# The interdependence between hospital choice and waiting time — with a case
study in urban China

Joris van de Klundert, Business School Universidad Adolfo Ibáñez  Roberto
Cominetti, Faculty of Engineering, Universidad Adolfo Ibáñez  Yun Liu, School
of Health Policy and Management, Erasmus University Rotterdam  Qingxia Kong,
Rotterdam School of Management, Erasmus University Rotterdam

Abstract: Hospital choice models often employ random utility theory and
include waiting time as a choice determinant. When applied to evaluate health
system improvement interventions, these models disregard that hospital choice
in turn is a determinant of waiting time. We present a novel, general model
capturing the endogeneous relationship between waiting time and hospital
choice, including the choice to opt out, and characterize the unique
equilibrium solution of the resulting convex problem.

We apply the general model in a case study on the urban Chinese health system,
specifying that patient choice follows a multinomial logit (MNL) model and
waiting times are determined by M/M/1 queues. The results reveal that analyses
which solely rely on MNL models overestimate the effectiveness of present
policy interventions and that this effectiveness is limited. We explore
alternative, more effective, improvement interventions.

Keywords: Hospital choice, waiting time, multinomial logit, queuing theory,
urban China

Funding: The third author was supported by the Chinese Scholarship Council
[grant number 201507720036]

##  1 Introduction

Long waiting presents a significant barrier to healthcare access in various
forms and in a variety of health services and settings across the globe [25].
It often disadvantages patients of lower socioeconomic status and is
negatively associated with patient outcomes such as mortality [25, 24, 14].

For patients, waiting times are an important determinant of their choice of
hospital and hospital level and may even cause them to avoid care and opt out
[38, 11, 21]. The significance of waiting time as a determinant of hospital
choice is evidenced by revealed preferences and by stated preferences obtained
in a number of discrete choice experiments (DCEs) conducted across the globe
[4, 3, 42, 20] (however not for all patient populations and contexts [33,
22]).

Long waiting can be viewed to signal a need for health system interventions
that address underlying performance issues. In urban China, for instance, long
waiting has led to great discontent and caused undesired avoidance of care
[20]. In this setting, long waiting times especially arise at overutilized
tertiary hospitals, as patients freely access hospitals at the level of their
choice and tend to leave primary care underutilized [20, 2]. Both policy
measures and modeling studies reported in the academic literature have sought
to improve the persistent utilization problems by strengthening primary care
[19, 20, 28]. The model studies regard waiting time as a hospital attribute,
or as a hospital level attribute, i.e., as an independent explanatory variable
determining patient choice.

As observed by Lee and Cohen, waiting times in hospitals may not be static,
independent, hospital (level) attributes but typically vary with the choices
made [17, 23, 41]. Increased choice probabilities can cause congestion and
result in longer waiting. Thus, the variable waiting time, which is presumed
to be independent in the aforementioned DCE models, is endogeneously related
to the dependent variable hospital choice. This form of endogeneity, i.e.
simultaneity involving a dependent and a presumed independent is essentially
different from endogeneity between independents, or between independents and
unobserved factors, for which econometric methods are readily available (see
[36] and the references therein).

Disregard of endogeneity in DCE models can imply inconsistency and
incorrectness in the model estimation and cause the model to be ill suited for
policy analysis [36, 1]. In the case of hospital choice in urban China, it may
explain why evidence based interventions to improve medical equipment and
skills of physicians in primary care have not resolved the utilization
problems [26, 35, 19, 18, 21]. For example, interventions to strengthen
primary care can increase the probability of choosing primary care and reduce
the probability to choose tertiary care. The revised choice probabilities
translate to a tertiary care waiting time decrease that in turn increases its
choice probability and thus may partially undo the intervention effects. A
similar effect may occur when interventions to reduce the opt out probability
translate into longer waiting times of hospitals, partially undoing the
relative increase in attractiveness.

In this study, we advance the theoretical analysis of the endogenous
relationship between waiting time and choice probability and present a case
study analysing policy interventions in urban China ([23, 41, 15]). The
presented general model incorporates the interdependence between waiting times
and hospital choice, including the choice to opt out. It employs random
utility theory to specify functions capturing how choice probabilities depend
on utilities and how these utilities subsequently depend on waiting times. In
addition, the model relies on a fairly general function to express how waiting
times depend on choice probabilities. We show that the model yields unique
equilibrium choice probabilities and waiting times, which can be characterized
and found by convex optimization.

In the case study, we specifically apply a model in which a multinomial logit
(MNL) model specifies choice probability as a function of waiting time (among
other factors) and a queuing model specifies waiting time as a function of
choice probability [23, 41]. This special case model resolves the disregard of
endogeneity in previously published preference and policy studies that
consider waiting time to be exogenous. It applies the presented model to
provide a consistent assessment of the quantitative effects of health system
interventions (such as interventions to strengthen primary care) that is more
accurate than assessments obtained by previous MNL based models[20, 42]. The
analysis reports on intervention effects obtained with the special case model
and on significance of differences in policy intervention effects obtained
through models presented in previous studies.

In the discussion, we extensively consider future research on hospital choice
behaviour and models.

##  2 Models and Equilibrium Solutions

The model deals with a finite set of healthcare facility types $i\in
I=\\{1,\ldots,n\\}$, each one characterized by a set of attributes such as
skills of medical staff, equipment, price of service, and capacity. The types
can refer to hospital levels, such as primary, secondary, and tertiary
hospitals.

On the other hand, we consider a finite set $K=\\{1,\ldots,m\\}$ of patient
types where each $k\in K$ represents a homogeneous patient population. The
homogeneity refers to patient characteristics such as symptoms, insurance
type, age, and occupation, that determine the utility $U^{k}_{i}$ attached to
facility type $i$ by patients of type $k$. If the locations of facilities and
patients are relevant choice determinants — as might be the case when
considering travel time — each patient and/or facility may constitute a
separate type by itself.

We model utilities in terms of random variables. Namely, for each patient type
$k\in K$ and facility type $i\in I$, the utility is described by a random
variable $U^{k}_{i}=u^{k}_{i}+\varepsilon^{k}_{i}$ where
$u^{k}_{i}\in\mathbb{R}$ is the expected utility level, and
$\varepsilon^{k}_{i}$ is a random deviation that captures the variability
within the $k$-th patient population with $\mathbb{E}(\varepsilon^{k}_{i})=0$.
The model also includes the possibility that patients choose to opt out. To
this purpose, we introduce a fictitious opt-out facility type $i=0$ with
utility $U^{k}_{0}=u^{k}_{0}+\varepsilon^{k}_{0}$ for patients of type $k\in
K$ [15].

Following random utility theory, we assume patients choose the facility that
yields the maximal utility and hence the probabilities
$\pi^{k}=(\pi^{k}_{i})_{i\in I}$ of choosing facility type $i$ are given by
the discrete choice model

$\pi^{k}_{i}=\mathbb{P}(U^{k}_{i}\geq U^{k}_{j},\,\forall j\in I\cup\\{0\\}).$

For simplicity we assume that the random vector
$\varepsilon^{k}=(\varepsilon^{k}_{i})_{i\in I}$ has a continuous
distribution.

Denoting $u^{k}=(u^{k}_{0},u^{k}_{1},\ldots,u^{k}_{n})$ and considering the
expected utility function

$\Phi^{k}(u^{k})=\mathbb{E}(\max\\{u^{k}_{0}+\varepsilon^{k}_{0},u^{k}_{1}+\varepsilon^{k}_{1},\ldots,u^{k}_{n}+\varepsilon^{k}_{n}\\}),$

we explicitly establish that:

######  Theorem 1.

$\Phi^{k}(\cdot)$ is smooth and convex, and its derivatives are precisely the
choice probabilities, namely

$\pi^{k}_{i}=\pi^{k}_{i}(u^{k})=\frac{\partial\Phi^{k}}{\partial
u^{k}_{i}}(u^{k})\qquad(\forall i\in I\cup\\{0\\}).\\\ $ (1)

The proof is presented in Appendix A. We refer to [9, 32, 39, 17] for related
(partial) results.

In the remainder, we adopt the common assumption that utility varies linearly
with the waiting time and that an increase (decrease) in waiting time
translates into a decrease (increase) in utility. More formally, we consider
utilities of the form

$u^{k}_{i}=\bar{u}^{k}_{i}-\alpha^{k}w_{i}$ (2)

where $\bar{u}^{k}_{i}$ is a reference utility, $w_{i}$ is the expected
waiting time at facility $i\in I$, and the parameter $\alpha^{k}>0$ represents
the sensitivity of patients of type $k\in K$ to waiting time. We reflect on
this common assumption in the discussion section as a topic for future
research.

