The Effect of Bundled Payment Schemes on Cost–Speed Tradeoff for Outpatient Service: A Queueing-Game Analysis

Abstract

In recent years, payment schemes in healthcare have garnered attention for their potential impact on service delivery and cost management. This paper explores the impact of the bundled payment scheme (BP) on hospital outpatient services, focusing on the cost–speed tradeoff. Specifically, a higher service rate increases patient demand but also raises medical costs. We consider a queueing-game theoretical model to analyze servers’ service rate behaviors under different payment schemes (fee-for-service and BP) and the payer’s optimal payment scheme setting. Our study shows that achieving the first-best outcome under centralized decision making using the BP requires specific conditions. When the medical budget is sufficiently high, the payer can guide hospitals toward the first-best decision by setting an optimal price under the BP. However, when the budget is at an intermediate level, hospitals may set slower equilibrium service rates to control costs. To address this issue, the payer can implement service level regulation based on the BP scheme to achieve the first-best outcome. This scheme encourages hospitals to choose higher service rates by limiting expected waiting times. When the budget is too low, hospitals may be unwilling to provide service due to unprofitability. Moreover, as competition between hospitals intensifies, it becomes easier to maximize social welfare under the BP scheme.

1. Introduction

The aging of the population and the evolution of the medical insurance system have led to an increase in demand for healthcare services, particularly for outpatient services. A key factor influencing the quality and satisfaction of outpatient services is consultation time (consultation time refers to the time that doctors spend on patients while providing outpatient consultation services). Unfortunately, many doctors still spend limited time with patients during consultations. For example, a study analyzing consultation times in 67 countries found that in 18 countries, about 50% of the global population experiences consultations of less than 5 min with primary care physicians [1]. In 2019, a report highlighted that waiting for up to two hours for a two-minute consultation with a doctor had become a major issue in the pediatric department of public hospitals (https://finance.sina.cn/chanjing/gdxw/2019-04-17/detail-ihvhiqax3547366.d.html (accessed on 15 October 2025)). In 2022, at a tertiary comprehensive hospital in Zhejiang Province, China, a doctor, during a 7 h workday, has to conduct consultations for up to 150 patients. The average consultation time allocated to each patient was less than 3 min (https://new.qq.com/rain/a/20220823A0976T00 (accessed on 15 October 2025)).

The short consultation time in outpatient care can reduce patients’ waiting time; however, it may also lead to increased medical costs. Outpatient services cover a wide range of diagnostic tests, prescriptions, treatment services, and minor outpatient surgical procedures when necessary [2]. When healthcare providers do not conduct careful inquiries and examinations due to time constraints, it can lead to a decline in the quality of diagnosis and treatment, including even misdiagnosis. As a result, during follow-up treatment, the service provider may need to perform more high-level medical tests, prescribe more effective drugs and equipment, and carry out more complex surgeries, resulting in higher medical costs for an entire episode of care [3]. Furthermore, Guan et al. discovered a correlation between inappropriate prescriptions and workload, indicating that as the workload increases, the incidence of inappropriate prescriptions is higher [4]. Therefore, the rapid pace of outpatient services poses a high risk of prescribing inappropriately, which may further increase medical costs.

A key factor contributing to short consultation times and high medical costs is the payment scheme used to reimburse healthcare providers. Payment schemes regulate provider behavior, control costs, and improve care quality [5]. The most commonly used medical insurance payment scheme is fee-for-service (FFS). Under FFS, service providers are paid based on the volume of services performed. This payment structure incentivizes providers to treat more patients quickly [6]. While FFS has been effective in reducing waiting times [7], it is also associated with lower service quality and higher medical costs [8].

Bundled payment (BP), a prospective payment, is currently a key medical insurance payment scheme being promoted in China. The principle is that the provider is compensated with one fixed lump sum for the entire episode of care, regardless of the specific tests and procedures conducted or any complications that may arise [9]. Under the BP scheme, hospitals assume all financial risks during the episode of care. If the patient is not cured within the allotted payment, the hospital covers the excess medical costs, while any surplus becomes the hospital’s profit [10]. Therefore, hospitals are motivated to provide high-quality medical services while controlling medical costs. The purpose of this payment scheme is to transfer the responsibility of controlling medical costs and ensuring service quality to hospitals, which would have an impact on the hospital’s operational management. The BP scheme originated in the United States in 1984, and, since then, many countries, including the Netherlands, Sweden, the United Kingdom, Canada, and China, have adopted the BP scheme as part of their medical reforms (https://healthcarefunding.ca/key-issues/bundle-test-2/ (accessed on 16 October 2025)). For instance, Sweden uses BP for total hip and knee replacement, and the National Health Services in the UK uses BP for cystic fibrosis and maternal care [11]. In China, the BP scheme for chronic and special critical diseases in outpatient settings implemented in Liuzhou City, Guangxi, has effectively controlled the unreasonable growth of outpatient medical expenses [12]; the implementation of the Big Data Diagnosis Intervention Packet in Guangzhou, Guangdong, and Diagnosis Related Groups in Shenzhen, Guangdong, have achieved the dual goals of controlling medical insurance fees and incentivizing medical institutions [13]. However, studies have shown that compared with FFS, the BP scheme may increase patients’ waiting time [14].

Under the BP scheme, the hospitals bear the total medical cost for a whole episode of care. As mentioned above, short outpatient consultation times may not only lead to high medical costs, but they also attract more patients at the same time because of the short wait time. As a result, under the BP, hospitals face a cost–speed tradeoff. Despite observed implementation effects in practice, the broader implications and impact of the BP scheme on healthcare operational efficiency, such as how to approach the cost–speed tradeoff, remain unclear. Therefore, many public healthcare systems continue to use the FFS.

To examine the impact of the BP scheme on the equilibrium behavior of hospitals providing outpatient elective care, we develop a three-stage Stackelberg game to characterize the public outpatient service system, which includes a payer, n hospitals, and a population of patients. The payer, acting as a social welfare maximizer or social cost minimizer, selects a payment scheme and designs its parameters. The hospitals consider the reimbursement they receive for providing medical services, including diagnostic services, medical tests, and drug therapy, as well as the total medical cost per patient for an entire episode of care, when making decisions about their service rates. The objective of the hospitals is to maximize profits, with higher service rates leading to higher equilibrium arrival rates and increased medical costs, while patients choose which hospital to seek medical care from based on the expected utility of each hospital, which takes into account the service value and the expected delay cost.

Based on these settings, we constructed a game-theoretical queueing model to capture the three-stage Stackelberg game and the interactions among all three parties. This framework allows us to analyze how hospitals choose service rates under different payment schemes and how patients respond through their hospital choices and compare these decentralized outcomes with a centralized benchmark. On this basis, the main contributions of this paper are summarized as follows:

(1) We incorporate a cost–speed tradeoff into a competitive public outpatient service system and develop a three-stage Stackelberg queueing-game model with one payer, n competing hospitals, and a pool of patients under both FFS and BP schemes. In our model, hospitals’ service rates simultaneously affect patients’ waiting time and the total medical cost for an entire episode of care.

(2) We analytically characterize the equilibrium service rates and patient arrival rates under FFS and BP and compare them with a centralized first-best benchmark in which the payer directly chooses hospitals’ service rates to minimize social cost. We show that FFS always induces excessively high service rates and can never achieve the first-best outcome, whereas the ability of BP to reach the first-best outcome critically depends on the level of the medical budget. We further identify how competition intensity among hospitals and the market size affect the payer’s optimal price and the resulting equilibrium service rates.

(3) Motivated by the limitations of BP, we propose and analyze a new payment mechanism, bundled payment with service level regulation (BPW), in which the payer combines a bundled price with an explicit waiting time guarantee. We derive the optimal price and waiting time guarantee, prove that BPW can implement the first-best outcome when the budget is at an intermediate level, and use numerical examples to illustrate how BPW improves total social cost and waiting performance compared with standard BP.

The remainder of this paper is organized as follows. Section 2 reviews the relevant literature. In Section 3, we introduce our queueing model and the problems of patients, hospitals, and the payer. In Section 4, we examine patients’ and hospitals’ equilibrium behavior and the payer’s pricing strategy under the BP scheme. Section 5 explores the equilibrium behavior of hospitals under the FFS scheme. Section 6 presents an analysis of a “benchmark” scenario—the first-best case and the performance comparison between the first-best case and the FFS and BP schemes. In Section 7, we provide a payment scheme, the BP scheme with service level regulation, which can achieve the first-best outcome. Conclusions are provided in Section 8.

2. Literature Review

Our paper is related to research on queueing economics, specifically focusing on competition in service levels (capacity levels). Hassin and Haviv have conducted extensive comprehensive studies and literature reviews in this field [15,16]. Chen and Wan [17] analyze equilibrium behavior when two identical service providers simultaneously compete on price and capacity. Allon and Federgruen [18] generalize the M/M/1 queueing model to a G/GI/s system, where firms compete based on price and waiting time. Lu et al. [19] build both analytical and empirical models to explore the impact of competition on the service quality of nursing homes under different payer settings. Han et al. [20] study quality competition between two hospitals under a payment scheme set by one insurer. However, our work differs from these studies as we focus on competition related to waiting time (service rate), a critical issue affecting patient choice in public healthcare service systems [21]. Therefore, considering waiting time competition is particularly relevant in outpatient care.

Queueing models are widely used in healthcare service systems to address patient waiting time problems. For example, Yu et al. [22] develop a queueing model to characterize the referral process in China’s tiered hospital system, which includes both general hospitals and community health centers, with a green referral channel based on priority theory. Lim et al. [23] use queueing theory to analyze staffing and operational efficiency in obstetric operating rooms, evaluating the impact of various staffing models on patient waiting times and operating room efficiency. Wang et al. [24] establish a queueing-game-theoretical model to capture the interactions between a comprehensive hospital and a primary hospital, examining the tradeoff between efficiency and quality. Guo et al. [14] use an M/M/1 queueing model with endogenous arrivals and revisits to investigate the impact of different payment schemes on the performance of a public healthcare system consisting of a public funder, a public healthcare provider, and a pool of patients. Their findings suggest that the size of the patient pool is an important factor affecting which payment scheme is more desirable for social welfare, revisit rate, and waiting time. Our contribution to this stream of literature is that we consider the impact of service rates on medical costs when building the queueing model to describe patient waiting times, which is crucial for customer-intensive services like outpatient care.

Our study also contributes to the literature on healthcare operation management and payment schemes. Many scholars have discussed the impact of BP and other payment schemes on healthcare system performance and the behavioral decisions of all parties involved in healthcare services. Adida et al. [9] compare the impact of FFS and BP on doctors’ behavior in patient selection and diagnosis quality. Andritsos and Tang [25] examine the incentive programs—FFS, Pay-for-Performance (P4P), and BP—for reducing readmissions when patient care is co-produced. Han et al. [20] examine quality competition between two hospitals under BP and FFS payment schemes. Ghamat et al. [26] study a healthcare system consisting of four parties (a payer, a hospital, a service provider, and patients) and analyze the impact of a “target price minimum quality” (TPMQ) bundled payment model on the hospital’s profit, the payer’s medical expense, and nursing quality, as well as the optimal conditions for this care coordination. Fan et al. [11] conduct an empirical study using insurance claim data from China to investigate the impact of bundled payment schemes on medical cost, efficiency, quality, and shift of care. Li et al. [27] conduct an experimental analysis of how internal compensation incentives based on medical insurance payment schemes—FFS and Diagnosis Related Group (DRG)—influence physician behavior. The results showed that different compensation structures significantly affect physicians’ service provision behaviors, thereby impacting patient welfare. In contrast to these operational management studies, this paper focuses on the impact of service rate competition on the equilibrium behavior of hospitals, patients, and the payer under the BP scheme, an issue widely observed in healthcare markets under BP [28].

This paper also relates to research on the quality–speed tradeoff in operations management, which considers the relationship between waiting time and service quality. A higher service rate reduces waiting time but simultaneously reduces service quality. Anand et al. [29] build a queueing model to explore the quality–speed tradeoff strategy of a service provider when demand is affected by service quality, waiting time, and price. In contrast to the static model of Anand et al., Kostami and Rajagopalan [30] build a dynamic model in a monopoly environment to describe such a quality–speed tradeoff and discuss the optimal decision of the supplier when customer demand is determined by the service quality of previous suppliers. The above two models assume that customers are rational. In contrast, Li et al. [31] expand the model to allow for boundedly rational customers. Furthermore, Li et al. [32] extend the service rate competition game with boundedly rational customers to the case with a general reward function and multiple servers. In customer-intensive services, Zhang and Wang [33], considering boundedly rational customers and the quality–speed tradeoff, analyze the optimal service rate and service price for revenue maximization and social welfare maximization. Sun et al. [34] consider the tradeoff between improving diagnostic accuracy (service quality) and waiting costs (service speed) in healthcare service systems and explore the impact of blockchain implementation on service quality, service price, and system congestion. In contrast to the aforementioned work on the quality–speed conundrum, we investigate the cost–speed tradeoff for hospitals in a competitive setting. The key difference is that our study focuses on the supply-side tradeoff (hospitals’ decisions related to medical costs and demand), while the quality–speed tradeoff is on the demand side (patients’ decisions related to waiting costs and service quality). Wang et al. [3] also consider the cost–speed tradeoff, specifically in terms of service rates and medical consumption costs in customer-intensive services. However, our research differs in that we focus on whether the BP scheme can achieve the first-best outcome (maximum social welfare) when hospitals face the cost–speed tradeoff. To the best of our knowledge, this is the first study to evaluate the BP scheme based on the relationship between medical costs and service rates.

As shown in

Table 1. Comparison of this study’s main features and previous related literature. The ✔ symbol indicates "features that were present in the study”.

