The Problem of Emergency Department Overcrowding: Agent-Based Simulation and Test by Questionnaire

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The Problem of Emergency Department Overcrowding
Agent-Based Simulation and Test by Questionnaire

• Roger A. McCain PhD,
• Richard Hamilton, M.D.
• Frank Linnehan PhD

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For presentation to Artificial Economics 2011

• The Seventh Conference,
• The Hague, Sept. 1-2

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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.

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Solution?

• One generally accepted solution to ED
  overcrowding and congestion is to in- crease
  capacity.
• If the game theory hypothesis is correct, then
  increasing capacity will merely reproduce the
  crowding problem on a larger scale.

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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.

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A Small-Scale Example
This is an anticoordination game, and presents
special problems for a learning model.
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At a Slightly Larger Scale
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Assumptions

• The patients arrive in a random order.
• This determines the patients place in line.
• Average waiting time is proportional to the place
  in line.
• Being one place further back in the line reduces
  this satisfaction by two
• The alternative to the ED provides a satisfaction
  level of five.

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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.

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Generalizing,

• In a real case, we would expect
• Just as in the two-person anticoordination game,
  equilibrium requires some agents to choose
  different strategies even if they themselves do
  not differ.
• 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.
• The equilibrium is not efficient, in general.

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Simulation

• To further extend the model and allow for 1) much
  larger numbers of potential pa- tients, 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.

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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.

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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.

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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.

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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.

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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 aXt-1 (1-a)Et-1.
• For the simulations reported a 1/2.

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A Representative Simulation
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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.

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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.

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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.

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Congestion
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Reported Experience Type 1
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Reported Experience Type 2
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Number of Users
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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

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Conclusions from the Simulations 2

1. 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
2. these results are uniform and predictable over
   simulations with a wide range of differing random
   inputs and detailed evolution.

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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.

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Questionnaire 2

• Eight survey items were used to assess patient
  satisfaction.
• Quantity of care,
• Promptness of care.
• Administrative staff effectiveness
• Medical staff capability
• Personal Care

6. Staff Time Spent 7. Overall Quality of
Care 8. Overall Satisfaction
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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

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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.

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Comparison
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Disappointment!
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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.

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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.

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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.