Let us now turn to expressing how choice probabilities determine waiting time.
To this purpose, we first define for each patient type $k\in K$ the arrival
rate $I^{k}>0$ as the number of patients of type $k$ arriving per time unit.
Then, for facility type $i\in I$ and choice probabilities $\pi^{k}_{i},k\in
K$, and assuming Poisson arrivals, the total arrival rate at facility type $i$
can be expressed as

$x_{i}=\sum_{k\in K}I^{k}\pi^{k}_{i}\qquad(\forall i\in I).$ (3)

The waiting time is assumed to be zero when opting out, i.e., $w_{0}\equiv 0$,
whereas for $i\geq 1$ we assume that it is given by a strictly increasing
continuous function $\theta_{i}:[0,\bar{x}_{i})\to(0,\infty)$ of the arrival
rates, namely

$w_{i}=\theta_{i}(x_{i})\qquad\hskip 14.2083pt(\forall i\in I),\\\ $ (4)

with $\bar{x}_{i}>0$ a saturation level at which waiting times diverge, that
is

$\lim_{x\to\bar{x}_{i}}\theta_{i}(x_{i})=\infty.$

The saturation level $\bar{x}_{i}$ can serve to model the finite capacity of
each of the facility types and the infinitely long queues that can form when
patient volumes are at or above capacity. However, the model allows for
$\bar{x}_{i}=\infty$ at some or all facility types (as in [17]).

Given the exogenous rates $I^{k}>0$ and the parameters $\bar{u}^{k}_{i}$ and
$\alpha^{k}$ in the utilities (2), we look for $w_{i}$ and $x_{i}$, $i\in I$,
simultaneously satisfying the system of equations (1)-(4). The equations (3)
and (4) extend the standard equations (1), (2) in random utility based choice
models, thus capturing the endogenous relationship between choice and waiting
time.

Remark. From (3) it follows that any solution satisfies $x_{i}\geq 0$ and
therefore $w_{i}\geq w^{0}_{i}$ where $w_{i}^{0}\triangleq\theta_{i}(0)$. We
observe that the equalities $x_{i}=0$ and $w_{i}=w^{0}_{i}$ can not be
excluded at equilibrium. This will occur for a facility type $i\in I$ whenever
the choice probabilities are zero for all $k\in K$, that is $\pi_{i}^{k}=0$,
which is the case if the random terms $\varepsilon_{i}^{k}$ have bounded
support and the facilities $j\neq i$ provide much higher utilities so that $i$
is never optimal.

Let us consider a continuous and strictly increasing extension of
$\theta_{i}(\cdot)$ to the negative reals with $\theta_{i}(x_{i})\to-\infty$
for $x_{i}\to-\infty$. Since any equilibrium satisfies $x_{i}\geq 0$, the
specific form of the extension of $\theta_{i}(\cdot)$ is irrelevant, and we
may simply set $\theta_{i}(x_{i})=\theta_{i}(0)+x_{i}$ for $x_{i}<0$. This
extended function has a well defined inverse
$\theta_{i}^{-1}:(-\infty,\infty)\to(-\infty,\bar{x}_{i})$ which is also
strictly increasing and continuous. Thus, we can eliminate $x_{i}$ from
(3)-(4) and express these equations in terms of the variables $w_{i}$ alone,
namely

$\theta_{i}^{-1}(w_{i})=\sum_{k\in K}I^{k}\pi^{k}_{i}\qquad(\forall i\in I).$
(5)

The equilibrium choice probabilities and waiting times can now be
characterized as follows:

######  Theorem 2.

The system of equations (1)-(4) has a unique solution. This equilibrium is the
unique minimizer of the strictly convex smooth function
$\Theta:\mathbb{R}^{d}\to\mathbb{R}$ given by

$\Theta(w)\triangleq\sum_{i\in I}H_{i}(w_{i})+\sum_{k\in
K}\mbox{$\frac{I^{k}}{\alpha^{k}}$}\,\Phi^{k}(\bar{u}^{k}_{0},\bar{u}^{k}_{1}-\alpha^{k}w_{1},,\ldots,\bar{u}^{k}_{n}-\alpha^{k}w_{n})$
(6)

where ${\displaystyle
H_{i}(w_{i})=\int_{w^{0}_{i}}^{w_{i}}\\!\\!\theta_{i}^{-1}(z)\,dz}$.

The proof is presented in Appendix A. We also present a refinement of Theorem
2 for the case in which there is no possibility to opt out:

######  Theorem 3.

Theorem 2 remains valid when there is no opt-out alternative, under the
following non-saturation condition:

$\sum_{i\in I}\bar{x}_{i}>\sum_{k\in K}I^{k}.\\\ $ (7)

The proof is presented in Appendix A.

Hence, we have established that the general model, which assumes that choice
probabilities follow random utility theory and that waiting times at
facilities strictly increase with the corresponding total inflows, can be
formulated as a strictly convex minimization problem. The model therefore has
a unique solution describing the equilibrium choice probabilities and waiting
times. The absence of alternative local minima reduces the interest to explore
alternative (non-cooperative) game theoretical equilibria.

Let us now consider several specific elaborations to which these general
results apply.

Example 1 – Multinomial Logit (MNL). A popular discrete choice model,
supported by Gnedenko’s theorem on extremal value distributions, considers
i.i.d. Gumbel variables $\\{\varepsilon^{k}_{i}:i\in I\\}$ centered at 0 and
with scale parameter $\beta_{k}$, in which case

$\displaystyle\Phi^{k}(u^{k})$ $\displaystyle=$
$\displaystyle\mbox{$\frac{1}{\beta_{k}}\ln\left(\sum_{j=0}^{n}\exp(\beta_{k}u^{k}_{j})\right)$},$
$\displaystyle\frac{\partial\Phi^{k}}{\partial u^{k}_{i}}$ $\displaystyle=$
$\displaystyle\frac{\exp(\beta_{k}u_{i}^{k})}{\sum_{j=0}^{n}\exp(\beta_{k}u^{k}_{j})}.$

Notice that the latter equation gives the choice probabilities as per (1).

Example 2 – Queuing models. We can use any queuing model for the waiting time
$w_{i}=\theta_{i}(x_{i})$ as a function of the arrival rate $x_{i}$. The
simplest one is an M/M/1 queue with
$w_{i}=\frac{x_{i}}{\mu_{i}(\mu_{i}-x_{i})}$. Similarly, for an M/M/$s$ queue
(as also analysed in [41]) we have
$w_{i}=\frac{1}{\mu_{i}}Q_{s}\\!\left(\frac{x_{i}}{\mu_{i}}\right)$ where

$Q_{s}(z)=\left(\sum_{k=0}^{s-1}\frac{z^{k}}{k!}+\frac{z^{s}}{s!}\frac{s}{(s-z)}\right)^{-1}\frac{z^{s}}{s!}\frac{s}{(s-z)^{2}}.$

Our general result can also accommodate queues with finite capacity such as
M/M/$s$/$K$ queues and queues with different arrival distributions, such as
M/G/$s$ queues (see also [23, 15] and the references therein). The only
restriction is that $\theta_{i}(\cdot)$ must be strictly increasing and
continuous.

##  3 Case Study in Urban China

###  3.1 Model Specification

The case considers hospital choice in urban China. In line with the health
system challenges and preceding research, the case study focuses on choice
probabilities and waiting times per facility level rather than per facility.
The indices $i\in\\{1,2,3\\}$ of facility types therefore will represent the
primary, secondary, and tertiary hospitals, and $i=0$ refers to opting out.

The patient choice related health system challenges center around the
perceived weakness of primary care facilities. The low utility attached to the
primary care level institutions by China’s urban population – and to a lesser
extent to the level of secondary hospitals [20] – causes 1) under-utilization
of primary care, 2) over-utilization of tertiary care, and 3) undesirable
avoidance of care, i.e., high opt-out rates [26, 35, 19, 18, 21]. This
problematic situation has resisted large scale healthcare reforms which mainly
aimed to change choice behavior by improving the skills level of the medical
staff and the quality of the medical equipment at primary care facilities.

According to the DCE study conducted by Liu et al., skills level of medical
staff and quality of medical equipment are indeed the two main determinants of
patient choice for the population of 24 million of Shanghai [20, 27]).

This case study uses the MNL model, data, and results obtained by the
aforementioned DCE study [20]. To capture the endogeneity between choice
probability and waiting time it additionally includes an M/M/1 queuing model.