Research	Queueing Model	Provider Competition	Healthcare Payment Schemes	Quality–Speed Tradeoff /Cost–Speed Tradeoff
Chen and Wan [17]	✔	✔	-	-
Allon and Federgruen [18]	✔	✔	-	-
Lu et al. [19]		✔	✔	-
Han et al. [20]		✔	✔	-
Yu et al. [22]	✔	-	-	-
Lim et al. [23]	✔	-	-	-
Wang et al. [24]	✔	-	-	-
Guo et al. [14]	✔	-	✔	-
Adida et al. [9]		-	✔	-
Andritsos and Tang [25]		-	✔	-
Ghamat et al. [26]		-	✔	-
Fan et al. [11]		-	✔	-
Li et al. [27]		-	✔	-
Anand et al. [29]	✔	-	-	✔
Kostami and Rajagopalan [30]		-	-	✔
Li et al. [31]	✔	-	-	✔
Li et al. [32]	✔	✔	-	✔
Zhang and Wang [33]	✔	-	-	✔
Sun et al. [34]	✔	-	-	✔
Wang et al. [3]	✔	-	✔	✔
This study	✔	✔	✔	✔

, the comparison between this study and previous related literature highlights the key features and distinctions of the research conducted.

3. Model Setup

3.1. Basic Assumptions

At the beginning of this section, we summarize the main notation used in the model in

Table 2. Notations.

Symbol	Description
$\mathrm{n}$	Number of hospitals in the public outpatient system
${\mathrm{H}}_{\mathrm{i}}$	Hospital ${\mathrm{H}}_{\mathrm{i}},\mathrm{i}=1,...,n$
$\Lambda$	Total potential arrival rate
${\lambda }_{\mathrm{i}}$	The initial arrival rate of patients who join hospital ${\mathrm{H}}_{\mathrm{i}}$
${\mu }_{\mathrm{i}}$	Service rate of hospital ${\mathrm{H}}_{\mathrm{i}}$
$\overline{\mathrm{\mu }}$	The upper bound of the service rate
$\mathrm{d}$	Unit delay cost/unit waiting cost
${\mathrm{W}}_{\mathrm{i}}$	Expected waiting time of patients at hospital ${\mathrm{H}}_{\mathrm{i}}$
$\overline{\mathrm{w}}$	An upper limit of the expected waiting time
$\mathrm{v}$	Service value that a patient receives after being served
$C_0$	The unit benchmark medical cost
$C_u$	The constant unit medical cost
$\mathrm{p}$	Service price for the whole episode of care
$\mathrm{B}$	Medical budget level
SC	Social cost of the whole healthcare system (waiting cost + medical cost)
${\mathrm{w}}_0$	Waiting time guarantee (service level requirement) under the BP with service level regulation (BPW) scheme
$\alpha$	A profit margin of α ∈ [0, 1] on the total medical cost under the FFS scheme

.

We consider a public healthcare system that consists of a payer, n identical hospitals denoted by ${\mathrm{H}}_{\mathrm{i}}, \mathrm{i}=1, \dots , n$, providing outpatient elective services (e.g., hernia repairs, knee and hip replacement), and a pool of homogeneous patients. The payer designs the payment scheme, the hospitals decide the service rates, and patients determine which hospital to seek treatment from. The assumption of homogeneous patients and hospitals is realistic. For example, in Germany and the United States, patients are divided into different diagnosis-related groups according to different symptoms. Patients in the same group need similar medical service resources [35]. This assumption is reasonable and supported by the literature on healthcare using the queueing economy model [14].

The hospitals provide diagnosis and treatment services of a particular disease for patients throughout a whole episode of care, including outpatient consultation services, diagnostic tests, treatment services, and minor surgical procedures [20]. We consider a super-M/M/1 system for each hospital with s physicians/servers. We assume that when s independent and identical single-server systems, each with an arrival rate ${\lambda }_{\mathrm{d}\mathrm{i}}$, and service rate ${\mu }_{\mathrm{d}\mathrm{i}}$, are pooled together, the resulting system of each hospital is another single server system with arrival rate ${\lambda }_{\mathrm{i}}=\mathrm{s}{\lambda }_{\mathrm{d}\mathrm{i}}$ and service rate ${\mu }_{\mathrm{i}}=\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}$ (refer to [36]). We model potential patient arrivals according to a Poisson process with a maximum rate of $\Lambda$. The arrival rate of hospital ${\mathrm{H}}_{\mathrm{i}}$ formed by those joining patients is called the initial arrival rate ${\lambda }_{\mathrm{i}}$. For developing analytical results and general insights, we consider the simpler super-M/M/1 model. We find that the outpatient service under the BP scheme is consistent with most practical settings. (There exists literature that considers the pooling of s M/M/1 systems as an M/M/s system. However, how to set a specific model mainly depends on the actual coordination mechanism of the servers. If s customers are independently served by s servers at the same time, then the system can be characterized as an M/M/s system. If a single customer is served by s servers jointly, the service configuration is a super-M/M/1 system rather than an M/M/s system). Under the BP scheme, multiple physicians who provide consultation, medical tests, medicine, or other procedures are all for one patient during a whole episode of care. Figure 1 provides a schematic representation of the model.

In this queueing model, hospital ${\mathrm{H}}_{\mathrm{i}}$ serves patients on a first-come–first-service (FCFS) basis, and the service takes an exponentially distributed service time at rate $\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}$. The Poisson arrival process and exponential service time have been well-tested in the healthcare operations management literature. For example, Kim et al. [37] empirically verify that the arrival process to a hospital intensive care unit follows a Poisson process, while Guo et al. [14] consider it theoretically. They also find that service time follows an exponential distribution.

Outpatient service is an example of customer-intensive services that require a high level of diligence and attention for good customer service [29]. The hospital/doctor can spend less time with patients to increase demand ([38], p. 27). Anand et al. [29] also propose that service value could be improved by increasing the time spent serving each customer. Therefore, hospitals can adjust servers’ service rates by adjusting the service time per patient, maximizing their profits. In other words, for more profit, hospitals shall induce more/fewer patients to join the queue while offering a shorter/longer service time. The adjustable service rate settings have been applied to a large variety of settings [3]. We assume that the service rates are bounded, i.e., $\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}\le \overline{\mu }$ for i = 1, …, n, where $\overline{\mathrm{\mu }}$ is the upper bound of the service rate with $0<\overline{\mu }<\mathrm{\infty }$. The rationale behind this assumption is three-fold. First, the case $\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}=\mathrm{\infty }$ could lead to trivial/nonexistent solutions. Second, in the context of outpatient service problems, it is difficult to conceive of an unlimited service rate. Third, excessive consultation services may result in high misdiagnosis rates, poor service quality, and low patient satisfaction. To ensure high-quality service and sufficient demand, the service rate must not be too high.

Due to the high value of healthcare services and the consideration that all medical expenses are covered by medical insurance, we assume that all patients would opt to enroll in this outpatient service system (or with no balking and reneging) to ensure the accessibility of treatment for all patients. That is, all patients would join and eventually complete their outpatient service for a whole episode of care in the system, i.e., ${\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\lambda }_{\mathrm{i}}=\Lambda$ [39]. This assumption is rational in medical practice, particularly for non-urgent or elective surgical cases. Taking some provinces in Canada as an example, almost all patients waiting for knee or hip replacement surgery join the wait lists of public hospitals, with an extremely low abandonment rate that can be ignored. This is mainly attributed to the fact that knee or hip problems do not pose a direct threat to life, and thus patients usually do not choose to seek faster but more expensive services elsewhere. Moreover, this assumption helps to simplify the analysis and obtain some interesting and important theoretical results in the three-tier service system. This assumption is consistent with public healthcare systems where patients face minimal out-of-pocket costs and have limited alternative providers, especially for non-urgent elective procedures (e.g., knee or hip replacements in Canada) [39]. It simplifies the analysis by ensuring market coverage and allows us to focus on the strategic interaction between payment schemes and hospital behavior. However, in systems where patients bear significant costs or have private alternatives, some may opt out, potentially moderating the demand effects. Our results should thus be interpreted as applicable to publicly funded, comprehensive coverage contexts.

The expected waiting time represents general queuing processes, the expected waiting plus service time in a steady state, which is denoted by $\mathrm{W}$. The physical meaning and calculation method are in line with the classic literature of queueing theory [40]. The expected waiting time of patients at hospital ${\mathrm{H}}_{\mathrm{i}}$ satisfies

$${\mathrm{W}}_{\mathrm{i}}({\mu }_{\mathrm{d}\mathrm{i}},{\lambda }_{\mathrm{i}})={\displaystyle \frac{1}{\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}-{\lambda }_{\mathrm{i}}}}.$$

(1)

To ensure the stability of the service system, we further assume $\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}>{\lambda }_{\mathrm{i}}$. Because patients’ waiting time cannot be infinitely long and customers will not join a queue if their expected waiting time exceeds their patience time [41], we consider that there is an upper limit to the expected waiting time $\overline{\mathrm{w}}$, which is a sufficiently large positive number; therefore, ${\mathrm{W}}_{\mathrm{i}}(\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}},{\lambda }_{\mathrm{i}})\le \overline{\mathrm{w}}$.

We adopt the classic Naor model [42] to characterize the utility function of patients in this outpatient service system. Patients receive a reward $\mathrm{v}$ after being served and incur a linear waiting cost with rate $\mathrm{d}$ when they join the queueing system. Specifically, if the patient joins the service system, after the entire episode of care, the patient will receive a reward $\mathrm{v}$ for being cured. We assume that all patients have the same service reward, which represents the perceived value of achieving the service goal. Throughout the outpatient service process, patients receive consultation and treatment services and are discharged from the hospital after being cured. Therefore, no matter which hospital the patient chooses to receive service at, the perceived value of a cured patient is equal. This assumption is consistent with the relevant literature on healthcare service systems (e.g., [39]).

When a patient is waiting in the queue, a linear cost with rate $\mathrm{d}$ is incurred. To ensure the model’s tractability, we adopt the conventional assumption that for homogeneous patients, $\mathrm{d}$ is a fixed value [43]. In general, waiting cost represents the opportunity cost of the patient in the waiting process, especially in the outpatient service, and it may also reflect the monetary value of the patient’s anxiety and pain before the consultation. Although waiting cost heterogeneity exists, it is not as apparent as the heterogeneity in the benefits of receiving outpatient service. For tractability purposes, we assume that patients are homogeneous in delay sensitivity. For the whole episode of care, we model the patient’s waiting cost as a linear function of waiting time. This assumption is common in the relevant literature on queueing systems (e.g., [14,44]). The patient’s expected waiting time is $\mathrm{W}$, and thus the expected waiting cost (delay cost) can be expressed as $\mathrm{d}\mathrm{W}$. We consider that patients’ medical out-of-pocket expenses are reimbursed by medical insurance, which we refer to as a “zero” co-payment. This situation has been studied in the healthcare operations management literature. For example, Qian et al. [44] assume that public healthcare services do not have any user fees but incur a delay cost. Therefore, the patient’s utility function when joining the hospital ${\mathrm{H}}_{\mathrm{i}},\mathrm{i}=\mathrm{1,2}, \dots , \mathrm{n}$ is

$${\mathrm{u}}_{\mathrm{i}}\left({\lambda }_{\mathrm{i}},\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}\right)=\mathrm{v}-\mathrm{d}{\mathrm{W}}_{\mathrm{i}}\left({\lambda }_{\mathrm{i}},\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}\right).$$

(2)

The arriving patients decide which hospital to join to maximize their expected utility, that is, the patient’s joining decision. When patients make their joining decisions, they observe the waiting time and anticipate the expected utility of receiving service from each hospital. An implicit assumption is that patients know the expected waiting time [40]. In practice, although patients do not know in advance their actual waiting time, they have an expectation based on past experiences of what the expected waiting time will be at each service provider. Knowledge of the expected waiting time can also be accumulated from experiences of repeated visits and various information channels, such as patient reviews, medical studies, and the news.

3.2. The Hospital’s Problem

Consider the medical costs of patients for the whole episode of care incurred by the hospital. We focus on a context where a longer service time with more inquiry and examination can reduce the total medical cost [3]. In other words, if the hospital provides slow healthcare outpatient service to patients, the probability of misdiagnosis and medical expense for the whole episode of care (e.g., service cost, medical test cost, and prescription cost) will be low. In line with the setting in Wang et al. [3], we model the expected medical cost of a patient for the whole episode of care in ${\mathrm{H}}_{\mathrm{i}},\mathrm{i}=\mathrm{1,2}, \dots , \mathrm{n}$ through a monotonic increasing function of the service rate. (For analytical tractability, although our medical cost model assumes that the medical cost $C(\mu )$ is linear in service rate $\mu$, our main conclusions and the major insights still hold if an S-shaped logistic function is considered).

$$C\left({\mu }_{\mathrm{d}\mathrm{i}}\right)=C_0+C_u\cdot \mathrm{s}\cdot {\mu }_{\mathrm{d}\mathrm{i}}$$

(3)

where $C_0>0$ represents the unit benchmark medical cost and $C_u>0$ is the constant unit medical cost, which represents the rate by which the medical cost changes at the actual service rate. (Although we do not discuss the results of a case where the unit medical cost is negative $C_{\mathrm{u}}<0$ in this study, meaning that a longer service time results in higher medical costs, our technical analysis carries over to that case, and the conclusions are mostly reversed).

Remark 1. 

Our model assumes that the expected medical cost is a monotonic increasing function of the service rate${\mu }_{di}$, as stated in Equation (3). This relationship is consistent with the healthcare operations literature [3,45,46], particularly Wang et al. [3], who model the expected medical cost as an increasing function of the service rate, where longer service times (i.e., slower service rates) lead to more thorough examination and reduced downstream medical consumption, such as fewer medical tests and prescriptions [3]. We adopt a linear form for analytical tractability, which is a common simplification in the literature. While we acknowledge that the true relationship between the service rate and medical costs may be nonlinear, Wang et al. [3] note that their key findings are robust to nonlinear forms, such as S-shaped logistic functions. This suggests that the primary insights of our model are not highly sensitive to the specific functional form.

