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 in- crease 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 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.

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.