The case study thus advances on the previous DCE study and policy analysis
therein which disregarded the endogeneity and included fixed waiting times as
choice determinants [20]. We thus aim to present a more accurate assessment of
the policy interventions implemented to remedy the health system challenges
and to present insight in the (lack of) effectiveness of these interventions.
Moreover the case explores the possible effectiveness (efficacy) of
alternative interventions.

###  3.2 MNL Model, Parameters and Data

The patient choice data were collected in a DCE study for which patients gave
their consent and for which ethics approval was obtained from the Shanghai
General Hospital Medical Ethical Review Committee (no 2017KY207) [20].

In the MNL model, the utility of a facility level is determined by the waiting
time and by the factors presented in the top half of Table 1 (cf. [20]).

| Primary | Secondary | Tertiary
---|---|---|---
Facility Attributes |  |  |
Facility Size | Small | Medium | Large
Out of Pocket Cost (RMB) | 59 | 88 | 105
Typical Skills Level | Junior | Senior | Expert
Status of Equipment | Outdated | Standard | Advanced
Travel Time (min.) | 15 | 15 | 15
Total Visit Time (hours) | 1 | 3 | 5
Parameters |  |  |
Service Rate (patients/hour) | 10 | 10 | 12
Visit Time Other Than Waiting (min.) | 34 | 88 | 87
Zero Wait Utility for Mild Patients | 0.207 | 0.417 | -0.259
Zero Wait Utility for Severe Patients | -0.257 | 0.089 | 0.773
Waiting Time Multipliers | 3 | 5 | 7
Number of Facilities | 1009 | 105 | 47
Number of Doctors per Facility | 6.76 | 40.8 | 98.9
Doctor Time Allocated to First Visits (%) | 50 | 50 | 50
Table 1: Attributes and parameters of facilities for all three levels

Focusing on the interplay between choice probability and waiting time per
intervention, we recast the utility functions to the form
$u^{k}_{i}=\bar{u}^{k}_{i}-\alpha^{k}w_{i}$, in which the only explicit factor
is the waiting time $w_{i}$ and the reference utility $\bar{u}^{k}_{i}$
incorporates the utilities from all other factors. Thus, $u^{k}_{i}$
represents the utility patients of type $k\in K$ attach to facilities of level
$i\in I$.

The original DCE study in [20] considered the factor sojourn time which is
composed of the waiting time plus the visit time. The latter is assumed not to
be affected by waiting time and therefore included in the reference utility
$\bar{u}^{k}_{i}$.

As the urban citizens of Shanghai typically live close to facilities of each
level and travel time is not among the most important factors determining
facility choice, we assume a generic travel time of 15 minutes and distinguish
patient types only by the evidence based factor perceived severity [22, 20].
More specifically, we consider two patient types: patients perceiving mild
disease and patients perceiving severe disease. The patient type dependent
utilities that differ significantly for different parameter levels are
presented in Table 2.

Factor Utilities | Mild | Severe
---|---|---
Opt Out Utility | 2.499 | -6.024
Waiting Time Sensitivities $\alpha^{k}$ | 0.232 | 0.0995
Junior Doctor Utility | -0.277 | -0.05
Senior Doctor Utility | 0.199 | -0.089
Expert Doctor Utility | 0.078 | 0.139
Obsolete Equipment Utility | -0.275 | -0.43
Advanced Equipment Utility | 0.275 | 0.43
Table 2: Data determining the utility per patient type

The sensitivity to waiting time varies depending on the disease severity,
i.e., across patient types $k\in K$, but is independent of facility level. Opt
out utilities also differ per patient type. The patient type dependent waiting
time sensitivities and opt out utilities are also included in 2.

Recalling (2) from Example 1 above, the MNL choice probabilities $\pi_{i}^{k}$
can now be expressed as:

$\pi^{k}_{i}=\frac{\exp(\beta_{k}(\bar{u}^{k}_{i}-\alpha^{k}w_{i}))}{\sum_{j=0}^{3}\exp(\beta_{k}(\bar{u}^{k}_{j}-\alpha^{k}w_{j}))}.$

###  3.3 M/M/1 Model, Parameters and Data

Focusing the case study on patients who seek care for a first visit and hence
without appointment with a pre-assigned physician, we assume the assignment of
patients to physicians happens as soon as the patient is registered, i.e.,
upon arrival at the facility. Moreover, we assume that this causes patients to
be evenly distributed among the doctors for each level. Thus, each registered
patient joins a single server queue. More specifically, we assume arrivals and
service rates are exponentially distributed and thus adopt an M/M/1 model and
consider alternative queuing models in the discussion. The data required for
the M/M/1 model are presented in the bottom half of Table 1, as further
outlined below.

As stated in Example 2 above, the M/M/1 model yields that expected waiting
time equals $w_{i}=\frac{x_{i}}{\mu_{i}(\mu_{i}-x_{i})}$ where $x_{i}$ is the
arrival rate of patients for a doctor at facility level $i\in I$ and $\mu_{i}$
their service rate.

Data on the service rates $\mu_{i}$ were provided by four medical doctors, two
from primary care facilities, one from a secondary hospital, and one from a
tertiary hospital in Shanghai in June 2019. The obtained data indicate that
maximum service rates are normally around 12 patients per hour for each of the
three facility levels. As tertiary hospitals are especially known for long
waits and short service times [5, 21], the base service rates are set at 12
patients per hour for tertiary care facilities and 10 patients per hour for
primary and secondary level facilities.

The same medical doctors also provided data on the average number of on duty
doctors per day and documentation on the total number of licensed medical
doctors per facility. The numbers of facilities per level were obtained from
[31].

The service capacity per level can then be determined by the number of
facilities, the number of physicians per facility, and the fraction of
physician time spent on first visit consults, which are all presented in Table
1. Moreover, we assume 261 working days of 8 hours per year.

The incidence of patients who perceive to have an illness and consider
visiting a healthcare facility is taken from the national census [16]. The
division of these patients over the two severity types is determined by
solving (1)-(4) for the base case model and data with the additional
requirement that the solution yields the evidence based reference waiting
times of the DCE on which it was based [20]. This yielded that 47.9 percent of
the 160,432,700 patients in Shanghai per year who consider to visit a
healthcare facility or to opt out instead, perceive their disease as minor.

The parameter values for the queuing model presented in the bottom half of
Table 1 additionally reflect that primary care consults typically require a
follow up visit to the hospital pharmacy and the cashier, secondary hospitals
typically perform diagnostic services such as imaging and lab tests, and
tertiary hospitals have the most elaborate and time consuming pathways [22,
19, 43]. Correspondingly, we assume that patients will join additional queues
for these follow-up services (e.g. at the pharmacy or for the lab test) as
indicated by the waiting time multipliers depicted in Table 1.

###  3.4 Comparative Intervention Analysis Methods

The case study evaluates the effects on choice probabilities and waiting times
of policy interventions which are modeled through modifications of the
(default) parameter values depicted in Tables 1, 2. Thus, each intervention is
modeled by specifying a set of values for the utilities of all choice
determinants with the exception of (the endogenous) waiting time and updating
the reference utilities $\bar{u}^{k}_{i}$ accordingly.

The comparative analysis presents both the results from solving (1)-(4) with
choice probabilities determined by an MNL model and waiting times determined
by an M/M/1 queuing model as explained above, and the results obtained by
using an MNL model only with the constant reference total visit times of 1,3,
and 5 hours for primary, secondary and tertiary care respectively from the DCE
study [20].

We compare the results for the base scenario and five policy interventions.
The first three of these interventions aim to strengthen primary care and
model the substantial efforts made as part of the national health reform to
upgrade the medical skills of the physicians and to upgrade the equipment of
primary care facilities. Skill and equipment are the factors evidenced to
impact choice probabilities most [19, 20, 28].

  * •

Upskill: Maximally upskill primary care physicians from the mostly junior
level to the mostly expert level of tertiary care physicians. To avoid
secondary care having less skilled physicians than primary care, the secondary
care physicians are upskilled to the mostly expert level as well. The required
upskilling effort can be considered to be unrealistically ambitious. Hence,
the relevance of the intervention results is not to model the effectiveness of
actual interventions but rather to estimate the maximum effect attainable by
these actively practiced upskilling interventions.

  * •

Upgrade: Maximally upgrade primary care equipment from the level obsolete to
the advanced level of tertiary care. To avoid secondary care having poorer
equipment than primary care, the secondary care equipment is upgraded to
advanced as well. Again, the required upgrading effort can be viewed as
unrealistic, e.g., for being too costly, and the relevance of the analysis is
to provide insight into the maximum effectiveness of the on going equipment
upgrading interventions.