The problem of hospital ${\mathrm{H}}_{\mathrm{i}}$ is to maximize its profit ${\pi }_{\mathrm{i}}$ by designing the service rates of servers ${\mathrm{\mu }}_{\mathrm{d}\mathrm{i}}$. Hospital ${\mathrm{H}}_{\mathrm{i}}$ earns income $\mathrm{p}$, referred to as the service price, by providing outpatient services for the whole episode of care. This can be formulated as follows:

$$\begin{array}{c}\begin{array}{c}\underset{{\mu }_{\mathrm{d}\mathrm{i}}}{\mathrm{m}\mathrm{a}\mathrm{x}}\hspace{1em}{\pi }_{\mathrm{i}}({\mu }_{\mathrm{d}\mathrm{i}})=(\mathrm{p}-C({\mu }_{\mathrm{d}\mathrm{i}})){\lambda }_{\mathrm{i}}^{\mathrm{e}}({\mu }_{\mathrm{d}\mathrm{i}},{\mu }_{-\mathrm{d}\mathrm{i}})\end{array}\\ \mathrm{s}.\mathrm{t}.\hspace{1em}{\pi }_{\mathrm{i}}({\mu }_{\mathrm{d}\mathrm{i}})\ge 0,\\ {\lambda }_{\mathrm{i}}^{\mathrm{e}}({\mu }_{\mathrm{d}\mathrm{i}},{\mu }_{-\mathrm{d}\mathrm{i}})+{\displaystyle \frac{1}{\overline{\mathrm{w}}}}\le \mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}\le \overline{\mu }.\end{array}$$

(4)

where ${\mathrm{\mu }}_{-\mathrm{d}\mathrm{i}}$ is the service rate of servers/physicians in any hospital other than ${\mathrm{H}}_{\mathrm{i}}$ and ${\lambda }_{\mathrm{i}}^{\mathrm{e}}({\mu }_{\mathrm{d}\mathrm{i}},{\mu }_{-\mathrm{d}\mathrm{i}})$ is the equilibrium arrival rate of patients at hospital ${\mathrm{H}}_{\mathrm{i}}$.

3.3. The Payer’s Problem

In this outpatient service system, the payer acts as a medical insurance bureau, funder, or social planner who is responsible for making decisions on medical insurance payment schemes, including choosing the appropriate scheme and setting parameters. The goal of the payer is to maximize social welfare. Based on the definition of social welfare in medical services, we define social welfare as the patients’ service reward minus their waiting cost and the hospitals’ medical costs. Combined with the previous assumption of patient reward homogeneity, the patient’s reward is a fixed value. Consequently, the goal of maximizing social welfare is equivalent to minimizing social cost.

The social cost is composed of two parts: (i) the expected waiting cost of patients, i.e., ${\displaystyle \frac{\mathrm{d}\Lambda }{\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}(\mathrm{p})-{\lambda }_{\mathrm{i}}^{\mathrm{e}}(\mathrm{p})}},\mathrm{i}=\mathrm{1,2}, \dots , \mathrm{n}$, where $\mathrm{d}$ is the unit delay cost/unit waiting cost; (ii) the medical costs incurred by the hospitals, i.e., $C\left({\mu }_{\mathrm{d}\mathrm{i}}\right)\Lambda$. Note that ${\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{e}}\left(\mathrm{p}\right)$ represents the optimal service rate of servers in hospital ${\mathrm{H}}_{\mathrm{i}}$. Taking the payer’s budget B for all patients into consideration [14], the optimization problem of the payer can be formulated as follows:

$$\begin{array}{c}\underset{\mathrm{p}}{\mathrm{m}\mathrm{i}\mathrm{n}}\hspace{1em}\mathrm{S}\mathrm{C}(\mathrm{p})={\displaystyle \frac{\mathrm{d}\Lambda }{\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{e}}(\mathrm{p})-{\lambda }_{\mathrm{i}}^{\mathrm{e}}(\mathrm{p})}}+\Lambda C({\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{e}}(\mathrm{p}))\\ \mathrm{s}.\mathrm{t}.\hspace{1em}\Lambda \mathrm{p}\le \mathrm{B}.\end{array}$$

(5)

This paper investigates a three-stage Stackelberg game that involves the payer, n hospitals, and a pool of patients. The sequence of decisions is as follows. First, the payer designs the payment scheme, taking into account the three different payment schemes proposed in the introduction—FFS, BP, and BP with service level regulation (BPW for simplicity in this paper). The objective of the payer is to minimize the social cost. Next, given the payment scheme setting and anticipating patients’ reactions, n hospitals decide their servers’ service rates ${\mathrm{\mu }}_{\mathrm{d}\mathrm{i}}$ simultaneously. The hospitals aim to maximize their respective profits. Finally, based on the service rates, the patients choose one of the hospitals to seek service at to maximize their utilities. Figure 2 describes the game sequence.

4. Bundled Payment

We consider the BP scheme under which hospitals are compensated with a fixed service price for a whole episode of care for a certain disease, regardless of what kind of examinations, tests, and procedures are adopted and irrespective of eventual complications [9]. Each hospital is paid a uniform price set by the payer ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}$ for providing outpatient services and bears the total medical cost incurred for their respective services. Therefore, for more profits, hospitals have the incentive to minimize their medical costs as much as possible.

We use the backward introduction to analyze the three-stage Stackelberg game. First, patients decide which hospital to seek service at based on their expected waiting time. Second, anticipating the patients’ arrival rates in equilibrium, hospitals determine their service rates simultaneously to maximize their profits in a competitive setting. Finally, we derive the payer’s payment scheme setting that minimizes social cost.

4.1. The Patients’ Joining Decision

From Equation (1), we know that patient demand depends solely on waiting time, meaning that patients choose the hospital with the shortest expected waiting time. We assume that except for hospital ${\mathrm{H}}_{\mathrm{i}},\mathrm{n}-1$ hospitals (unified as ${\mathrm{H}}_{-\mathrm{i}}$) adopt a symmetric strategy $\mathrm{s}{\mu }_{-\mathrm{d}\mathrm{i}}$ and the responding arrival rate is ${\lambda }_{-\mathrm{i}}$. For patients, if the expected waiting times of n services are equal, there is no difference between their choices, leading to $\mathrm{W}\left({\mu }_{\mathrm{d}\mathrm{i}},{\lambda }_{\mathrm{i}}\right)=\mathrm{W}\left({\mu }_{-\mathrm{d}\mathrm{i}},{\lambda }_{-\mathrm{i}}\right)$ in equilibrium. Therefore, the patients’ equilibrium arrival rates satisfy

$$\left\{\begin{array}{c}{\mathrm{W}}_{\mathrm{i}}({\mu }_{\mathrm{d}\mathrm{i}},{\lambda }_{\mathrm{i}})={\mathrm{W}}_{-\mathrm{i}}({\mu }_{-\mathrm{d}\mathrm{i}},{\lambda }_{-\mathrm{i}}),\hfill \\ {\lambda }_{\mathrm{i}}+(\mathrm{n}-1){\lambda }_{-\mathrm{i}}=\Lambda ,\hfill \\ \mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}>{\lambda }_{\mathrm{i}},\mathrm{s}{\mu }_{-\mathrm{d}\mathrm{i}}>{\lambda }_{-\mathrm{i}},\hfill \\ \mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}\le \overline{\mu },\mathrm{s}{\mu }_{-\mathrm{d}\mathrm{i}}\le \overline{\mu },\hfill \\ {\displaystyle \frac{1}{\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}-{\lambda }_{\mathrm{i}}}}\le \overline{\mathrm{w}},{\displaystyle \frac{1}{\mathrm{s}{\mu }_{-\mathrm{d}\mathrm{i}}-{\lambda }_{-\mathrm{i}}}}\le \overline{\mathrm{w}}.\hfill \end{array}\right.$$

(6)

The second equation of (6) ensures that all patients will join the public outpatient service system. The values of $\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}>{\lambda }_{\mathrm{i}}\hspace{0.33em}$ and $\mathrm{s}{\mu }_{-\mathrm{d}\mathrm{i}}>{\lambda }_{-\mathrm{i}}$ are set to ensure the stability of the queueing system. The last inequality of (6) is the regulation of waiting time. The expected waiting time of patients at hospital ${\mathrm{H}}_{\mathrm{i}}$ or hospital ${\mathrm{H}}_{-\mathrm{i}}$ cannot exceed $\overline{\mathrm{w}}$.

Denote ${\lambda }_{\mathrm{i}}^{\mathrm{e}}$ and ${\lambda }_{-\mathrm{i}}^{\mathrm{e}}$ as the equilibrium arrival rates of hospitals ${\mathrm{H}}_{\mathrm{i}}$ and ${\mathrm{H}}_{-\mathrm{i}}$, respectively. According to (6), ${\lambda }_{\mathrm{i}}^{\mathrm{e}}$ and ${\lambda }_{-\mathrm{i}}^{\mathrm{e}}$ mainly depend on the service rates of the hospitals, the number of the hospitals, and the market size. In Proposition 1, we provide the equilibrium arrival rates for different conditions of the service rate ${\mu }_{\mathrm{d}\mathrm{i}}$.

Proposition 1. 

The equilibrium arrival rates ${\lambda }_{\mathrm{i}}^{\mathrm{e}} and {\lambda }_{-\mathrm{i}}^{\mathrm{e}}$can be given as follows:

(i)

$\mathrm{I}\mathrm{f}\hspace{0.33em}{\mu }_{\mathrm{d}\mathrm{i}}\in \left.[0,{\mathrm{Y}}_1\left({\mu }_{-\mathrm{d}\mathrm{i}}\right)\right),{\lambda }_{\mathrm{i}}^{\mathrm{e}}={\lambda }_{-\mathrm{i}}^{\mathrm{e}}=0;$

(ii)

$\mathrm{I}\mathrm{f}\hspace{0.33em}{\mu }_{\mathrm{d}\mathrm{i}}\in \left.{[\mathrm{Y}}_1\left({\mu }_{-\mathrm{d}\mathrm{i}}\right),{\mathrm{Y}}_2\left({\mu }_{-\mathrm{d}\mathrm{i}}\right)\right),{\lambda }_{\mathrm{i}}^{\mathrm{e}}=0,{\lambda }_{-\mathrm{i}}^{\mathrm{e}}={\displaystyle \frac{\Lambda }{\mathrm{n}-1}};$

(iii)

$\mathrm{I}\mathrm{f}\hspace{0.33em}{\mu }_{\mathrm{d}\mathrm{i}}\in [\left.{\mathrm{Y}}_2\left({\mu }_{-\mathrm{d}\mathrm{i}}\right),{\mathrm{Y}}_3\left({\mu }_{-\mathrm{d}\mathrm{i}}\right)\right),{\lambda }_{\mathrm{i}}^{\mathrm{e}}={\displaystyle \frac{\Lambda +\mathrm{s}(\mathrm{n}-1)\left({\mu }_{\mathrm{d}\mathrm{i}}-{\mu }_{-\mathrm{d}\mathrm{i}}\right)}{\mathrm{n}}},{\lambda }_{-\mathrm{i}}^{\mathrm{e}}={\displaystyle \frac{\Lambda +\mathrm{s}{\mu }_{-\mathrm{d}\mathrm{i}}-\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}}{\mathrm{n}}};\hspace{0.33em}$

(iv)

$\mathrm{I}\mathrm{f}\hspace{0.33em}{\mu }_{\mathrm{d}\mathrm{i}}\in \left.[{\mathrm{Y}}_3\left({\mu }_{-\mathrm{d}\mathrm{i}}\right),\overline{\mu }\right),{\lambda }_{\mathrm{i}}^{\mathrm{e}}=\Lambda ,{\lambda }_{-\mathrm{i}}^{\mathrm{e}}=0.$

Note. The functions ${\mathrm{Y}}_1\left({\mu }_{-\mathrm{d}\mathrm{i}}\right),{\mathrm{Y}}_2\left({\mu }_{-\mathrm{d}\mathrm{i}}\right)\hspace{0.33em}$ and ${\mathrm{Y}}_3\left({\mu }_{-\mathrm{d}\mathrm{i}}\right)$ are defined as piecewise functions, which make their expressions relatively complex and lengthy. To maintain the readability and flow of the main text, we provide the detailed definitions and derivations in Appendix A, proof of Proposition 1, and Figure 3. However, its graphics can be found in Figure 3.

Proof. 

See Appendix A. □

Combining Proposition 1 and Figure 3, we find that when the service rate is low enough (case (i)), no patient will be willing to enter the queue to wait for treatment service, that is, ${\lambda }_{\mathrm{i}}^{\mathrm{e}}={\lambda }_{-\mathrm{i}}^{\mathrm{e}}=0$; when the hospitals’ service rate $\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}$ is low enough while the others $\mathrm{s}{\mu }_{-\mathrm{d}\mathrm{i}}$ are high (case (ii)), then no patient will enter this hospital to wait for service, and patient demands will be evenly distributed among other n − 1 hospitals, i.e., ${\lambda }_{\mathrm{i}}^{\mathrm{e}}=0,{\lambda }_{-\mathrm{i}}^{\mathrm{e}}={\displaystyle \frac{\Lambda }{\mathrm{n}-1}}$; conversely, when the hospital’s service rate is high enough, while the others’ is low (case (iv)), all patients will choose to join because the expected waiting cost will be low enough, i.e., ${\lambda }_{\mathrm{i}}^{\mathrm{e}}=\Lambda ,{\lambda }_{-\mathrm{i}}^{\mathrm{e}}=0$; only when the service rate of all hospitals is high enough (case (iii)) will each hospital have patients choose to join.

