offer different types of medicines. Thus, future research can extend our work by considering heterogeneity among medicines (e.g. branded vs generic).
In addition, to keep the model tractable, our study considers the platform as a monopolist so that we do not need to model competition. This assumption is also grounded in our extensive research about 111.com, which is by far the dominating healthcare platform that offers both diagnosis services and medicines in China. However, in other countries or a different platform context, there could be multiple online healthcare platforms competing for doctors and patients, or online platforms competing with brick-and-mortar pharmacies. Therefore, a promising future research direction is to consider examining how different types of competitions affect pricing and service investment decisions.
Finally, we do not consider the risk profiles of human decision makers in this study. Instead, we assume all parties are risk neutral. However, on an online healthcare platform, there could exist different types of risks faced by the platform, doctors and patients. For instance, since the diagnosis service and medicines are directly related to patients' health, there exist health risks for patients and liability risks for both doctors and a healthcare platform, which could be seen from our interviews. Therefore, future studies could examine how risk aversion affect the joint optimisation outcomes of an online platform's pricing and investment decisions as well as participating behaviours of doctors and patients. Tingting Yan is a professor in the Marketing and Global Supply Chain department in Mike Ilitch School of Business, Wayne State University. She is a Charles H. Gershenson Distinguished Faculty Fellow. Her research is primarily around answering one question: How could a firm better innovate by leverage resources in its supply network as well as those in suppliers' extended networks? She used a wide diversity of research methodologies and data sources, such as surveys, interviews, secondary quantitative data, experiments (lab, vignette, field intervention), simulations, etc. She is a Co-Editor in Chief for Journal of Supply Chain Management, a leading academic journal in the field of supply chain management. She also serves as an associate editor for Journal of Operations Management, and is an Editorial Board reviewer for