  * •

Upskill and Upgrade: The third intervention combines the previous two and
therefore gives insight into the maximum effect attainable by combined
upskilling and upgrading.

As both evidence from practice and the comparative analysis results below
illustrate, the (maximum) effectiveness of upskilling and upgrading
interventions is limited. A first main underlying cause appears to be the high
utility patients perceiving a mild disease attach to opting out (self care).
Patients perceiving a mild disease attach a utility difference between opting
out and a zero wait primary care visit of 2.3, more than twice the utility
gain from maximum upskilling and upgrading (0.903) and equivalent to 10 hours
of waiting time reduction (0.232 per hour) (see Tables 2 and 1). As a result
more than three out of four such patients opt out in expectation, and this
choice probability is only weakly impacted by upskilling and upgrading.
Intervention four therefore models a health promotion campaign that reduces
the utility of self care (opting out) by half.

A second important underlying cause is that utility is quite insensitive to
waiting times for patients perceiving a severe disease. One hour of waiting
time reduction reduces the utility of a facility by 0.1 (approximately), while
the utility difference between primary and tertiary care is more than ten
times larger for these patients. For realistic waiting times which correspond
to being serviced on the day of visiting, the probability of choosing primary
care therefore remains low and the probability to choose tertiary care remains
high for these patients.

One may argue that the utility of waiting should only depend on the person and
not on the severity of the illness. Then, the current difference between the
waiting time utilities for perceiving mild and severe disease are an artifact
that may reflect other more lowly valued characteristics of primary and
secondary care not already captured by the utility factors presented in Table
1. Uniform waiting time sensitivities, which disregard perceived disease
severity, then more realistically reflect waiting time sensitivities and may
apply to the aspired future state of the health system in which it provides
balanced access to care.

Thus we have two more interventions:

  * •

Health Promotion Campaign: This campaign discourages self care for patients
perceiving a mild disease and halves the utility of opting out from 2.499 to
1.250.

  * •

Uniform Waiting Time Sensitivity: This intervention does not reflect a health
reform but rather adjusts the model assuming that the current long waiting
times may no longer be valid in the future and adjust the present low
sensitivities to waiting time for patients perceiving severe disease to become
equal to the waiting time sensitivity of patients perceiving mild disease,
0.232.

Appendix B additionally presents analysis results for an intervention in which
upskilling is only for primary care physicians and thus from mostly junior to
mostly senior, which is the skills level for secondary care. Moreover, the
same appendix presents results obtained when combining the upskilling and
upgrading interventions with interventions four and five.

As the data on service use and hospital capacity are not based on strong
evidence, we conducted a sensitivity analysis for each of the interventions
studied. For each intervention study, we generated 1,000 problem instances in
which, for each of the facility levels, the waiting time multipliers (demand)
and fraction of on duty doctors per day (supply) experience a uniformly
distributed random perturbation of between -10% and 10%.

This sensitivity analysis provides insights into the robustness of the
results. Moreover, it allows to test whether the differences between the
results obtained by solving (1)-(4) are significantly different from the
solutions obtained when solving the MNL model only. We present results on two
statistical tests for the significance of difference. First, we conducted a
sign test, testing the null hypothesis that there are no differences in choice
probabilities and waiting times between the two models. For each of these
parameters and for 1000 instances, this hypothesis is rejected if the number
of instances for which the difference is positive (negative) is at least 526
(at most 473) because this number has a probability of less than 0.05. These
results are indicated with ‘+’ and ‘-’, in case of rejection. A stronger
variant of the same test rejects the hypothesis if the number of instances for
which the difference is positive (negative) is at least 537 (at most 462),
which has a probability of less than 0.01 under the null hypothesis.

Even though the service use and capacity perturbations are sampled from
uniform distributions which are symmetric around the mean values of the base
scenario, the effects on waiting time can be highly nonlinear and asymmetric,
if only because M/M/1 waiting time is superlinear in the arrival rate per
doctor ($w_{i}=\frac{x_{i}}{\mu_{i}(\mu_{i}-x_{i})}$). Thus, a second test
rejects the null hypothesis of zero difference for choice probability or
waiting time, if the value is outside of a 95 % confidence interval around the
mean of the variable under consideration. This occurs when at most 25 of the
1000 problem instances report a difference below zero or above zero. Notice
that this second test is much more demanding. Both tests are robust as they
make no assumptions about the distributions of the (differences between)
outcome parameters.

The Python code used for the computational studies is available on Github [7].

###  3.5 Comparative Intervention Analysis Results

Figure 3 presents the results of the base case with the originally collected
data and the results of 1000 perturbed model instances. Columns two and three
present the choice probabilities and waiting times for the MNL model and for
model (1)-(4) respectively for the unperturbed base instance. These columns
present the choice probabilities $\pi_{i}^{k}=\mathbb{P}(i|k)$ denoted in
column 1, starting with the probability of opting out by patients perceiving
mild disease $\mathbb{P}(OO|M)$, to primary care for patients perceiving mild
disease $\mathbb{P}(1|M)$, et cetera, to tertiary care for patients perceiving
severe disease $\mathbb{P}(3|S)$. These probabilities confirm that the
calibrated base instance of model (1)-(4) accurately models the empirical data
on which the DCE study is based. This is further confirmed by the virtually
equal waiting times reported by both models for each of the three hospital
levels in the corresponding bottom rows of columns two and three.

Baseline | MNL | 1-4 |  $\mu$ MNL |  $\mu$ 1-4 | Sign Test | Nonzero
---|---|---|---|---|---|---
$\mathbb{P}(OO|M)$ | 0.7757 | 0.7759 | 0.7757 | 0.7740 | \- - |
$\mathbb{P}(1|M)$ | 0.0784 | 0.0784 | 0.0784 | 0.0783 | \- - |
$\mathbb{P}(2|M)$ | 0.0967 | 0.0966 | 0.0967 | 0.0972 | \+ + |
$\mathbb{P}(3|M)$ | 0.0492 | 0.0491 | 0.0492 | 0.0505 | \+ + |
$\mathbb{P}(OO|S)$ | 0.0006 | 0.0006 | 0.0006 | 0.0006 | \- - |
$\mathbb{P}(1|S)$ | 0.1917 | 0.1918 | 0.1917 | 0.1906 | \- - |
$\mathbb{P}(2|S)$ | 0.2709 | 0.2709 | 0.2709 | 0.2701 | \- - |
$\mathbb{P}(3|S)$ | 0.5368 | 0.5367 | 0.5368 | 0.5387 | \+ + |
$W(1)$ | 0.43 | 0.43 | 0.43 | 0.43 | \- - |
$W(2)$ | 1.53 | 1.52 | 1.50 | 1.48 | \- - |
$W(3)$ | 3.54 | 3.54 | 3.58 | 3.43 | \+ + |
Table 3: Baseline choice probabilities and waiting times

Columns four and five of Table 3 present the average values over 1000 randomly
perturbed problem instances for the MNL model and model (1)-(4) respectively.
The choice probabilities in column four are identical to the choice
probabilities in the second column as the MNL model does not consider effects
of any changes in waiting time on the choice probabilities. The bottom rows
show that the average waiting times differ slightly from the unperturbed base
instance. However, one might argue that the column two and four waiting times
should be disregarded as the MNL model assumes there is no change in the
utility from waiting times and hence no change in waiting times. Thus, below
we focus on the choice probabilities in the model comparison. Column five
shows that the choice probabilities and waiting times differ slightly from
those of column three when using model (1)-(4).

Column six of Table 3 shows the result of the sign test, where ‘++’ and‘$--$’
mean that the null hypothesis of zero difference holds with probability less
than 0.01 and single ‘+’, ‘-’ indicate the same yet with probability at most
0.05. Thus ‘(+)+’s indicate that (1)-(4) results are significantly higher than
MNL results and ‘(-)-’s indicate that these are significantly smaller. Empty
cells in column six indicate that there are no significant differences. Even
for this base case for which the model (1)-(4) is calibrated to reproduce the
MNL outcomes and evidence-based DCE assumptions, the perturbations lead to
differences in outcomes between the two models that cause the sign test to
reject that hypothesis of equal outcomes ($p\leq 0.01$) for all choice
probabilities and waiting times.

An ‘X’ in column seven of Table 3 indicates that the second test described in
the methods section is rejected and any empty cell indicates it is not
rejected, which is the case for all outcome variables in this base scenario.

Next, we turn to considering the model results for the upskilling and
upgrading interventions. These results are presented in Table 4.