4.2. The Hospitals’ Service Rate Decision

The objective of each hospital is to set its service rate for servers to maximize its profit within the competitive setting. From the hospital’s profit function (given by model (4)), we find that under the BP scheme with a fixed service price determined by the payer, the hospital’s profit depends on the medical cost and the equilibrium demand. Recall the situation that when the hospital serves more slowly and spends more time on patient consultation, the total medical cost decreases. However, slow service also results in longer waiting time and decreased demand (Li et al., [31]). Therefore, under the BP scheme with a given service price, the hospitals face a cost–speed tradeoff. The problem of hospital ${\mathrm{H}}_{\mathrm{i}}$ can be shown as model (4) in the model’s setup, as follows:

$$\begin{array}{c}\begin{array}{c}\underset{{\mu }_{\mathrm{d}\mathrm{i}}}{\mathrm{m}\mathrm{a}\mathrm{x}}\hspace{1em}{\pi }_{\mathrm{i}}({\mu }_{\mathrm{d}\mathrm{i}})=(\mathrm{p}-C({\mu }_{\mathrm{d}\mathrm{i}})){\lambda }_{\mathrm{i}}^{\mathrm{e}}({\mu }_{\mathrm{d}\mathrm{i}},{\mu }_{-\mathrm{d}\mathrm{i}})\end{array}\\ \mathrm{s}.\mathrm{t}.\hspace{1em}{\pi }_{\mathrm{i}}({\mu }_{\mathrm{d}\mathrm{i}})\ge 0,\\ {\lambda }_{\mathrm{i}}^{\mathrm{e}}({\mu }_{\mathrm{d}\mathrm{i}},{\mu }_{-\mathrm{d}\mathrm{i}})+{\displaystyle \frac{1}{\overline{\mathrm{w}}}}\le \mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}\le \overline{\mu }.\end{array}$$

(7)

The equilibrium service rate of each server in hospital ${\mathrm{H}}_{\mathrm{i}},{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}},\mathrm{i}=\mathrm{1,2}, \dots , \mathrm{n}$ for a given service price under the BP is provided in the following proposition.

Proposition 2. 

Under the BP scheme, for a given service price ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}$, the optimal service rates of the servers in hospital ${\mathrm{H}}_{\mathrm{i}}$ are

$${\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}=\left\{\begin{array}{c}\begin{array}{c}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{i}\mathrm{f}\hspace{0.33em}{\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [0,{\mathrm{p}}_1),\hfill \\ {\displaystyle \frac{\Lambda }{\mathrm{s}\mathrm{n}}}+{\displaystyle \frac{1}{\mathrm{s}\overline{\mathrm{w}}}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{i}\mathrm{f}\hspace{0.33em}{\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [{\mathrm{p}}_1,{\mathrm{p}}_4),\hfill \end{array}\\ {\displaystyle \frac{{\mathrm{p}}^{\mathrm{B}\mathrm{P}}-C_0}{\mathrm{s}{\mathrm{C}}_{\mathrm{u}}}}-{\displaystyle \frac{\Lambda }{\mathrm{s}(\mathrm{n}-1)}}\hspace{1em}\hspace{1em}\mathrm{i}\mathrm{f}\hspace{0.33em}{\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [{\mathrm{p}}_4,+\mathrm{\infty }).\hfill \end{array}\right.$$

(8)

The equilibrium arrival rate of hospital ${\mathrm{H}}_{\mathrm{i}}$ is

$${\lambda }_{\mathrm{i}}^{\mathrm{B}\mathrm{P}}=\left\{\begin{array}{c}0 \mathrm{i}\mathrm{f}\hspace{0.33em}{\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [0,{\mathrm{p}}_1),\hfill \\ {\displaystyle \frac{\Lambda }{\mathrm{n}}} \mathrm{i}\mathrm{f}\hspace{0.33em}{\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [{\mathrm{p}}_1,+\mathrm{\infty }).\end{array}\right.$$

where ${\mathrm{p}}_1=C_0+{\displaystyle \frac{C_u}{\overline{\mathrm{w}}}},{\mathrm{p}}_4=C_0+{\displaystyle \frac{(2\mathrm{n}-1)C_u\Lambda }{\mathrm{n}(\mathrm{n}-1)}}$+${\displaystyle \frac{{\mathrm{C}}_{\mathrm{u}}}{\overline{\mathrm{w}}}}$.

Proof. 

See Appendix A. □

Proposition 2 provides the optimal service rates of servers and equilibrium demands for a given service price ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}$. When the service price is low enough, i.e., ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in \left.[0,{\mathrm{p}}_1\right)$, hospitals will not provide services. When the service price is within a moderate range, i.e., ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [\left.{\mathrm{p}}_1,{\mathrm{p}}_4\right)$, hospitals’ strategies are independent of the service price. However, when the price is high $,{\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [\left.{\mathrm{p}}_4,\mathrm{\infty }\right),$ the optimal service rates increase the service price and are at least as high as those in the previous case. The rationale for the behavior of hospitals under different service price levels is as follows. When the service price is very low, hospitals are unprofitable and therefore opt to withdraw from the market. When the price is within a moderate range, hospitals would not earn much profit from serving patients. Therefore, the only way for them to be profitable is to slow down the service rate to lower the medical cost. In this case, this results in very high patient waiting times. However, when the price is sufficiently high, hospitals would earn more profit by serving more patients. The optimal strategy in this case is to improve the service rate to shorten the waiting time and attract more patients.

When the service price is sufficiently low, i.e., ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [\left.0,{\mathrm{p}}_1\right),$ no hospitals are willing to provide healthcare service and therefore no patients will enter the market, i.e., ${\lambda }_{\mathrm{i}}^{\mathrm{B}\mathrm{P}}=0$. When the service price is high due to the homogeneous hospitals and focusing on symmetrical equilibrium, all hospitals will provide services and share the total market demand equally ${\lambda }_{\mathrm{i}}^{\mathrm{B}\mathrm{P}}={\displaystyle \frac{\Lambda }{\mathrm{n}}}$. As the competition intensity $\mathrm{n}$ increases, the demand for each hospital decreases.

4.3. The Payer’s Pricing Strategy

Under the BP scheme, the goal of the payer is to minimize the total social cost SC by setting a unified price ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}$ for the hospitals. Recall Proposition 2, which states that for any ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [\left.0,{\mathrm{p}}_1\right),$ the marginal profit of the hospitals is negative, and hence no hospitals are willing to provide service; thus, we focus on the situation where ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [\left.{\mathrm{p}}_4,\mathrm{\infty }\right)$. In addition, according to the value range of the price, the budget needs to be high enough, i.e., $\mathrm{B}\in [\left.\Lambda {\mathrm{p}}_1,+\mathrm{\infty }\right)$. Therefore, combining the problem of the payer (5) and the equilibrium strategy of the hospital ${\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}$ in Proposition 2, the specific optimization model of the payer can be given by

$$\begin{array}{c}\underset{{\mathrm{p}}^{\mathrm{B}\mathrm{P}}}{\mathrm{m}\mathrm{i}\mathrm{n}}\hspace{1em}\mathrm{S}\mathrm{C}({\mathrm{p}}^{\mathrm{B}\mathrm{P}})\\ \hspace{1em}\mathrm{s}.\mathrm{t}.\hspace{0.33em}\hspace{0.33em}{\mathrm{p}}_1\le {\mathrm{p}}^{\mathrm{B}\mathrm{P}}\le {\displaystyle \frac{\mathrm{B}}{\Lambda }}.\end{array}$$

(9)

where

$$\mathrm{S}\mathrm{C}({\mathrm{p}}^{\mathrm{B}\mathrm{P}})=\left\{\begin{array}{c}\begin{array}{c}\mathrm{d}\Lambda \overline{\mathrm{w}}+(C_0+C_u({\displaystyle \frac{\Lambda }{\mathrm{n}}}+{\displaystyle \frac{1}{\overline{\mathrm{w}}}}))\Lambda \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}\mathrm{i}\mathrm{f}\hspace{0.33em}{\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [{\mathrm{p}}_1,{\mathrm{p}}_4),\end{array}\\ {\displaystyle \frac{\mathrm{d}\Lambda }{\frac{{\mathrm{p}}^{\mathrm{B}\mathrm{P}}-C_0}{C_u}-\frac{(2\mathrm{n}-1)\Lambda }{\mathrm{n}(\mathrm{n}-1)}}}+({\mathrm{p}}^{\mathrm{B}\mathrm{P}}-{\displaystyle \frac{C_u\Lambda }{\mathrm{n}-1}})\Lambda \hspace{1em}\hspace{1em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{1em}\hspace{1em}\mathrm{i}\mathrm{f}\hspace{0.33em}{\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [{\mathrm{p}}_4,+\mathrm{\infty }).\end{array}\right.$$

By solving the model (9), we can obtain the following Proposition 3.

Proposition 3. 

Under the BP scheme, the optimal service price of the payer that minimizes social cost ${\tilde{\mathrm{p}}}^{\mathrm{B}\mathrm{P}}$and the corresponding equilibrium service rates ${\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}$ can be given as follows:

$${\tilde{\mathrm{p}}}^{\mathrm{B}\mathrm{P}}=\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{B}}{\Lambda }}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{i}\mathrm{f}\hspace{0.33em}\mathrm{B}\in [\Lambda {\mathrm{p}}_1,{\mathrm{B}}_1),\\ C_0+C_u(\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}+{\displaystyle \frac{(2\mathrm{n}-1)\Lambda }{\mathrm{n}(\mathrm{n}-1)}})\hspace{1em}\hspace{0.33em}\hspace{0.33em}\mathrm{i}\mathrm{f}\hspace{0.33em}\mathrm{B}\in [{\mathrm{B}}_1,+\mathrm{\infty }).\end{array}\right.$$

(10)

$${\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}=\left\{\begin{array}{c}\begin{array}{c}{\displaystyle \frac{\Lambda }{\mathrm{s}\mathrm{n}}}+{\displaystyle \frac{1}{\mathrm{s}\overline{\mathrm{w}}}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hspace{0.33em}\hspace{0.33em}\mathrm{if}\hspace{0.33em}B\in [\Lambda {\mathrm{p}}_1,{\mathrm{B}}_0),\\ {\displaystyle \frac{\frac{\mathrm{B}}{\Lambda }-C_0}{\mathrm{s}{C}_u}}-{\displaystyle \frac{\Lambda }{\mathrm{s}(\mathrm{n}-1)}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if}\hspace{0.33em}B\in [{\mathrm{B}}_0,{\mathrm{B}}_1),\end{array}\\ {\displaystyle \frac{1}{\mathrm{s}}}\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}+{\displaystyle \frac{\Lambda }{\mathrm{s}\mathrm{n}}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}\mathrm{i}\mathrm{f}\hspace{0.33em}\mathrm{B}\in [{\mathrm{B}}_1,+\mathrm{\infty }).\end{array}\right.$$

(11)

where ${\mathrm{B}}_0=\Lambda {\mathrm{p}}_4=\Lambda (C_0+{\displaystyle \frac{C_u\Lambda (2\mathrm{n}-1)}{\mathrm{n}(\mathrm{n}-1)}}+{\displaystyle \frac{Cu}{\overline{\mathrm{w}}}}),{\mathrm{B}}_1=\Lambda (C_0+C_u(\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}+{\displaystyle \frac{(2\mathrm{n}-1)\Lambda }{\mathrm{n}(\mathrm{n}-1)}}))$.

Proof of Proposition 3. 

When ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [{\mathrm{p}}_1,{\mathrm{p}}_4)$, the social cost function $\mathrm{S}\mathrm{C}({\mathrm{p}}^{\mathrm{B}\mathrm{P}})$ is not affected by ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}$ and is larger than that for ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [{\mathrm{p}}_4,+\mathrm{\infty })$ due to sufficiently large $\overline{\mathrm{w}}$; therefore, we next focus on the case ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [{\mathrm{p}}_4,+\mathrm{\infty })$. In this case, the second-derivative of the social cost function is

$${\displaystyle \frac{{\partial }^2\mathrm{S}\mathrm{C}}{\partial ({\mathrm{p}}^{\mathrm{B}\mathrm{P}}{)}^2}}={\displaystyle \frac{2\mathrm{d}\Lambda }{{C_u}^2[\frac{{\mathrm{p}}^{\mathrm{B}\mathrm{P}}-C_0}{C_u}-\frac{\Lambda }{\mathrm{n}(\mathrm{n}-1)}{]}^3}}>0.$$

Hence, $\mathrm{S}\mathrm{C}({\mathrm{p}}^{\mathrm{B}\mathrm{P}})$ is convex in ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}$. The first-order condition ${\displaystyle \frac{\partial \mathrm{S}\mathrm{C}}{\partial {\mathrm{p}}^{\mathrm{B}\mathrm{P}}}}=-{\displaystyle \frac{\mathrm{d}\Lambda }{C_u[\frac{{\mathrm{p}}^{\mathrm{B}\mathrm{P}}-C_0}{C_u}-\frac{\Lambda }{\mathrm{n}(\mathrm{n}-1)}{]}^2}}+\Lambda =0$ leads to ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}=C_0+C_u(\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}+{\displaystyle \frac{(2\mathrm{n}-1)\Lambda }{\mathrm{n}(\mathrm{n}-1)}})$, noted by ${\mathrm{p}}_{\mathrm{d}}$, and so we have ${\mathrm{p}}_{\mathrm{d}}>{\mathrm{p}}_4$.

