The Problem of Emergency
Department Overcrowding:
Agent-Based Simulation and Test
by Questionnaire
Roger A. McCain PhD,
Richard Hamilton, M.D.
Frank Linnehan PhD
For presentation to Artificial
Economics 2011
The Seventh Conference,
The Hague, Sept. 1-2
The Problem
• Overcrowding of hospital emergency departments
is a recognized problem in the U. S. A.
• Patients seek healthcare in the Emergency
Department for a variety of reasons.
– a portion have an acute emergency
– a portion seeks care because of lack of an acceptable
alternative.
• Drawing on noncooperative game theory, we
argue that ED overcrowding is the equilibrium
state of the current health care system.
Solution?
• One generally accepted solution to ED
overcrowding and congestion is to increase capacity.
• If the game theory hypothesis is correct,
then increasing capacity will merely
reproduce the crowding problem on a larger
scale.
Our Study
• We address this issue by means of
– Examples from noncooperative game theory
– Agent-based simulations
• The scale is larger
• Boundely rational learning is incorporated
– A Questionnaire Study
• The agent-based simulations and the questionnaire
study are coordinated.
A Small-Scale Example
This is an anticoordination game, and presents
special problems for a learning model.
At a Slightly Larger Scale
Assumptions
1. The patients arrive in a random order.
2. This determines the patient’s place in line.
3. Average waiting time is proportional to the
place in line.
– Being one place further back in the line
reduces this satisfaction by two
4. The alternative to the ED provides a
satisfaction level of five.
Noncooperative Solution
• This game has a large number of solutions.
• All are defined by the same condition,
however: just six choose the ED while the
other four choose their alternative.
Generalizing,
• In a real case, we would expect:
1. Just as in the two-person anticoordination game,
equilibrium requires some agents to choose different
strategies even if they themselves do not differ.
2. When the strategies are modes of service, the number
choosing the different services in equilibrium will be
such that the different services yield the same benefits, in
expected value terms.
3. The equilibrium is not efficient, in general.
Simulation
• To further extend the model and allow for
1) much larger numbers of potential patients, 2) heterogeneity of health states,
experience, and expectation, 3) boundedly
rational learning, and 4) initialization
effects, dynamic adjustment and transients,
we undertook agent-based computer
simulation.
Agents Are Patients
• For these simulations, the agents are
potential patients, while the ED is not a
player in the game but a mechanism that
mediates the interaction of the agents.
• It is assumed that (at each iteration of the
simulation) agents are randomly sorted into
four health states.
Health States
• Agents in states 1 and 3 have health concerns such
that treatment in the ED offers a higher benefit
than the alternative in the absence of congestion.
• For agents in state 2, there is a health concern such
that treatment through the alternative mode offers
higher expected benefit than treatment by the ED,
even in the absence of congestion.
• Others have no need for health care.
Some Details 1
• In the simulations, there are 10,000 agents, and at each
iteration they are sorted into health states such that about
60 % will seek health care from one source or another.
• Each agent makes the decision based on an expected
benefit variable, with a normally distributed pseudorandom
error.
• Qualification: For technical reasons having to do with the
trial-and-error learning process, at least 5 % of those
agents who seek health care choose the Emergency
Department regardless of their expectations.
Some Details 2
• After all these decisions have been registered, the
congestion of the emergency department is computed by
comparing its capacity parameter to the number of users.
• Congestion exists only when the number of users is greater
than capacity;
• The experiences for all agents who choose the ED are then
computed on the basis of congestion together with the
parameters of their specific health states, with a
pseudorandom variate to capture the uncertainty inherent
in medical treatment.
Some Details 3
• Numerical indicators of experience are roughly calibrated
to the five-point Likert scale used in the questionnaire
survey reported below.
• Expected benefits are then updated.
• The updating formula is the Koyck lag formula,
Et = αXt−1 +(1−α)Et−1.
• For the simulations reported α = 1/2.