Upskill | MNL | 1-4 |  $\mu$ MNL |  $\mu$ 1-4 | Sign Test | Nonzero
---|---|---|---|---|---|---
$\mathbb{P}(OO|M)$ | 0.7587 | 0.7545 | 0.7587 | 0.7526 | \- - |
$\mathbb{P}(1|M)$ | 0.1094 | 0.1080 | 0.1094 | 0.1079 | \- - |
$\mathbb{P}(2|M)$ | 0.0838 | 0.0814 | 0.0838 | 0.0822 | \- - |
$\mathbb{P}(3|M)$ | 0.0481 | 0.0560 | 0.0481 | 0.0573 | \+ + | $\times$
$\mathbb{P}(OO|S)$ | 0.0005 | 0.0005 | 0.0005 | 0.0005 | \- - |
$\mathbb{P}(1|S)$ | 0.2087 | 0.2020 | 0.2087 | 0.2009 | \- - | $\times$
$\mathbb{P}(2|S)$ | 0.3068 | 0.2949 | 0.3068 | 0.2944 | \- - | $\times$
$\mathbb{P}(3|S)$ | 0.4840 | 0.5026 | 0.4840 | 0.5042 | \+ + | $\times$
$W(1)$ | 0.47 | 0.46 | 0.47 | 0.46 | \- - | $\times$
$W(2)$ | 1.77 | 1.62 | 1.72 | 1.57 | \- - | $\times$
$W(3)$ | 2.41 | 2.85 | 2.36 | 2.76 | \+ + | $\times$
Upgrade | MNL | 1-4 |  $\mu$ MNL |  $\mu$ 1-4 | Sign Test | Nonzero
$\mathbb{P}(OO|M)$ | 0.7129 | 0.7046 | 0.7129 | 0.7029 | \- - | $\times$
$\mathbb{P}(1|M)$ | 0.1249 | 0.1194 | 0.1249 | 0.1193 | \- - | $\times$
$\mathbb{P}(2|M)$ | 0.1170 | 0.1110 | 0.1170 | 0.1110 | \- - |
$\mathbb{P}(3|M)$ | 0.0452 | 0.0650 | 0.0452 | 0.0657 | \+ + | $\times$
$\mathbb{P}(OO|S)$ | 0.0004 | 0.0004 | 0.0004 | 0.0004 | \- - | $\times$
$\mathbb{P}(1|S)$ | 0.3220 | 0.3003 | 0.3220 | 0.2994 | \- - | $\times$
$\mathbb{P}(2|S)$ | 0.2960 | 0.2752 | 0.2960 | 0.2755 | \- - | $\times$
$\mathbb{P}(3|S)$ | 0.3816 | 0.4241 | 0.3816 | 0.4247 | \+ + | $\times$
$W(1)$ | 0.62 | 0.58 | 0.61 | 0.57 | \- - | $\times$
$W(2)$ | 2.03 | 1.69 | 1.98 | 1.65 | - | $\times$
$W(3)$ | 1.49 | 1.91 | 1.46 | 1.86 | \+ + | $\times$
Upskill & |  |  |  |  |  |
Upgrade | MNL | 1-4 |  $\mu$ MNL |  $\mu$ 1-4 | Sign Test | Nonzero
$\mathbb{P}(OO|M)$ | 0.6856 | 0.6791 | 0.6856 | 0.6776 | \- - | $\times$
$\mathbb{P}(1|M)$ | 0.1713 | 0.1611 | 0.1713 | 0.1611 | \- - | $\times$
$\mathbb{P}(2|M)$ | 0.0997 | 0.0927 | 0.0997 | 0.0938 | \- - | $\times$
$\mathbb{P}(3|M)$ | 0.0435 | 0.067 | 0.0435 1 | 0.0676 | \+ + | $\times$
$\mathbb{P}(OO|S)$ | 0.0003 | 0.0004 | 0.0004 | 0.0004 | \- - | $\times$
$\mathbb{P}(1|S)$ | 0.3403 | 0.3158 | 0.3403 | 0.3151 | \- - | $\times$
$\mathbb{P}(2|S)$ | 0.3254 | 0.3005 | 0.3254 | 0.3011 | \- - | $\times$
$\mathbb{P}(3|S)$ | 0.3339 | 0.3833 | 0.3339 | 0.3834 | \+ + | $\times$
$W(1)$ | 0.72 | 0.66 | 0.71 | 0.65 | \- - | $\times$
$W(2)$ | 2.26 | 1.79 | 2.20 | 1.73 | - | $\times$
$W(3)$ | 1.26 | 1.61 | 1.24 | 1.50 | \+ + | $\times$
Table 4: Choice probabilities and waiting times after upskilling and upgrading
interventions

The most notable policy observation from these three modeled intervention
studies might be that even full upskilling and upgrading only reduces the opt
out probability of mild patients from 0.78 to 0.67 (or to 0.68 according to
the MNL model which ignores waiting time effects). However, the relative
changes in choice probabilities among the hospital levels tend to be larger.
For instance, the original probability of patients perceiving severe disease
to choose tertiary care was 0.54 and decreases to 0.50 (upskill), to 0.42
(upgrade), and to 0.38 (upskill and upgrade) for the successive interventions
studied. The MNL model finds this probability to be as low as 0.33 and both
significance tests reveal that this reduction appears significantly
overestimated.

More generally we can observe that differences between the policy effects
obtained by the models are always significant according to the sign test, and
significant for the far majority on the second test as well. As a rule, the
MNL model significantly overestimates the policy effects. There are some
irregularities, for instance, because the utility severe patients attach to
physicians is not increasing with the skills level. They value senior doctors
the least. The results also indicate that the combination of full upskilling
and upgrading goes a long way to balance the flow of patients perceiving
severe disease but is hardly effective to prevent patients perceiving mild
disease from opting out.

The Health Promotion intervention models the reduction of the opt out utility
for patients perceiving mild disease. The modeled intervention results are
presented in Table 5. Columns three and five show the effectiveness of this
intervention to reduce opting out among patients perceiving mild disease from
0.78 to around 0.54. The increased opting in leads to considerable increases
in waiting times, especially for secondary and tertiary for which patients
arrival rates were already close to facility level capacity.

For this health promotion intervention, the limitations of the MNL model again
cause an overestimation of the effectiveness. Moreover, Table 5 reveals
another main shortcoming of the MNL model. It may yield choice probabilities
that cause patient flows above capacity and thus unstable queues with infinite
waiting times. Column four shows that only 739 out of the 1000 perturbed
instances had feasible solutions. In fact, the feasibility limit applied is
slightly more restrictive than requiring finite waiting times. All instances
with a waiting time of more than ten hours are considered as infeasible. Such
because regular hospital opening hours are limited to 10 hours and longer
waiting might imply waiting until the next day. The estimated sensitivities of
the DCE are unlikely to be valid for such situations, despite the fact that
such exceptionally long waiting has occurred in Chinese urban health systems
in the recent past. Appendix B shows that when combining the interventions of
upgrading equipment and health promotion, even the undisturbed results (column
two) are infeasible.

Finally, the same table shows the effects of changing the waiting time
sensitivities of patients perceiving severe disease to equalize the
sensitivities of mild patients. These effects are comparable to the effect of
complete upskilling. Hence, they are relatively modest, with the exception of
a substantial reduction in the choice probabilities of tertiary care by
patients perceiving severe disease and the corresponding tertiary care waiting
times. Because the MNL model assumes waiting times to be fixed, it fails to
register any effects of modified waiting time sensitivities on choice
probabilities.

For both these additional interventions, the sign test qualifies all
differences between the outcomes of the two models as highly significant, as
is mostly confirmed by the stricter second test.