By considering the budget constraint $\Lambda {\mathrm{p}}^{\mathrm{B}\mathrm{P}}\le \mathrm{B}$,${\mathrm{p}}^{\mathrm{B}\mathrm{P}}\in [{\mathrm{p}}_4,{\displaystyle \frac{\mathrm{B}}{\Lambda }})$. Next, we analyze the optimal price based on the relationship between ${\mathrm{p}}_{\mathrm{d}}$ and ${\displaystyle \frac{\mathrm{B}}{\Lambda }}$. When $\mathrm{B}<\Lambda {\mathrm{p}}_{\mathrm{d}}$ (below, we note that ${\mathrm{B}}_1=\Lambda {\mathrm{p}}_{\mathrm{d}}$), $\mathrm{S}\mathrm{C}({\mathrm{p}}^{\mathrm{B}\mathrm{P}})$ is strictly decreasing in ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}$; thus, the optimal price is ${\tilde{\mathrm{p}}}^{\mathrm{B}\mathrm{P}}={\displaystyle \frac{\mathrm{B}}{\Lambda }}$ and the corresponding equilibrium service rate ${\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}={\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}({\displaystyle \frac{\mathrm{B}}{\Lambda }})={\displaystyle \frac{\frac{\mathrm{B}}{\Lambda }-C_0}{\mathrm{s}{C}_u}}-{\displaystyle \frac{\Lambda }{\mathrm{s}(\mathrm{n}-1)}}$; when $\mathrm{B}\ge {\mathrm{B}}_1$, $\mathrm{S}\mathrm{C}({\mathrm{p}}^{\mathrm{B}\mathrm{P}})$ is first decreasing and then increasing in ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}$, and thus ${\tilde{\mathrm{p}}}^{\mathrm{B}\mathrm{P}}={\mathrm{p}}_{\mathrm{d}}={\mathrm{C}}_0+C_u(\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}+{\displaystyle \frac{(2\mathrm{n}-1)\Lambda }{\mathrm{n}(\mathrm{n}-1)}})$,${\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}={\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}({\mathrm{p}}_{\mathrm{d}})={\displaystyle \frac{1}{\mathrm{s}}}\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}+{\displaystyle \frac{\Lambda }{\mathrm{s}\mathrm{n}}}$. □

We can further observe the impact of parameters (the number of hospitals $\mathrm{n}$, the market size $\Lambda ,$ and the unit medical cost $C_u$) on equilibrium results (the optimal service price ${\tilde{\mathrm{p}}}^{\mathrm{B}\mathrm{P}}$ and service rates of the hospitals ${\tilde{\mu }}_{\mathrm{i}}^{\mathrm{B}\mathrm{P}}=\mathrm{s}{\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}$). We formalize our discussion in Corollary 1 and

Table 3. The impact of parameters $\mathrm{n},\Lambda ,C_u$ on the equilibrium ${\tilde{\mathrm{p}}}^{\mathrm{B}\mathrm{P}},{\tilde{\mu }}_{\mathrm{i}}^{\mathrm{B}\mathrm{P}}$.

	$(1) \mathbf{B}\in \left.[\mathbf{\Lambda }{\mathbf{p}}_1,{\mathbf{B}}_0\right)$			$(2) \mathbf{B}\in \left.[{\mathbf{B}}_0,{\mathbf{B}}_1\right)$			$(3) \mathbf{B}\in [\left.{\mathbf{B}}_1,+\mathbf{\infty }\right)$
	$\mathbf{n}$	$\mathbf{\Lambda }$	${\mathit{C}}_{\mathit{u}}$	$\mathbf{n}$	$\mathbf{\Lambda }$	${\mathit{C}}_{\mathit{u}}$	n	$\mathbf{\Lambda }$	${\mathit{C}}_{\mathit{u}}$
${\tilde{p}}^{BP}$	$-$	$\downarrow$	$-$	$-$	$\downarrow$	$-$	$\downarrow$	$\uparrow$	$\uparrow$
${\tilde{\mu }}_i^{BP}$	$\downarrow$	$-$	$\uparrow$	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$

The symbol “–” indicates no change, “$\uparrow$” indicates a monotonic increase, and “$\downarrow$” indicates a monotonic decrease.

.

Corollary 1. 

Under the BP scheme,

As the number of hospitals$n$increases, the optimal service price is unaffected if the budget is low $\mathrm{B}\in \left.[\Lambda {\mathrm{p}}_1,{\mathrm{B}}_1\right)$ and decreases if the budget is relatively high $\mathrm{B}\in [\left.{\mathrm{B}}_1,+\mathrm{\infty }\right)$.

As the market size $\Lambda$ increases, the optimal service price decreases if the budget is low $\mathrm{B}\in \left.[\Lambda {\mathrm{p}}_1,{\mathrm{B}}_1\right)$ and increases if $\mathrm{B}\in \left.[{\mathrm{B}}_1,+\mathrm{\infty }\right)$;

As the unit medical cost $C_u$ increases, the optimal service price is unaffected if the budget is low $\mathrm{B}\in \left.[\Lambda {\mathrm{p}}_1,{\mathrm{B}}_1\right)$ and increases if $\mathrm{B}\in [\left.{\mathrm{B}}_1,+\mathrm{\infty }\right)$.

Proof. 

See Appendix A. □

According to case (i) of Corollary 1 and Table 3, the impact of the number of hospitals n on equilibrium results is closely related to the budget. When the budget is low $\mathrm{B}\in [\left.\Lambda {\mathrm{p}}_1,{\mathrm{B}}_1\right)$, the payer sets a service price that is not influenced by n, because the pricing is constrained by the budget. However, as n increases, for low budget $\mathrm{B}\in [\left.\Lambda {\mathrm{p}}_1,{\mathrm{B}}_0\right)$, hospitals will provide slower services to achieve positive profits while still meeting the basic waiting time constraint; for relatively high budget $\mathrm{B}\in \left.[{\mathrm{B}}_0,{\mathrm{B}}_1\right)$, hospitals will choose higher service rates to compete for demand, as this behavior reduces patient waiting costs. When the budget is high enough $\mathrm{B}\in \left.[{\mathrm{B}}_1,+\mathrm{\infty }\right)$, as n increases, the greater the competition, and the payer sets a lower service price to control the service rate of hospitals, thereby reducing the hospitals’ total medical costs.

Consider case (ii) next. When the budget is low $\mathrm{B}\in \left.[\Lambda {\mathrm{p}}_1,{\mathrm{B}}_1\right)$, the impact of Λ on medical costs dominates that of waiting costs. Hence, as Λ increases, the payer sets a lower price to control the hospitals’ service rate, thereby reducing medical costs. However, as n increases, for $\mathrm{B}\in [\left.\Lambda {\mathrm{p}}_1,{\mathrm{B}}_0\right)$, the hospitals would choose higher service rates for achieving the basic waiting time guarantee and reducing the waiting cost; for $\mathrm{B}\in [\left.{\mathrm{B}}_0,{\mathrm{B}}_1\right)$, the hospitals would serve more slowly to reduce medical costs. When the budget is high enough $\mathrm{B}\in \left.[{\mathrm{B}}_1,+\mathrm{\infty }\right)$, the impact of Λ on the waiting cost dominates that of the medical cost. Hence, as the market size increases, to control the total waiting cost of the patients, the payer would set a high price to encourage the hospitals to increase the service rate.

From case (iii) and Table 3, when the budget is low $\mathrm{B}\in [\left.\Lambda {\mathrm{p}}_1,{\mathrm{B}}_1\right)$, the pricing is constrained by the budget, and thus the optimal price setting is only related to the unit budget ${\displaystyle \frac{\mathrm{B}}{\Lambda }}$, regardless of the unit medical cost $C_u$. When the budget is high, the pricing is no longer limited by the budget, and thus as the unit medical cost $C_u$ becomes higher, the payer would set a higher price to ensure the profit of the hospitals. Moreover, for a low enough budget $\mathrm{B}\in \left.[\Lambda {\mathrm{p}}_1,{\mathrm{B}}_0\right)$, the equilibrium service rate is not affected by the unit medical cost; for $\mathrm{B}\in \left.[{\mathrm{B}}_0,+\mathrm{\infty }\right)$, the equilibrium service rate decreases because to control the total medical cost, the hospitals will be willing to spend more time on consulting patients.

5. Fee-for-Service

After analyzing the BP scheme, we now turn to examine the fee-for-service (FFS) scheme, where patients’ equilibrium arrival rates remain the same as in the BP case, as shown in Proposition 1. Next, we would like to examine the equilibrium service rate decision of the hospitals under the FFS scheme.

The FFS scheme is currently the predominant and influential payment scheme in healthcare system, under which the payment depends on service item and the number of services. Assuming that the hospitals have a profit margin of α ∈ [0, 1] on the total medical cost [26], i.e., $\mathrm{p}=\left(1+\alpha \right)C\left({\mu }_{\mathrm{i}}\right)$. Therefore, along with (3), the problem of the hospital $\mathrm{H}\mathrm{i}$ is to maximize the profit by designing service rates for its servers, which can be formulated as follows:

$$\begin{array}{c}\begin{array}{c}\underset{{\mu }_{\mathrm{d}\mathrm{i}}}{\mathrm{m}\mathrm{a}\mathrm{x}}\hspace{1em}{\pi }_{\mathrm{i}}({\mu }_{\mathrm{d}\mathrm{i}})=\alpha C({\mu }_{\mathrm{d}\mathrm{i}}){\lambda }_{\mathrm{i}}^{\mathrm{e}}({\mu }_{\mathrm{d}\mathrm{i}},{\mu }_{-\mathrm{d}\mathrm{i}})\end{array}\\ \hspace{1em}\hspace{1em}\mathrm{s}.\mathrm{t}.\hspace{1em}{\pi }_{\mathrm{i}}({\mu }_{\mathrm{d}\mathrm{i}})\ge 0,\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\lambda }_{\mathrm{i}}^{\mathrm{e}}({\mu }_{\mathrm{d}\mathrm{i}},{\mu }_{-\mathrm{d}\mathrm{i}})+{\displaystyle \frac{1}{\overline{\mathrm{w}}}}\le \mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}\le \overline{\mu }.\end{array}$$

(12)

Solving problem (12) leads to an optimal service rate for the hospital, as shown in Proposition 4.

Proposition 4. 

Under the FFS, the optimal service rate of the servers in hospital ${\mathrm{H}}_{\mathrm{i}}$ is ${\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}}={\displaystyle \frac{\overline{\mu }}{\mathrm{s}}}$.

Proof of Proposition 4. 

Following from that, both the medical cost function $C({\mu }_{\mathrm{d}\mathrm{i}})$ and the equilibrium demand function ${\lambda }_{\mathrm{i}}^{\mathrm{e}}({\mu }_{\mathrm{d}\mathrm{i}})$ are strictly increasing in ${\mu }_{\mathrm{d}\mathrm{i}}$, and therefore the optimal service rate is the maximum, ${\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}}=\overline{\mu }$. □

Proposition 4 shows that under the FFS scheme, the hospitals will choose the maximum service rate. Because the service rate has the same positive effect on medical cost and demand, the higher the hospital’s service rate, the higher the misdiagnosis rate and the higher the follow-up medical cost, and, at the same time, the greater the patients’ demand because of the lower waiting cost. Therefore, it follows that the higher the total medical cost, the greater the hospital’s profit. As a result, the hospitals will provide higher service rates for more demand under FFS.

Having established the optimal service rate under the FFS scheme, we next conduct a comparative analysis of service rates and system performance between the FFS and BP schemes.

Proposition 5. 

${\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}}>{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}$,W(${\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}}$) < W(${\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}$),$C({\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}})>C({\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}){,\mathrm{S}\mathrm{C}}^{FFS}\ge {\mathrm{S}\mathrm{C}}^{BP}$.

Proof of Proposition 5. 

Under FFS, ${\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{l} \mathrm{s}\mathrm{e}\mathrm{r}\mathrm{v}\mathrm{i}\mathrm{c}\mathrm{e} \mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e} \mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}}=\overline{\mu }$ which is the upper bound of the service rate. However, the optimal service rate under the BP scheme is a piecewise function. $\mathrm{I}\mathrm{f}\hspace{0.33em}\mathrm{B}\in [\Lambda {\mathrm{p}}_1,{\mathrm{B}}_0),{\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}={\displaystyle \frac{\Lambda }{\mathrm{s}\mathrm{n}}}+{\displaystyle \frac{1}{\mathrm{s}\overline{\mathrm{w}}}};\mathrm{i}\mathrm{f}\hspace{0.33em}\mathrm{B}\in [{\mathrm{B}}_0,{\mathrm{B}}_1),{\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}={\displaystyle \frac{\frac{\mathrm{B}}{\Lambda }-C_0}{\mathrm{s}{C}_u}}-{\displaystyle \frac{\Lambda }{\mathrm{s}(\mathrm{n}-1)}};\mathrm{i}\mathrm{f}\hspace{0.33em}\mathrm{B}\in [{\mathrm{B}}_1,+\mathrm{\infty }),{\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}={\displaystyle \frac{1}{\mathrm{s}}}\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}+{\displaystyle \frac{\Lambda }{\mathrm{s}\mathrm{n}}}$, where $\overline{\mathrm{w}}$ is an upper limit of the expected waiting time.

Because the waiting time is $\mathrm{W}({\mu }_{\mathrm{d}\mathrm{i}},{\lambda }_{\mathrm{i}})={\displaystyle \frac{1}{\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}-{\lambda }_{\mathrm{i}}}}$, a larger ${\mu }_{\mathrm{d}\mathrm{i}}$ leads to a shorter waiting time, so W(${\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}}$) < W(${\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}$); because the expected medical cost $C\left({\mu }_{di}\right)=C_0+C_u\cdot \mathrm{s}\cdot {\mu }_{\mathrm{d}\mathrm{i}}$, a larger ${\mu }_{\mathrm{d}\mathrm{i}}$ leads to a larger medical cost $\mathrm{C}({\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}})>\mathrm{C}({\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}})$; because the social cost $\mathrm{S}\mathrm{C}=\mathrm{d}\Lambda \mathrm{W}({\mu }_{\mathrm{d}\mathrm{i}},{\lambda }_{\mathrm{i}})$ + $C\left({\mu }_{di}\right),$ $\mathrm{i}\mathrm{f}\hspace{0.33em}\mathrm{B}\in [\Lambda {\mathrm{p}}_1,{\mathrm{B}}_0)$, the waiting cost under BP is $\mathrm{d}\Lambda \overline{\mathrm{w}}$, and the medical cost under FFS is also the upper bound, so it can be approximated that two social costs are equal and both are very large, ${\mathrm{i}.\mathrm{e}.,\mathrm{S}\mathrm{C}}^{\mathrm{F}\mathrm{F}\mathrm{S}}={\mathrm{S}\mathrm{C}}^{\mathrm{B}\mathrm{P}}$; $\mathrm{i}\mathrm{f}\hspace{0.33em}\mathrm{B}\in [{\mathrm{B}}_0,+\mathrm{\infty })$, the social cost under BP is a fixed positive value, but it is very large under FFS, and thus ${\mathrm{S}\mathrm{C}}^{\mathrm{F}\mathrm{F}\mathrm{S}}>{\mathrm{S}\mathrm{C}}^{\mathrm{B}\mathrm{P}}$. □

Proposition 5 shows that FFS induces hospitals to choose higher service rates, resulting in shorter waiting times but higher medical and social costs, whereas BP internalizes medical costs and leads to lower overall inefficiency. The result highlights that a higher service rate does not necessarily imply greater efficiency, and cost-internalizing payment schemes, such as BP, are more effective in balancing service rates and cost controls.