A Representative Simulation
A Complication
• These agents form their expectations as to the
benefit from ED care on the basis of their own
past experience plus an error.
• Those who choose the ED are the ones who most
overestimate the benefits of the ED.
• This is shown by the upper black line.
• Nevertheless experienced benefit from the ED
converges to the experienced benefit of the
alternative.
Why this Wrinkle?
• It is crucial that the agents learn only from their
own experience. In an anticoordination game
imitative learning will not converge to a Nash
equilibrium.
• An early version of the simulations had no errors.
• The questionnaire study indicated that the average
ED patient was disappointed.
• The experience-plus-error model retrodicts that
result.
More Simulations
• For this study 18 distinct simulations were recorded.
• Two simulations were run using each of 9 random number
seeds.
• For one series of 9 simulations the capacity of the
Emergency Department was set at 500, while for others it
was set at 1000.
• The simulations were run for 200 iterations.
• The next slide shows the recorded Emergency Department
congestion for the 18 simulations run.
Congestion
Reported Experience – Type 1
Reported Experience – Type 2
Number of Users
Conclusions from the
Simulations 1
1. As in the small-N models, an equilibrium or stable state
corresponds to congestion sufficient to reduce the benefits
of users of the ED to approximate equal- ity with the
benefits from alternative service;
2. In these simulations with a large but finite number of
agents and boundedly rational learning, the approximation
to Emergency Department Overcrowdiing to the
alternative benefit may not be perfect and may vary
somewhat with parameters and initialization, so that
Conclusions from the
Simulations 2
3. An expansion of ED capacity can result in some
slight improvement in congestion and patient
experience, despite very substantial deterioration
of the experience due to congestion, and finally
4. these results are uniform and predictable over
simulations with a wide range of differing
random inputs and detailed evolution.
Questionnaire 1
• A telephone survey was conducted of patients who
visited the emergency department of the hospital
of the Drexel University School of Medicine.
• These telephone surveys were conducted by an
independent research group who were given a list
of all patients who had visited the ED during
summer, 2007.
• Names and telephone numbers were randomly
chosen from this list to complete 301 interviews.
Questionnaire 2
• Eight survey items were
used to assess patient
satisfaction.
1. Quantity of care,
2. Promptness of care.
3. Administrative staff
effectiveness
4. Medical staff capability
5. Personal Care
6. Staff Time Spent
7. Overall Quality of Care
8. Overall Satisfaction
Questionnaire 3
• A five point, Likert- type response scale
was used for each item, ranging from 1 =
Very satisfied to 5 = Very dissatisfied.
• The same facets of satisfaction were also
used to assess the patients expected
experience in the ED and the patients
expected experience with an alternative
mode of care
Questionnaire 4
• Paired t-tests were used to assess
differences in satisfaction levels between
what the patients experienced at the ED
and expected satisfaction with the
alternative (Table 3), as well as the
difference in the expected and
experienced satisfaction with the ED.
Comparison
Disappointment!
Concluding Summary 1
The project reported in this paper was highly
interdisciplinary, drawing ideas and techniques from several
sources. There are novel contributions for each.
•For health care policy, we have specified, tested, and verified
a Nash equilibrium hypothesis of the cause and nature of
emergency room overcrowding. This hypothesis implies that
increasing emergency room capacity may have little or no
impact on overcrowding, in the absence of important changes
in access to the alternative modes of medical care.
Concluding Summary 2
• For game theory, we have provided an example of testing a
game-theoretic equilibrium model by questionnaire
methods, using a realistically scaled agent-based computer
simulation with boundedly rational learning to extend the
insights of two- and small N-person game models to
generate hypotheses for the survey.
Concluding Summary 3
• For questionnaire methods, we have provided an example
of application to hypotheses from game theory and some
evidence of the importance and consequences of
heterogeneity, and the possibility of modeling
heterogeneity explicitly by means of agent-based computer
simulation.