Health |  |  |  |  |  |
---|---|---|---|---|---|---
Promotion | MNL | 1-4 |  $\mu$ MNL |  $\mu$ 1-4 | Sign Test | Nonzero
$\mathbb{P}(OO|M)$ | 0.4980 | 0.5405 | 0.4980 | 0.5348 | \+ + | $\times$
$\mathbb{P}(1|M)$ | 0.1755 | 0.1859 | 0.1755 | 0.1844 | \+ + | $\times$
$\mathbb{P}(2|M)$ | 0.2164 | 0.1818 | 0.2164 | 0.1851 | \- - | $\times$
$\mathbb{P}(3|M)$ | 0.1101 | 0.0917 | 0.1101 | 0.0957 | \- - |
$\mathbb{P}(OO|S)$ | 0.0006 | 0.0006 | 0.0006 | 0.0006 | \+ + | $\times$
$\mathbb{P}(1|S)$ | 0.1917 | 0.2079 | 0.1917 | 0.2053 | \+ + | $\times$
$\mathbb{P}(2|S)$ | 0.2709 | 0.2661 | 0.2709 | 0.2656 | \- - |
$\mathbb{P}(3|S)$ | 0.5368 | 0.5254 | 0.5368 | 0.5285 | \- - |
$W(1)$ | 0.51 | 0.54 | 0.51(739) | 0.53 | \+ + | $\times$
$W(2)$ | 4.08 | 2.62 | 4.19(739) | 2.50 | \- - | $\times$
$W(3)$ | 6.92 | 4.66 | 5.24(739) | 4.46 | \- - |
Uniform |  |  |  |  |  |
Waiting Time |  |  |  |  |  |
Sensitivities | MNL | 1-4 |  $\mu$ MNL |  $\mu$ 1-4 | Sign Test | Nonzero
$\mathbb{P}(OO|M)$ | 0.7757 | 0.7697 | 0.7757 | 0.7678 | \- - | $\times$
$\mathbb{P}(1|M)$ | 0.0784 | 0.0773 | 0.0784 | 0.0773 | \- - | $\times$
$\mathbb{P}(2|M)$ | 0.0967 | 0.0951 | 0.0967 | 0.0961 | \- - |
$\mathbb{P}(3|M)$ | 0.0492 | 0.0577 | 0.0492 | 0.0588 | \+ + | $\times$
$\mathbb{P}(OO|S)$ | 0.0006 | 0.0007 | 0.0006 | 0.0007 | \+ + | $\times$
$\mathbb{P}(1|S)$ | 0.1917 | 0.2252 | 0.1917 | 0.2227 | \+ + | $\times$
$\mathbb{P}(2|S)$ | 0.2709 | 0.2753 | 0.2709 | 0.2750 | \+ + |
$\mathbb{P}(3|S)$ | 0.5368 | 0.4986 | 0.5368 | 0.5015 | \- - | $\times$
$W(1)$ | 0.43 | 0.46 | 0.43 | 0.45 | \+ + | $\times$
$W(2)$ | 1.52 | 1.55 | 1.48 | 1.50 | \+ + |
$W(3)$ | 3.54 | 2.80 | 3.55 | 2.72 | \- - |
Table 5: Choice probabilities and waiting times after intervening on opt out
utility and waiting time sensitivity

##  4 Discussion and Conclusion

We discuss the results in the reverse order of presenting them, first
interpreting case study results from urban China and then zooming out to the
presented models, as well as present and future theoretical contributions on
the relationship between patient choice and waiting time.

The inclusion of the mutual dependence between choice probabilities in the
modeling of health policy intervention effects strongly suggests that the
ongoing efforts to upskill doctors and upgrade equipment can contribute to
alleviating the persistent problems of underutilization of primary care and
overutilization of tertiary care. At the same time, the analysis reveals that
these interventions likely fall short of resolving these problems. In
comparison to the previous MNL based policy analyses, the increase in utility
of primary and secondary care facilities – and thus the probability to choose
facilities at these levels instead of opting out – that results from these
interventions are significantly smaller when accounting for endogeneity
between waiting time and choice probability by solving (1)-(4). An intuitive
interpretation is that the utility increases caused by the interventions yield
higher choice probabilities, which in turn imply longer waiting times that
partially undo the initial utility increases.

While the opt in probabilities remain low for patients perceiving mild disease
for both models, the differences in effects of the intervention to upgrade
equipment and upskill medical doctors are highly significant and larger in
relative terms. For example, the MNL model estimates their probability to
attend tertiary care is 0.0435 while solving (1)-(4) yields a 0.0241 higher
probability. Thus, the relative difference is more than 50 percent.

The patients perceiving severe disease almost all opt in and the equally
significant differences between model outcomes of upgrading and upskilling now
also translate into larger absolute differences in the choice probabilities.
For example, the MNL model estimates their probability to attend tertiary care
after upgrading and upskilling is 0.3339 while solving (1)-(4) gives a
probability of 0.3833. This difference translates to around 4 million first
visits to tertiary care per year.

Based on the above observations, we analysed two alternative interventions.
The first suggestion is a health promotion campaign that diminishes the
utility attached to self care and opting out. Our analysis reveals that such a
campaign has much more efficacy potential than upskilling and upgrading.
Moreover, our findings suggest that MNL based analysis significantly errs on
the resulting choice probability changes and that its disregard of endogeneity
can even yield choice probabilities that are infeasible because they violate
hospital capacity limits.

The MNL model is incapable of evaluating interventions which cause changes in
waiting time sensitivities as it assumes waiting times to be constant. Hence,
the newly developed model is a welcome advancement to provide insight into the
effect of remedying the waiting time insensitivity reported for patients
perceiving severe disease [20]. The analysis shows that the effects of
interventions to adjust waiting time sensitivities of patients perceiving
severe disease to those of patients perceiving mild disease, resulting in
uniform patient dependent sensitivities, are comparable to those obtained by
full upskilling of physicians. This intervention follows the universal health
coverage principles of the WHO which reaffirm the importance of access to
strong primary care facilities for all and referral to higher levels as needed
[12]. Further research into the determinants of waiting time (in)sensitivity
and interventions to resolve the insensitivity is called for.

The case study analysis and in particular the analysis comparing the results
obtained by the newly developed model with the results obtained by the MNL
model only shows that the newly presented model can significantly more
accurately capture the effects of interventions aimed at altering patient
choice and corresponding health system utilization and performance. This
confirms the importance of capturing the endogeneous relationships in the
model for intervention analysis [36, 1].

Regarding the data collection for the case study at hand, one may wonder
whether revealed preference data would have yielded more accurate results. The
choice set of the DCE limited waiting times to 1, 3, and 5 hours when
eliciting stated preferences. Revealed preferences can potentially include a
wider variety of combinations of choices and waiting times, and more realistic
ones based on data per hospital rather than per hospital level and for
different moments in time. Notice that this also requires obtaining
corresponding revealed opt out choices, which is not trivial.

Another difficulty with revealed preference data is that the attributes
equipment, skills level, out of pocket costs, and hospital size, will be
highly correlated as these are regulated to vary among hospital levels but not
between hospitals of the same level. This renders the data to be unsuited for
the analysis of the policy interventions targeting the individual attributes
considered in our study. Combined with earlier evidence that stated
preferences can predict revealed choices with a high degree of confidence for
healthcare choices, we believe that stated preferences elicited through
appropriately designed (MNL) models and corresponding choice sets may be most
valuable to collect data for policy intervention studies in waiting time and
choice interactions [8].

Regarding the choice of queuing model for the case study at hand, one may
argue that the M/M/1 queuing model which assumes patients distribute uniformly
over the physicians per level has limited validity and moreover limits
generalization of the results to other settings. Indeed, service rates or
arrival rates may not be exponentially distributed and the service regime may
be different in other settings. Example 2 of the modeling section already
discusses the alternatives of M/M/$s$, /M/M/$s$/$K$ and M/G/$s$ queuing
models. These examples illustrate that the proposed mathematical model is much
more general than the specific model of the case study. It accommodates all
queuing regimes in which waiting times are increasing in the choice
probabilities, in combination with a variety of random utility based choice
models. Theorem 3 extends the model to contexts in which opting out is not
permitted.

The general model presented still offers a variety of research directions
worthwhile pursuing to strengthen and extend it. A main area of improvement
lies in the assumption that hospital utility is a linear function of expected
waiting time. We are not aware of evidence supporting this assumption, which
is for instance implied by the MNL model. There are several indications that
this relationship may not be linear. For travel time, for instance, there is
evidence that any travel time less than 30 minutes is acceptable [37]. More
generally, it may appear incorrect that a waiting time increase from 15
minutes to 45 minutes has the same effect on utility as a waiting time
increase from 5 hours and 15 minutes to 5 hours and 45 minutes. Thus
generalizations of the proofs of Theorems 2 and 3 to such alternative models,
possibly with multiple equilibrium solutions, are called for.

On the same thread, the utility of waiting time likely not only depends on
expected waiting time but also on other moments such as the variance, or on
the probability of long waiting. This is especially important as waiting times
may vary considerably in practice for reasons of variability in demand and
capacity as modelled in the perturbed instances of the computational study.
Thus, we call for studies which include uncertainty of waiting time and
nonlinear relations between expected waiting time and utility, for example
using prospect theory based value functions which have been recently explored
in a variety of studies on patient preferences [30, 34].