6. First-Best Case and Performance Comparison

To better explore the impact of different payment schemes on the operational efficiency of the outpatient service system, we establish a “benchmark” as a comparison standard under ideal conditions, which is called the first-best case. In this section, we first analyze the optimal service rates of each server in hospitals in the first-best case and then compare the results with those under different schemes.

6.1. First-Best Case

In the first-best case, the entire outpatient service system is fully owned by the payer in a centralized public outpatient service system, and the payer sets the optimal service rates for the servers to minimize social costs. Based on the previous assumptions of homogeneous hospitals and no balking behavior of patients, the payer equally distributes the patients’ demand, that is, the arrival rate of patients to each hospital is ${\displaystyle \frac{\Lambda }{\mathrm{n}}}$. According to the definition of social costs in model setup, the optimization problem of the payer is

$$\begin{array}{c}\underset{{\mu }_{\mathrm{d}\mathrm{i}}}{\mathrm{m}\mathrm{i}\mathrm{n}}\hspace{1em}\mathrm{S}\mathrm{C}({\mu }_{\mathrm{d}\mathrm{i}})={\displaystyle \frac{\mathrm{d}\Lambda }{\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}-\frac{\Lambda }{\mathrm{n}}}}+C({\mu }_{\mathrm{d}\mathrm{i}})\Lambda \\ \hspace{1em}\hspace{1em}\mathrm{s}.\mathrm{t}.\hspace{0.33em}{\displaystyle \frac{\Lambda }{\mathrm{n}}}+{\displaystyle \frac{1}{\overline{\mathrm{w}}}}\le \mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}\le \overline{\mu }.\end{array}$$

(13)

By solving the optimization model (13), we have the first-best optimal service rate of each server in hospital ${\mathrm{H}}_{\mathrm{i}}$, ${\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{f}}$, as shown in the following proposition.

Proposition 6. 

The first-best solution is to set ${\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{f}}={\displaystyle \frac{1}{\mathrm{s}}}\cdot \sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}+{\displaystyle \frac{\Lambda }{\mathrm{s}\cdot \mathrm{n}}}$.

Proof of Proposition 6. 

According to the optimization model (13), the first-derivative and the second-derivative of the social cost function SC with respect to ${\mu }_{\mathrm{d}\mathrm{i}}$ are

$${\displaystyle \frac{\partial \mathrm{S}\mathrm{C}}{\partial {\mu }_{\mathrm{d}\mathrm{i}}}}=-{\displaystyle \frac{\mathrm{s}\mathrm{d}\Lambda }{(\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}-\frac{\Lambda }{\mathrm{n}}{)}^2}}+\mathrm{s}{C}_u\Lambda ,\hspace{1em}{\displaystyle \frac{{\partial }^2\mathrm{S}\mathrm{C}}{\partial {{\mu }_{\mathrm{d}\mathrm{i}}}^2}}={\displaystyle \frac{2{\mathrm{s}}^2\mathrm{d}\Lambda }{(\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}-\frac{\Lambda }{\mathrm{n}}{)}^3}}.$$

Because ${\displaystyle \frac{2{\mathrm{s}}^2\mathrm{d}\Lambda }{(\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}-\frac{\Lambda }{\mathrm{n}}{)}^3}}>0$, $\mathrm{S}\mathrm{C}({\mu }_{\mathrm{d}\mathrm{i}})$ is a convex function with respect to ${\mu }_{\mathrm{d}\mathrm{i}}$, the first-order condition leads to ${\mu }_{\mathrm{d}\mathrm{i}}={\displaystyle \frac{1}{\mathrm{s}}}\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}+{\displaystyle \frac{\Lambda }{\mathrm{s}\cdot \mathrm{n}}}$ and the value is in the domain $[{\displaystyle \frac{\Lambda }{\mathrm{s}\mathrm{n}}}+{\displaystyle \frac{1}{\mathrm{s}\overline{\mathrm{w}}}},{\displaystyle \frac{\overline{\mu }}{\mathrm{s}}}]$; hence, the first-best solution ${\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{f}}={\displaystyle \frac{1}{\mathrm{s}}}\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}+{\displaystyle \frac{\Lambda }{\mathrm{s}\cdot \mathrm{n}}}$. □

Proposition 6 shows that the first-best service rate decreases in the number of hospitals and the unit medical cost while it increases in the unit waiting cost and the market size. The results are driven by the impact of the service rate on the medical cost and the waiting cost; as the provider increases the service rate, the medical cost increases, and the waiting cost decreases. Therefore, the larger the number of hospitals or the higher the unit medical cost, the payer sets a lower service rate for the servers to control the medical cost. As the unit waiting cost or the market size increases, the payer would set a higher service rate for the servers to reduce the total waiting cost.

6.2. Performance Comparison

In this subsection, we examine whether the first-best outcome can be achieved under the FFS and BP schemes by comparing the equilibrium service rates of the hospitals under the above two schemes and the first-best solution.

Proposition 7. 

If $\mathrm{B}\in [\Lambda {\mathrm{p}}_1,{\mathrm{B}}_1),{\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}<{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{f}}<{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}};if\hspace{0.33em}\mathrm{B}\in [{\mathrm{B}}_1,+\mathrm{\infty }),{\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}={\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{f}}<{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}}.$

Proof of Proposition 7. 

If $\mathrm{B}\in [\Lambda {\mathrm{p}}_1,{\mathrm{B}}_1)$, ${\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}-{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{f}}=({\displaystyle \frac{\frac{\mathrm{B}}{\Lambda }-C_0}{\mathrm{s}{C}_u}}-{\displaystyle \frac{\Lambda }{\mathrm{s}(\mathrm{n}-1)}})-{\displaystyle \frac{1}{\mathrm{s}}}\sqrt{{\displaystyle \frac{\mathrm{d}}{{\mathrm{C}}_{\mathrm{u}}}}}+{\displaystyle \frac{\Lambda }{\mathrm{s}\mathrm{n}}}$. Let $\mathrm{f}({\mathrm{p}}^{\mathrm{B}\mathrm{P}})={\displaystyle \frac{{\mathrm{p}}^{\mathrm{B}\mathrm{P}}-C_0}{\mathrm{s}{C}_u}}-{\displaystyle \frac{\Lambda }{\mathrm{s}(\mathrm{n}-1)}}$, then ${\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}=\mathrm{f}({\displaystyle \frac{\mathrm{B}}{\Lambda }}),{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{f}}=\mathrm{f}({\mathrm{p}}_{\mathrm{d}})$, and $\mathrm{f}({\mathrm{p}}^{\mathrm{B}\mathrm{P}})$ is strictly increasing in ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}$. Because $\mathrm{B}<\Lambda {\mathrm{p}}_{\mathrm{d}}$, we have ${\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}-{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{f}}=\mathrm{f}({\displaystyle \frac{\mathrm{B}}{\Lambda }})-\mathrm{f}({\mathrm{p}}_{\mathrm{d}})<0$. Thus, ${\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}<{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{f}}<{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}}$.

If $\mathrm{B}\in [{\mathrm{B}}_1,+\mathrm{\infty })$, ${\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}={\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{f}}=\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}+{\displaystyle \frac{\Lambda }{\mathrm{n}}}<{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}}$.□

Proposition 7 reveals that under the FFS payment scheme, the first-best outcome cannot be implemented. In addition, higher healthcare service rate brings high medical costs with the FFS scheme, which is consistent with Calsyn and Lee [8]. For the BP scheme, it turns out that only when the budget is high enough $\mathrm{B}\in \left.{[\mathrm{B}}_1,+\mathrm{\infty }\right)$ can the payer regulate the hospitals’ service rate decision to the first-best solution by adjusting the price. However, when the budget is low $\mathrm{B}\in [\left.\Lambda {\mathrm{p}}_1,{\mathrm{B}}_1\right)$, the service rate under the BP scheme is slower than the first-best solution.

Compared with BP, we find that FFS is effective in increasing the service rate (i.e., ${\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}<{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}}$) and reducing waiting time, which is consistent with the results shown in Guo et al. [14]. In other words, the BP scheme leads to a slower service rate of the servers and more waiting time for patients. At the same time, it also realizes lower medical costs for the whole episode of care because the hospital pays more time and patience to serve patients, improving the efficiency and quality of diagnosis and treatment. To minimize the total waiting cost and medical cost, in the next section, we design a coordination mechanism, the BP with service level regulation (BPW), to adjust the hospitals’ behavior.

7. Bundled Payment with Service Level Regulation

For many service industries, providers often advertise a kind of service level regulation, the expected waiting time guarantee of their services [47]. For example, for non-emergency medical services, many countries have implemented some form of a maximum waiting time. Taking Canada as an example, in 2005, provinces agreed to establish common waiting time guarantees nationwide to reduce patient waiting times [48]. Ameritrade promises that trade takes no more than 10 s to be executed. Some call centers guarantee that customers can be serviced within an hour. Some studies characterize this waiting time guarantee through the queueing model [47,49]. Therefore, in line with the above commonly observed phenomenon regarding the waiting time guarantee, we assume that the expected waiting time of patients needs to be shorter than a waiting time guarantee ${\mathrm{w}}_0$, and it is set by the payer. Because $\overline{\mathrm{w}}$ is a sufficiently large positive number, and thus ${\mathrm{w}}_0<\overline{\mathrm{w}}$.

Combining the waiting time guarantee setting and the basic assumptions, the patients’ equilibrium arrival rates satisfy the following condition:

$$\left\{\begin{array}{c}\begin{array}{c}\begin{array}{c}\mathrm{W}({\mu }_{\mathrm{d}\mathrm{i}},{\lambda }_{\mathrm{i}})=\mathrm{W}({\mu }_{-\mathrm{d}\mathrm{i}},{\lambda }_{-\mathrm{i}}),\end{array}\\ {\lambda }_{\mathrm{i}}+(\mathrm{n}-1){\lambda }_{-\mathrm{i}}=\Lambda ,\end{array}\\ \begin{array}{c}\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}>{\lambda }_{\mathrm{i}},\mathrm{s}{\mu }_{-\mathrm{d}\mathrm{i}}>{\lambda }_{-\mathrm{i}},\\ \mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}\le \overline{\mu },\mathrm{s}{\mu }_{-\mathrm{d}\mathrm{i}}\le \overline{\mu },\\ {\displaystyle \frac{1}{\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}-{\lambda }_{\mathrm{i}}}}\le {\mathrm{w}}_0,{\displaystyle \frac{1}{\mathrm{s}{\mu }_{-\mathrm{d}\mathrm{i}}-{\lambda }_{-\mathrm{i}}}}\le {\mathrm{w}}_0.\end{array}\end{array}\right.$$

(14)

Let ${\lambda }_{\mathrm{i}}^{\mathrm{e}\mathrm{w}}$ and ${\lambda }_{-\mathrm{i}}^{\mathrm{e}\mathrm{w}}$ denote the equilibrium arrival rates of hospitals ${\mathrm{H}}_{\mathrm{i}}$ and ${\mathrm{H}}_{-\mathrm{i}}$ under the BPW scheme, respectively. By deducing the above model, we have the ${\lambda }_{\mathrm{i}}^{\mathrm{e}\mathrm{w}}$ and ${\lambda }_{-\mathrm{i}}^{\mathrm{e}\mathrm{w}}$ specified in following Proposition 8.

Proposition 8. 

Under the BPW scheme, the equilibrium arrival rates of the hospital ${\mathrm{H}}_{\mathrm{i}}$ and the other n − 1 hospitals ${\mathrm{H}}_{-\mathrm{i}}$ are as follows:

(i)

$\mathrm{I}\mathrm{f}\hspace{0.33em}{\mu }_{\mathrm{d}\mathrm{i}}\in [0,{\mathrm{Y}}_1^{\mathrm{w}}({\mu }_{-\mathrm{d}\mathrm{i}})),\hspace{0.33em}{\lambda }_{\mathrm{i}}^{\mathrm{e}\mathrm{w}}={\lambda }_{-\mathrm{i}}^{\mathrm{e}\mathrm{w}}=0;$

(ii)

$\mathrm{i}\mathrm{f}\hspace{0.33em}{\mu }_{\mathrm{d}\mathrm{i}}\in [{\mathrm{Y}}_1^{\mathrm{w}}({\mu }_{-\mathrm{d}\mathrm{i}}),{\mathrm{Y}}_2^{\mathrm{w}}({\mu }_{-\mathrm{d}\mathrm{i}})),\hspace{0.33em}{\lambda }_{\mathrm{i}}^{\mathrm{e}\mathrm{w}}=0,{\lambda }_{-\mathrm{i}}^{\mathrm{e}\mathrm{w}}={\displaystyle \frac{\Lambda }{\mathrm{n}-1}};$

(iii)

$\mathrm{i}\mathrm{f}\hspace{0.33em}{\mu }_{\mathrm{d}\mathrm{i}}\in [{\mathrm{Y}}_2^{\mathrm{w}}({\mu }_{-\mathrm{d}\mathrm{i}}),{\mathrm{Y}}_3^{\mathrm{w}}({\mu }_{-\mathrm{d}\mathrm{i}})),\hspace{0.33em}{\lambda }_{\mathrm{i}}^{\mathrm{e}\mathrm{w}}={\displaystyle \frac{\Lambda +\mathrm{s}(\mathrm{n}-1)({\mu }_{\mathrm{d}\mathrm{i}}-{\mu }_{-\mathrm{d}\mathrm{i}})}{\mathrm{n}}},{\lambda }_{-\mathrm{i}}^{\mathrm{e}\mathrm{w}}={\displaystyle \frac{\Lambda +\mathrm{s}{\mu }_{-\mathrm{d}\mathrm{i}}-\mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}}{\mathrm{n}}};$

(iv)

$\mathrm{i}\mathrm{f}\hspace{0.33em}{\mu }_{\mathrm{d}\mathrm{i}}\in [{\mathrm{Y}}_3^{\mathrm{w}}({\mu }_{-\mathrm{d}\mathrm{i}}),\overline{\mu }),\hspace{0.33em}{\lambda }_{\mathrm{i}}^{\mathrm{e}\mathrm{w}}=\Lambda ,{\lambda }_{-\mathrm{i}}^{\mathrm{e}\mathrm{w}}=0.$

Proof. 

See Appendix A.

Note: The definitions of ${\mathrm{Y}}_1^{\mathrm{w}}({\mu }_{-\mathrm{d}\mathrm{i}}),{\mathrm{Y}}_2^{\mathrm{w}}({\mu }_{-\mathrm{d}\mathrm{i}}),{ \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{Y}}_3^{\mathrm{w}}({\mu }_{-\mathrm{d}\mathrm{i}})$) are presented in Appendix A. □

Compared with the BP without service level regulation, under the BP with service level regulation, the equilibrium demands of the patients are the same, while the regional boundaries are different. Due to the higher requirement for the expected waiting time, only when the server’s service rate is higher will patients choose to join the service.

7.1. The Hospitals’ Service Rate Decision

After observing the given service price ${\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}$ and the waiting time guarantee ${\mathrm{w}}_0$, the hospitals maximize their profits by determining the service rates. Given the n − 1 hospitals’ service rates decisions ${\mu }_{-\mathrm{d}\mathrm{i}}$ and the corresponding arrival rate ${\lambda }_{\mathrm{i}}^{\mathrm{e}\mathrm{w}}$, the hospital ${\mathrm{H}}_{\mathrm{i}}$’s maximization problem is

$$\begin{array}{c}\underset{{\mu }_{\mathrm{d}\mathrm{i}}}{\mathrm{m}\mathrm{a}\mathrm{x}}\hspace{1em}\hspace{1em}{\pi }_{\mathrm{i}}({\mu }_{\mathrm{d}\mathrm{i}})=({\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}-C_0-C_u·{\mathrm{s}\mu }_{\mathrm{d}\mathrm{i}}){\lambda }_{\mathrm{i}}^{\mathrm{e}\mathrm{w}}({\mu }_{\mathrm{d}\mathrm{i}},{\mu }_{-\mathrm{d}\mathrm{i}})\\ \mathrm{s}.\mathrm{t}.\hspace{1em}{\pi }_{\mathrm{i}}({\mu }_{\mathrm{d}\mathrm{i}})\ge 0,\\ \hspace{1em}\hspace{1em}\hspace{1em}{\lambda }_{\mathrm{i}}^{\mathrm{e}\mathrm{w}}({\mu }_{\mathrm{d}\mathrm{i}},{\mu }_{-\mathrm{d}\mathrm{i}})+{\displaystyle \frac{1}{{\mathrm{w}}_0}}\le \mathrm{s}{\mu }_{\mathrm{d}\mathrm{i}}\le \overline{\mu }.\end{array}$$

(15)

Applying the same analytical method to that under the BP scheme, we can derive the equilibrium under BPW, ${\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}\mathrm{w}}$, and the corresponding equilibrium arrival rate ${\lambda }_{\mathrm{i}}^{\mathrm{e}\mathrm{w}}$ as follows.

Proposition 9. 

Under BPW, the equilibrium service rate of the servers in hospital ${\mathrm{H}}_{\mathrm{i}}$  is

$${\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}\mathrm{w}}=\left\{\begin{array}{c}\begin{array}{c}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{i}\mathrm{f}\hspace{0.33em}{\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}\in [0,{\mathrm{p}}_1^{\mathrm{w}}),\\ {\displaystyle \frac{\Lambda }{\mathrm{s}\mathrm{n}}}+{\displaystyle \frac{1}{\mathrm{s}{\mathrm{w}}_0}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{i}\mathrm{f}\hspace{0.33em}{\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}\in [{\mathrm{p}}_1^{\mathrm{w}},{\mathrm{p}}_4^{\mathrm{w}}),\end{array}\\ {\displaystyle \frac{{\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}-C_0}{\mathrm{s}{C}_u}}-{\displaystyle \frac{\Lambda }{\mathrm{s}(\mathrm{n}-1)}}\hspace{1em}\hspace{1em}\mathrm{i}\mathrm{f}\hspace{0.33em}{\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}\in [{\mathrm{p}}_4^{\mathrm{w}},+\mathrm{\infty }).\end{array}\right.$$

(16)

the corresponding equilibrium arrival rate is

$${\lambda }_{\mathrm{i}}^{\mathrm{e}\mathrm{w}}=\left\{\begin{array}{l}0\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{i}\mathrm{f}\hspace{0.33em}{\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}\in [0,{\mathrm{p}}_1^{\mathrm{w}}),\\ {\displaystyle \frac{\Lambda }{\mathrm{n}}}\hspace{1em}\hspace{1em}\hspace{0.33em}\mathrm{i}\mathrm{f}\hspace{0.33em}{\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}\in [{\mathrm{p}}_1^{\mathrm{w}},+\mathrm{\infty }).\end{array}\right.$$

(17)

where ${\mathrm{p}}_1^{\mathrm{w}}=C_0+{\displaystyle \frac{C_u}{{\mathrm{w}}_0}},{\mathrm{p}}_4^{\mathrm{w}}=C_0+{\displaystyle \frac{(2\mathrm{n}-1)C_u\Lambda }{\mathrm{n}(\mathrm{n}-1)}}+{\displaystyle \frac{C_u}{{\mathrm{w}}_0}}$.

Proof. 

See Appendix A. □

Compared with the BP (Proposition 3), under the BPW, the equilibrium service rate of the servers in hospital ${\mathrm{H}}_{\mathrm{i}}$ is different in the following ways. (1) The service price interval changes, and the equilibrium service rates shift to the right according to the segment interval of the price (${\mathrm{p}}_1^{\mathrm{w}}>{\mathrm{p}}_1,{\mathrm{p}}_4^{\mathrm{w}}>{\mathrm{p}}_4$). It can be found that the payer needs to provide a higher service price and the hospitals are willing to provide outpatient services. The reason is that because of the setting of the waiting time guarantee, the hospitals need to increase the service rate to meet the guarantee, which leads to higher medical costs. Therefore, the hospitals will only provide services when the price is higher. (2) When the price is low enough, ${\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}\in [0,{\mathrm{p}}_1^{\mathrm{w}})$, the equilibrium service rate is higher (${\displaystyle \frac{\Lambda }{\mathrm{s}\mathrm{n}}}+{\displaystyle \frac{1}{\mathrm{s}{\mathrm{w}}_0}}>{\displaystyle \frac{\Lambda }{\mathrm{s}\mathrm{n}}}+{\displaystyle \frac{1}{\mathrm{s}\overline{\mathrm{w}}}}$). Due to the regulation of waiting time, the hospitals need to improve the service rate while guaranteeing non-negative profits. In this case, the waiting time of the patients just meets the guarantee ${\mathrm{w}}_0$.

7.2. The Payer’s Price and Waiting Time Guarantee Strategy

Under the BPW, the payer designs the service price ${\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}$ and waiting time guarantee ${\mathrm{w}}_0$ to minimize the total social cost. The total social cost includes the delay cost of the patients and the medical cost of the hospitals. We focus on cases where the hospitals are willing to provide services, that is, when the service price is high enough, i.e., ${\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}\in [{\mathrm{p}}_1^{\mathrm{w}},+\mathrm{\infty })$. Accordingly, we consider that the budget is high enough, i.e., $\mathrm{B}\in [{\mathrm{B}}_1^{\mathrm{w}},+\mathrm{\infty })$, where ${\mathrm{B}}_1^{\mathrm{w}}=\Lambda {\mathrm{p}}_1^{\mathrm{w}}$. In this case, substituting the equilibrium service rates ${\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}\mathrm{w}}$ in Proposition 8 into the model (5), we have the problem of the payer as follows:

$$\begin{array}{c}\underset{{\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}},{\mathrm{w}}_0}{\mathrm{M}\mathrm{i}\mathrm{n}}\hspace{1em}\mathrm{S}\mathrm{C}({\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}},{\mathrm{w}}_0)\\ \begin{array}{c}\hspace{0.33em}\hspace{0.33em}\mathrm{s}.\mathrm{t}.{\mathrm{p}}_1^{\mathrm{w}}\le {\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}\le {\displaystyle \frac{\mathrm{B}}{\Lambda }},\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}\hspace{0.33em}{\mathrm{w}}_0\ge 0.\end{array}\end{array}$$

(18)

where the social cost function is

$$\mathrm{S}\mathrm{C}({\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}},{\mathrm{w}}_0)=\left\{\begin{array}{c}\begin{array}{c}\mathrm{d}\Lambda {\mathrm{w}}_0+(C_0+C_u({\displaystyle \frac{\Lambda }{\mathrm{n}}}+{\displaystyle \frac{1}{{\mathrm{w}}_0}}))\Lambda \hspace{1em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}\mathrm{i}\mathrm{f}\hspace{0.33em}{\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}\in [{\mathrm{p}}_1^{\mathrm{w}},{\mathrm{p}}_4^{\mathrm{w}}),\end{array}\\ {\displaystyle \frac{\mathrm{d}\Lambda }{\frac{{\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}-C_0}{C_u}-\frac{(2\mathrm{n}-1)\Lambda }{\mathrm{n}(\mathrm{n}-1)}}}+({\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}-{\displaystyle \frac{C_u\Lambda }{\mathrm{n}-1}})\Lambda \hspace{1em}\hspace{0.33em}\hspace{0.33em}\hspace{1em}\mathrm{i}\mathrm{f}\hspace{0.33em}{\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}\in [{\mathrm{p}}_4^{\mathrm{w}},+\mathrm{\infty }).\end{array}\right.$$

(19)

We solve the payer’s social cost minimization problem in two steps. First, we find the optimal service price ${\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}({\mathrm{w}}_0)$ for a given waiting time guarantee ${\mathrm{w}}_0$. Then, using ${\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}({\mathrm{w}}_0)$, we find the optimal waiting time guarantee, which minimizes the social cost ${\mathrm{w}}_0^{\mathrm{*}}$, and further the optimal service price ${\tilde{\mathrm{p}}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}$. The following proposition presents the outcomes.

Proposition 10. 

Under the BPW, the payer’s optimal strategies for service price and waiting time guarantee $({\tilde{\mathrm{p}}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}},{\mathrm{w}}_0^{\mathrm{*}})$, and the corresponding service rate ${\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}\mathrm{w}}$ satisfies the following:

$$\mathrm{i}\mathrm{f}\hspace{0.33em}\mathrm{t}\mathrm{h}\mathrm{e} \mathrm{b}\mathrm{u}\mathrm{d}\mathrm{g}\mathrm{e}\mathrm{t}\hspace{0.33em}\mathrm{B}\in \left[{\mathrm{B}}_1^{\mathrm{w}},{\mathrm{B}}_1\right),{\mathrm{w}}_0^{\mathrm{*}}=\sqrt{{\displaystyle \frac{C_u}{\mathrm{d}}}},{\tilde{\mathrm{p}}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}=C_0+\sqrt{\mathrm{d}{C}_u},\hspace{0.33em}{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}\mathrm{w}}={\displaystyle \frac{\Lambda }{\mathrm{s}\mathrm{n}}}+{\displaystyle \frac{1}{\mathrm{s}}}\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}};$$

$$\begin{gathered} \mathrm{i}\mathrm{f}\hspace{0.33em}\mathrm{t}\mathrm{h}\mathrm{e}\hspace{0.33em}\mathrm{b}\mathrm{u}\mathrm{d}\mathrm{g}\mathrm{e}\mathrm{t}\hspace{0.33em}\mathrm{B}\in [{\mathrm{B}}_1,+\mathrm{\infty }),{\mathrm{w}}_0^{\mathrm{*}}\in [\sqrt{{\displaystyle \frac{C_u}{\mathrm{d}}}},+\mathrm{\infty }),{\tilde{\mathrm{p}}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}=C_0+C_u(\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}+{\displaystyle \frac{(2\mathrm{n}-1)\Lambda }{\mathrm{n}(\mathrm{n}-1)}}), \\ {\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}\mathrm{w}}={\displaystyle \frac{\Lambda }{\mathrm{s}\mathrm{n}}}+{\displaystyle \frac{1}{\mathrm{s}}}\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}. \end{gathered}$$

Proof. 

See Appendix A. □

Proposition 10 indicates that under the BPW scheme, the optimal strategy is related to the medical budget. When the budget is immediate, the optimal waiting time guarantee increases in unit medical cost and decreases in patient delay sensitivity. This is because as the unit medical cost increases, the payer sets more a lenient waiting time guarantee to relax constraints on the hospital service rates. This encourages the hospitals to provide slower services to control overall medical costs. When patients are more sensitive to waiting times, the payer needs to set a stricter time guarantee to incentivize higher service rates, thereby enhancing patient satisfaction with waiting times. With a higher budget, there is a broader range of choices for optimal waiting time guarantee setting. However, from the perspective of patient satisfaction with waiting time, the same value as the immediate budget scenario can be set. In this case, patients can maximize their waiting time savings. The optimal service price under the immediate budget is lower than that under the high budget because the lower budget restricts setting of the price. Therefore, the payer needs to carefully consider the medical budget when establishing healthcare payment schemes.

Comparing the optimal service rates of the servers under the three payment schemes (FFS, BP, BPW) and the first-best service rate, we have the result in the following proposition.

Proposition 11. 

$If\hspace{0.33em}\mathrm{B}\in [{\mathrm{B}}_1^{\mathrm{w}},{\mathrm{B}}_1),{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}}>{\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}\mathrm{w}}={\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{f}}>{\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}};\mathrm{i}\mathrm{f}\hspace{0.33em}\mathrm{B}\in [{\mathrm{B}}_1,+\mathrm{\infty }),{\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}}>{\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}\mathrm{w}}={\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}={\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{f}}.$

Proof of Proposition 11. 