Among the interesting extensions of the general model, choices and queuing
beyond the first visit are of interest, as well as models in which service
tasks and times vary depending on patient type (severity).

Another valuable direction for extension is formed by the modeling of the
learning dynamics of patients. We have assumed that equilibrium choice and
waiting time will materialize but policy interventions intend to disturb the
equilibrium, for instance, to reduce waiting times. To analyse the
effectiveness of interventions, it is therefore also of interest to know how
patients learn about waiting times and how newly learned waiting times
translate into updated utilities and choices. This matter is especially
challenging as the expected waiting time is not easy to observe. The queue
length at a given moment in time may be observable, but this does not easily
translate into an accurate assessment of expected queue length or subsequent
expected waiting time. Moreover, patients may typically have few waiting time
observations from personal visits. Added to the fact that learning processes
may be slow and not fully rational, the envisioned equilibrium effects of
policy interventions may not be realized until much later, if at all [10].
Further research into the underlying strategic learning dynamics of hospital
choice and how they can be promoted is thus an important area of future
research [40, 13]. This research can borrow from related research in
transportation science (see for instance [6] and references therein).

## References

  * [1] John Antonakis, Samuel Bendahan, Philippe Jacquart, and Rafael Lalive.  On making causal claims: A review and recommendations.  The leadership quarterly, 21(6):1086–1120, 2010.
  * [2] Catherine Arsenault, Min Kyung Kim, Amit Aryal, Adama Faye, Jean Paul Joseph, Munir Kassa, Tizta Tilahun Degfie, Talhiya Yahya, and Margaret E Kruk.  Hospital-provision of essential primary care in 56 countries: determinants and quality.  Bulletin of the World Health Organization, 98(11):735, 2020.
  * [3] Adugnaw Berhane and Fikre Enquselassie.  Patients’ preferences for attributes related to health care services at hospitals in amhara region, northern ethiopia: a discrete choice experiment.  Patient preference and adherence, 9:1293, 2015.
  * [4] Paul Brown, Laura Panattoni, Linda Cameron, Stephanie Knox, Toni Ashton, Tim Tenbensel, and John Windsor.  Hospital sector choice and support for public hospital care in new zealand: Results from a labeled discrete choice survey.  Journal of health economics, 43:118–127, 2015.
  * [5] Jianqian Chao, Boyang Lu, Hua Zhang, Liguo Zhu, Hui Jin, and Pei Liu.  Healthcare system responsiveness in jiangsu province, china.  BMC health services research, 17:1–7, 2017.
  * [6] Roberto Cominetti, Emerson Melo, and Sylvain Sorin.  A payoff-based learning procedure and its application to traffic games.  Games and Economic Behavior, 70(1):71–83, 2010.
  * [7] Roberto Cominetti and Joris Van de Klundert.  Python code for equilibrium computation.  https://github.com/JorisAtWork/HospitalChoiceandWaitingTimeEquilbria/blob/main/Copy_of_HospitalChoice.ipynb.
  * [8] Esther W de Bekker-Grob, Bas Donkers, Michiel CJ Bliemer, Jorien Veldwijk, and Joffre D Swait.  Can healthcare choice be predicted using stated preference data?  Social science & medicine, 246:112736, 2020.
  * [9] Thomas A Domencich and Daniel McFadden.  Urban travel demand-a behavioral analysis.  North-Holland Publishing Company Limited, Oxford, England, 1975.
  * [10] Ido Erev and Alvin E Roth.  Predicting how people play games: Reinforcement learning in experimental games with unique, mixed strategy equilibria.  American economic review, pages 848–881, 1998.
  * [11] Sophia Fischer, Stefanie Pelka, and René Riedl.  Understanding patients’ decision-making strategies in hospital choice: Literature review and a call for experimental research.  Cogent Psychology, 2(1):1116758, 2015.
  * [12] Tedros Adhanom Ghebreyesus, Henrietta Fore, Yelzhan Birtanov, and Zsuzsanna Jakab.  Primary health care for the 21st century, universal health coverage, and the sustainable development goals.  The Lancet, 392(10156):1371–1372, 2018.
  * [13] Sergiu Hart.  Adaptive heuristics.  Econometrica, 73(5):1401–1430, 2005.
  * [14] Simon Jones, Chris Moulton, Simon Swift, Paul Molyneux, Steve Black, Neil Mason, Richard Oakley, and Clifford Mann.  Association between delays to patient admission from the emergency department and all-cause 30-day mortality.  Emergency Medicine Journal, 39(3):168–173, 2022.
  * [15] Beste Kucukyazici, Yue Zhang, Amir Ardestani-Jaafari, and Lijie Song.  Incorporating patient preferences in the design and operation of cancer screening facility networks.  European Journal of Operational Research, 287(2):616–632, 2020\.
  * [16] Sheng Laiyun et al.  An Analysis Report of National Health Services Survey in China.  Center for Health Statistics and Information, 2013.
  * [17] Hau Leung Lee and Morris A Cohen.  Equilibrium analysis of disaggregate facility choice systems subject to congestion-elastic demand.  Operations research, 33(2):293–311, 1985.
  * [18] Xi Li, Harlan M Krumholz, Winnie Yip, Kar Keung Cheng, Jan De Maeseneer, Qingyue Meng, Elias Mossialos, Chuang Li, Jiapeng Lu, Meng Su, et al.  Quality of primary health care in china: challenges and recommendations.  The Lancet, 395(10239):1802–1812, 2020.
  * [19] Xi Li, Jiapeng Lu, Shuang Hu, KK Cheng, Jan De Maeseneer, Qingyue Meng, Elias Mossialos, Dong Roman Xu, Winnie Yip, Hongzhao Zhang, et al.  The primary health-care system in china.  The Lancet, 390(10112):2584–2594, 2017.
  * [20] Yun Liu, Qingxia Kong, Shan Wang, Liwei Zhong, and Joris van de Klundert.  The impact of hospital attributes on patient choice for first visit: evidence from a discrete choice experiment in shanghai, china.  Health policy and planning, 35(3):267–278, 2020.
  * [21] Yun Liu, Qingxia Kong, Shasha Yuan, and Joris van de Klundert.  Factors influencing choice of health system access level in china: a systematic review.  PLoS One, 13(8):e0201887, 2018.
  * [22] Yun Liu, Liwei Zhong, Shasha Yuan, and Joris van de Klundert.  Why patients prefer high-level healthcare facilities: a qualitative study using focus groups in rural and urban china.  BMJ global health, 3(5):e000854, 2018.
  * [23] Vladimir Marianov, Miguel Ríos, and Manuel José Icaza.  Facility location for market capture when users rank facilities by shorter travel and waiting times.  European Journal of Operational Research, 191(1):32–44, 2008.
  * [24] Diego A Martinez, Haoxiang Zhang, Magdalena Bastias, Felipe Feijoo, Jeremiah Hinson, Rodrigo Martinez, Jocelyn Dunstan, Scott Levin, and Diana Prieto.  Prolonged wait time is associated with increased mortality for chilean waiting list patients with non-prioritized conditions.  BMC public health, 19(1):1–11, 2019.
  * [25] Daniel McIntyre and Clara K Chow.  Waiting time as an indicator for health services under strain: a narrative review.  INQUIRY: The Journal of Health Care Organization, Provision, and Financing, 57:0046958020910305, 2020.
  * [26] Qingyue Meng, Anne Mills, Longde Wang, and Qide Han.  What can we learn from china’s health system reform?  bmj, 365, 2019.
  * [27] Natural Bureau of Statistics of China.  China Statistical Yearbook 2019.  China Statistics Press, 2019.
  * [28] National Health Commission of the People’s Republic of China.  Notice of the national health commission on comprehensively promoting the construction of community health centers, 2020.
  * [29] R Tyrrell Rockafellar.  Convex analysis, volume 11.  Princeton university press, 1997.
  * [30] Thomas Rouyard, Arthur Attema, Richard Baskerville, José Leal, and Alastair Gray.  Risk attitudes of people with manageable chronic disease: an analysis under prospect theory.  Social Science & Medicine, 214:144–153, 2018.
  * [31] MSB Shanghai.  Shanghai statistical yearbook.  China Statistics Press, 2018.
  * [32] Yosef Sheffi and Carlos F Daganzo.  Another paradox of traffic flow.  Transportation Research, 12(1):43–46, 1978.
  * [33] Honora Smith, Christine Currie, Pornpimol Chaiwuttisak, and Andreas Kyprianou.  Patient choice modelling: how do patients choose their hospitals?  Health care management science, 21(2):259–268, 2018.
  * [34] Aline C Stolk-Vos, Arthur E Attema, Michele Manzulli, and Joris J van de Klundert.  Do patients and other stakeholders value health service quality equally? a prospect theory based choice experiment in cataract care.  Social Science & Medicine, 294:114730, 2022.
  * [35] Yuqi Ta, Yishan Zhu, and Hongqiao Fu.  Trends in access to health services, financial protection and satisfaction between 2010 and 2016: Has china achieved the goals of its health system reform?  Social Science & Medicine, 245:112715, 2020.
  * [36] Kenneth E Train.  Discrete choice methods with simulation.  Cambridge university press, 2009.
  * [37] Marco Varkevisser, Stéphanie A van der Geest, and Frederik T Schut.  Assessing hospital competition when prices don’t matter to patients: the use of time-elasticities.  International Journal of Health Care Finance and Economics, 10:43–60, 2010.
  * [38] Aafke Victoor, Diana MJ Delnoij, Roland D Friele, and Jany JDJM Rademakers.  Determinants of patient choice of healthcare providers: a scoping review.  BMC health services research, 12(1):1–16, 2012.
  * [39] Huw CWL Williams.  On the formation of travel demand models and economic evaluation measures of user benefit.  Environment and planning A, 9(3):285–344, 1977.
  * [40] H Peyton Young.  Strategic learning and its limits.  OUP Oxford, 2004.
  * [41] Yue Zhang and Derek Atkins.  Medical facility network design: User-choice and system-optimal models.  European Journal of Operational Research, 273(1):305–319, 2019\.
  * [42] Jingrong Zhu, Jinlin Li, Zengbo Zhang, Hao Li, and Lingfei Cai.  Exploring determinants of health provider choice and heterogeneity in preference among outpatients in beijing: a labelled discrete choice experiment.  BMJ open, 9(4):e023363, 2019.
  * [43] DD ZOU, ZF ZHANG, et al.  Analysis on outpatient stay in tertiary public hospitals in shanghai and relative patient satisfaction degree.  Chinese Health Resources, 20(6):464–468, 2017.