When $\mathrm{B}\in [{\mathrm{B}}_1^{\mathrm{w}},{\mathrm{B}}_1),$

$${\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}={\displaystyle \frac{\frac{\mathrm{B}}{\Lambda }-C_0}{\mathrm{s}{C}_u}}-{\displaystyle \frac{\Lambda }{\mathrm{s}(\mathrm{n}-1)}}<{\displaystyle \frac{\frac{{\mathrm{B}}_1}{\Lambda }-C_0}{\mathrm{s}{C}_u}}-{\displaystyle \frac{\Lambda }{\mathrm{s}(\mathrm{n}-1)}}={\displaystyle \frac{{\mathrm{p}}_{\mathrm{d}}-C_0}{\mathrm{s}{C}_u}}-{\displaystyle \frac{\Lambda }{\mathrm{s}(\mathrm{n}-1)}}={\displaystyle \frac{\Lambda }{\mathrm{s}\mathrm{n}}}+{\displaystyle \frac{1}{\mathrm{s}}}\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}={\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}\mathrm{w}}={\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{f}}<{\displaystyle \frac{\overline{\mu }}{\mathrm{s}}}={\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}}$$

when $\mathrm{B}\in [{\mathrm{B}}_1,+\mathrm{\infty }),{\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}}={\tilde{\mu }}_{\mathrm{d}\mathrm{i}}^{\mathrm{B}\mathrm{P}\mathrm{w}}={\displaystyle \frac{\Lambda }{\mathrm{s}\mathrm{n}}}+{\displaystyle \frac{1}{\mathrm{s}}}\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}={\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{f}}<{\displaystyle \frac{\overline{\mu }}{\mathrm{s}}}={\mu }_{\mathrm{d}\mathrm{i}}^{\mathrm{F}\mathrm{F}\mathrm{S}}$. □

Proposition 11 shows that regardless of the budget value, the optimal service rate of the servers under the FFS is the highest. The reason behind this is as follows. Under the FFS, the hospitals obtain income according to a certain proportion of the total medical cost, and hence the hospitals will aim to maximize the service rate to attract patient demand and increase medical costs to maximize revenue. Under the BP, the optimal service rate is lower than the first-best outcome when the medical budget is in the middle, and it achieves the first-best service rate when the budget is high enough. Under the BPW, the optimal service rates of hospitals can achieve optimal social welfare outcomes.

To facilitate our exposition, we numerically illustrate the comparison results here—the comparison of optimal service rates, waiting cost, medical cost, and social cost under first-best, BP, and BPW. Because the extreme results under FFS are easy to compare, the numerical analysis does not discuss the FFS case. From our numerical analysis, shown in Figure 4a and Proposition 11, we observe the optimal service rates’ comparison.

When the budget is immediate with $\mathrm{B}\in [{\mathrm{B}}_1^{\mathrm{w}},{\mathrm{B}}_1)$, the hospitals are motivated to provide services under the BPW, and the first-best outcome can be picked. In addition, the optimal service rate is higher than that under the BP scheme. The corresponding social cost under BP is higher than that under first-best and BPW. The reasons behind this can be explained by Figure 4c,d. Under BP, the hospitals provide slower services. Although the medical cost is relatively low, the waiting cost is much higher than that under BPW because the setting of the waiting time regulation can urge the hospitals to improve the service rate under BPW. Therefore, the BPW scheme dominates the BP scheme in terms of the social cost and the waiting cost. This result indicates that for the payer, reducing patient waiting time and waiting cost and improving patient satisfaction should be some of the important indicators for evaluating bundled payment schemes.

When the budget is high enough with $\mathrm{B}\in [{\mathrm{B}}_1,+\mathrm{\infty })$, the optimal service rates under both two BP schemes (BP and BPW) are the same, which is equal to the first-best service rate (Figure 4a). At the same time, the medical cost and waiting cost of the social costs under both BP schemes are the same as those in the first-best case (Figure 4b–d). The reason is that in this case, the price interval is wider. Even if there is no service level regulation, the payer can stimulate the hospitals to improve the service rate by increasing the service price.

To further assess the robustness of our results, we conduct additional numerical experiments by varying key system parameters over wider ranges, including the total potential arrival rate $\Lambda$, the number of hospitals n, the unit waiting cost d, and so on. The qualitative patterns observed in Figure 4 remain unchanged across all tested parameter ranges. For brevity, the detailed robustness results are reported in Appendix B.

According to Proposition 11, we have the optimal payment scheme strategy of the payer shown in Theorem 1.

Theorem 1. 

When the budget $\mathrm{B}\in [{\mathrm{B}}_1^{\mathrm{w}},{\mathrm{B}}_1)$, the payer can implement the first-best outcome only by introducing the BPW; when the budget $\mathrm{B}\in [{\mathrm{B}}_1,+\mathrm{\infty })$, the first-best outcome can be achieved by the BP scheme. However, the first-best outcome cannot be achieved when $\mathrm{B}\in [0,{\mathrm{B}}_1^{\mathrm{w}})$.

Proof of Theorem 1. 

Following from Proposition 9, we know that when $\mathrm{B}\in [0,{\mathrm{B}}_1^{\mathrm{w}})$, no hospitals will provide service, and thus the first-best solution cannot be picked; when $\mathrm{B}\in [{\mathrm{B}}_1^{\mathrm{w}},{\mathrm{B}}_1),$ under the BP with service level regulation, by setting ${\mathrm{p}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}=C_0+\sqrt{\mathrm{d}{C}_u}$ and ${\mathrm{w}}_0={\mathrm{w}}_0^{\mathrm{*}}$, the first-best solution can be picked; when $\mathrm{B}\in [{\mathrm{B}}_1,+\mathrm{\infty })$, under the BP scheme, by setting ${\mathrm{p}}^{\mathrm{B}\mathrm{P}}={\mathrm{p}}_{\mathrm{d}}$, the first-best outcome can be picked. □

From Theorem 1 and Figure 5, we find that when the budget is low enough $\mathrm{B}\in [0,{\mathrm{B}}_1^{\mathrm{w}})$, the hospitals are not profitable because the service price is too low, and hence no hospital is willing to provide service. In this case, the first-best solution cannot be picked. When the budget is intermediate, from Proposition 11, under the BP scheme, the payer sets a uniform price, which will make the service rate too slow. Meanwhile, under the BPW scheme, the payer sets the price ${\tilde{\mathrm{p}}}_{\mathrm{w}}^{\mathrm{B}\mathrm{P}}=C_0+\sqrt{\mathrm{d}{C}_u}$ and sets waiting time constraint ${\mathrm{w}}_0^{\mathrm{*}}=\sqrt{{\displaystyle \frac{C_u}{\mathrm{d}}}}$ to encourage the hospitals to increase the service rate to achieve the first-best outcome. When the budget is high enough, the payer can implement the first-best outcome through the BP scheme because the price can be set higher to induce a high service rate. Our study provides management insight for the payer when making payment scheme decisions.

Next, we analyze the impact of the parameters on the boundaries ${\mathrm{B}}_1$ and ${\mathrm{B}}_1^{\mathrm{w}}$. Recall that ${\mathrm{B}}_1=\Lambda [C_0+C_u(\sqrt{{\displaystyle \frac{\mathrm{d}}{C_u}}}+{\displaystyle \frac{(2\mathrm{n}-1)\Lambda }{\mathrm{n}(\mathrm{n}-1)}})],{\mathrm{B}}_1^{\mathrm{w}}=\Lambda (C_0+{\displaystyle \frac{C_u}{{\mathrm{w}}_0}})$. Let $\Delta \mathrm{B}={\mathrm{B}}_1-{\mathrm{B}}_1^{\mathrm{w}}$ denote the difference between the two boundaries, which represents the area where the first-best solution cannot be implemented with only the BP scheme and the BP with service level regulation is required.

Corollary 2. 

For the number of hospitals n, ${\mathrm{B}}_1$ and $\Delta \mathrm{B}$ are decreasing in $\mathrm{n} \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{B}}_1^{\mathrm{w}}$ is independent in $\mathrm{n}$, i.e., ${\displaystyle \frac{\partial {\mathrm{B}}_1}{\partial \mathrm{n}}}<0,{\displaystyle \frac{\partial {\mathrm{B}}_1^{\mathrm{w}}}{\partial \mathrm{n}}}=0,{\displaystyle \frac{\partial \Delta \mathrm{B}}{\partial \mathrm{n}}}<0$;

For the market size $\Lambda ,{\mathrm{B}}_1,{\mathrm{B}}_1^{\mathrm{w}}\hspace{0.33em}and\hspace{0.33em}\Delta \mathrm{B}$ are increasing in Λ, i.e., ${\displaystyle \frac{\partial {\mathrm{B}}_1}{\partial \Lambda }}>0,{\displaystyle \frac{\partial {\mathrm{B}}_1^{\mathrm{w}}}{\partial \Lambda }}>0,{\displaystyle \frac{\partial \Delta \mathrm{B}}{\partial \Lambda }}>0$.

Corollary 2 reveals that as the number of hospitals n increases, the budget boundary that the payer needs for added wait time regulation to implement the first-best outcome ${\mathrm{B}}_1$ decreases, the budget boundary of the hospital’s willingness to serve ${\mathrm{B}}_1^{\mathrm{w}}$ keeps constant, and the area where only by adding service level regulation based on the BP scheme can first-best outcomes be implemented ∆B decreases. The reasons can be given as follows. Under the BP scheme with an intermediate budget $\mathrm{B}\in [{\mathrm{B}}_1^{\mathrm{w}},{\mathrm{B}}_1]$, as the competition becomes greater, the hospitals would increase the service rate and the first-best service rate becomes smaller; hence, in such cases, the first-best outcome can be picked for the relatively high budget in $[{\mathrm{B}}_1^{\mathrm{w}},{\mathrm{B}}_1]$. That is, ${\mathrm{B}}_1$ decreases. In addition, this also means that under the BP scheme, it is easier to realize the first-best outcomes, which also corresponds to the conclusion that the area that needs added waiting time regulation is smaller.

Corollary 2 shows that as the market size $\Lambda$ increases, two budget boundaries and the area where the BPW can implement the first-best outcomes increase. Because under the BP scheme with an intermediate budget $\mathrm{B}\in [{\mathrm{B}}_1^{\mathrm{w}},{\mathrm{B}}_1]$, as the market size becomes larger, the hospitals would serve more slowly and the first-best service rate becomes larger; hence, in such cases, it is more difficult to realize the first-best outcomes, which also corresponds to the conclusion that the area that needs added waiting time regulation is larger.

8. Conclusions

This paper investigates the impact of the FFS and BP schemes on service rate decisions of hospitals and the performance of the service system. Hospitals face a cost–speed tradeoff when making decisions, where accelerating service rates increases medical costs for the whole episode of care but can attract more patient demand. In such a situation, we focus on how to configure healthcare payment schemes to achieve the first-best outcome—a benchmark scenario that maximizes social welfare ideally. The objective is to explore the inherent mechanisms and performance of the BP scheme, aiming to offer theoretical guidance for operational management strategies for the payer and hospitals. To the best of our knowledge, this paper is the first to explore the impact of the BP scheme on hospital service rates and system performance from an operations management perspective, considering the dependency of medical costs on service rates.

This paper employs a game-theoretic queueing model to investigate a Stakelberg game involving three participants: (1) the payer, responsible for deciding the payment scheme; (2) hospitals, which determine their service rates in a competitive environment under different schemes; and (3) patients, who choose one hospital for treatment. By comparing the performance of FFS, BP, and the first-best case, the study yields the following managerial insights. First, as expected, FFS is unlikely to be an optimal payment scheme, as rapid increases in service rates lead to lower quality of care and higher medical costs. Furthermore, whether the BP scheme can achieve the minimum sum of medical and waiting costs is mainly related to the medical budget. When the budget is low, hospitals are unwilling to provide services, because the payer cannot set a price that allows hospitals to be profitable. When the budget is moderate, the hospitals slow down service rates to control marginal medical costs, resulting in lower medical costs but higher and more significant waiting costs for patients. In such cases, we propose a coordination mechanism—BP with service level regulation—whereby setting guarantees for average waiting time can achieve optimal service rates, minimizing social costs. However, when the budget is sufficiently high, the payer can achieve the same incentivizing effect by setting a uniform optimal service price under the BP scheme. This is because a high budget corresponds to a high service price, achieving an optimal service rate that minimizes overall social costs. Therefore, when formulating a healthcare payment scheme, the payer needs to carefully consider the healthcare budget.

In addition, several factors influence the service rate strategies of the hospitals under BP, including the intensity of competition and market size. The results reveal that intense competition facilitates a more straightforward achievement of the first-best outcome under BP. Intense competition leads to higher service rates, and the socially optimal service rate is slower, even with a lower budget, making it possible to minimize social costs. On the other hand, a larger market size hinders BP from achieving the first-best outcome, necessitating a higher budget for hospitals to be willing to provide services and incentivize hospitals to reach the optimal service rate.

There are several possible directions along which the models in the paper can be extended. Our paper implicitly assumes risk neutrality, meaning that the hospitals aim to maximize expected profits. While this assumption is standard in the operations management literature, extending our analysis to the risk-averse utility functions of the hospitals would be useful. We assume no information asymmetry. Considering asymmetric information, where the hospitals may have better information about medical costs and patient waiting time than the payer, is also a valuable extension to our work. Moreover, our model assumes that all patients join the system, which is a reasonable approximation for publicly funded, non-urgent elective care systems with minimal patient out-of-pocket costs. In systems where patients face significant financial or waiting time barriers, patient attrition could moderate demand responses and affect equilibrium outcomes. Future work could incorporate balking or reneging assumptions to further generalize the analysis. Lastly, while we use a linear cost function for simplicity, we acknowledge that the true relationship between service rate and medical cost may be nonlinear in some settings. Nonlinearities could shift quantitative outcomes but are unlikely to alter the key qualitative conclusions of the model.