Appendix A: Proofs

###### Proof of Theorem 1.

Let $U_{i}=u_{i}+\varepsilon_{i}$. For each realization of the random
variables $\varepsilon_{i}$, the expression

$\varphi(u,\varepsilon)=\max\\{u_{1}+\varepsilon_{1},\ldots,u_{n}+\varepsilon_{n}\\}$

is a maximum of linear functions of $u$, hence it is a convex polyhedral
function. Therefore its expectation, being an average of these, is also convex

$\Phi(u)=\int_{\mathbb{R}^{n}}\varphi(u,\varepsilon)\,d\mathbb{P}(\varepsilon).$

Now, let $e_{i}$ be the $i$-th canonical vector in $\mathbb{R}^{n}$. Then,
when $t\to 0$ with $t>0$ one can readily check that the differential quotients
of $\varphi(\cdot,\varepsilon)$ converge monotonically towards an indicator
function, namely

$\frac{\varphi(u+te_{i},\varepsilon)-\varphi(u,\varepsilon)}{t}\to\left\\{\begin{array}[]{cl}1&\mbox{if
}u_{i}+\varepsilon_{i}\geq u_{j}+\varepsilon_{j},\forall j\\\
0&\mbox{otherwise},\end{array}\right.$

and therefore Lebesgue’s monotone convergence theorem yields

$\lim_{t\to
0^{+}}\frac{\Phi(u+te_{i})-\Phi(u)}{t}=\int_{\mathbb{R}^{n}}\mathbbm{1}_{\\{u_{i}+\varepsilon_{i}\geq
u_{j}+\varepsilon_{j},\forall
j\\}}\,d\mathbb{P}(\varepsilon)=\mathbb{P}(U_{i}\geq U_{j},\forall j).$

Similarly, for $t\to 0$ with $t<0$ we have

$\lim_{t\to
0^{-}}\frac{\Phi(u+te_{i})-\Phi(u)}{t}=\int_{\mathbb{R}^{n}}\mathbbm{1}_{\\{u_{i}+\varepsilon_{i}>u_{j}+\varepsilon_{j},\forall
j\\}}\,d\mathbb{P}(\varepsilon)=\mathbb{P}(U_{i}>U_{j},\forall j).$

Since the random variables $\varepsilon_{i}$ are assumed continuous these two
limits coincide and then

$\frac{\partial\Phi}{\partial u_{i}}(u)=\mathbb{P}(U_{i}\geq U_{j},\forall
j)=\mathbb{P}(U_{i}>U_{j},\forall j).$

As every convex function which is Gateaux differentiable is automatically of
class $C^{1}$ [29], this completes the proof. ∎

###### Proof of Theorem 2.

Since the expected utilities $\Phi^{k}(\cdot)$ are smooth and convex, while
the functions $\theta_{i}^{-1}(\cdot)$ are strictly increasing, it follows
that $\Theta(\cdot)$ is strictly convex and smooth, with

$\frac{\partial\Theta}{\partial w_{i}}(w)=\theta_{i}^{-1}(w_{i})-\sum_{k\in
K}I^{k}\,\frac{\partial\Phi^{k}}{\partial
u^{k}_{i}}(\bar{u}^{k}_{0},\bar{u}^{k}_{1}-\alpha^{k}w_{1},\ldots,\bar{u}^{k}_{n}-\alpha^{k}w_{n}).$
(8)

Using (1)-(2) it follows that (5) is equivalent to $\nabla\Theta(w)=0$, and
therefore the equilibria coincide with the minimizers of $\Theta(\cdot)$.

Since $\Theta(\cdot)$ is strictly convex it has at most one minimizer, and
therefore it remains to establish that the minimum is attained. This follows
by showing that the recession function satisfies $\Theta^{\infty}(d)>0$ for
all $d\in\mathbb{R}^{n}\setminus\\{0\\}$. Indeed, let us take $d\neq 0$. From
Lebesgue’s monotone convergence theorem, we have

$(\Phi^{k})^{\infty}(d)=\max\\{0,-\alpha^{k}d_{1},\ldots,-\alpha^{k}d_{n}\\}=-\alpha^{k}\min\\{0,d_{1},\ldots,d_{n}\\},$

and then using standard rules for computing recession functions we get

$\Theta^{\infty}(d)=\sum_{i\in I}H_{i}^{\infty}(d_{i})-\sum_{k\in
K}I^{k}\min\\{0,d_{1},\ldots,d_{k}\\}$

with

$H_{i}^{\infty}(d_{i})=\lim_{t\to\infty}\theta_{i}^{-1}(td_{i})d_{i}=\left\\{\begin{array}[]{cl}\bar{x}_{i}d_{i}&\mbox{if
}d_{i}>0\\\ 0&\mbox{if }d_{i}=0\\\ +\infty&\mbox{if
}d_{i}<0.\end{array}\right.$

It follows that $\Theta^{\infty}(d)=+\infty$ whenever $d_{i}<0$ for some $i\in
I$. Otherwise, when $d_{i}\geq 0$ for all $i\in I$, we get
$\Theta^{\infty}(d)=\sum_{i\in I}\bar{x}_{i}\,d_{i}>0$ which follows since
$\bar{x}_{i}>0$ for all $i\in I$ and $d\neq 0$. This completes the proof. ∎

###### Proof of Theorem 3.

After removing the opt-out facility $i=0$ from the expression of the expected
utilities

$\Phi^{k}(u^{k})=\mathbb{E}(\max\\{u^{k}_{1}+\varepsilon^{k}_{1},\ldots,u^{k}_{n}+\varepsilon^{k}_{n}\\}),$

the characterization of equilibria as the unique minimizer of the strictly
convex smooth function $w\mapsto\Theta(w)$ remains valid with the same proof.

For the existence of equilibria, we observe that we still have
$\Theta^{\infty}(d)=+\infty$ when $d_{i}<0$ for some $i\in I$, and therefore
we only need to check that

$\sum_{i\in I}\bar{x}_{i}\,d_{i}>\sum_{k\in
K}I^{k}\min\\{d_{1},\ldots,d_{n}\\}\qquad(\forall\,d\geq 0,d\neq 0).$ (9)

Let $m=\min\\{d_{1},\ldots,d_{n}\\}$. The inequality (9) is again trivial when
$m=0$, whereas for $m>0$ we have that the expression on the left is minimal
when $d_{i}=m$ for all $i\in I$, in which case (9) reduces precisely to (7). ∎

Appendix B: Intervention Study Results

See pages - of appendix2montecarloresults.pdf