This is a blog post describing a paper Izabela Spernaes, Jeff Griffiths and I had published last year. I am going to start using this blog to write about papers I publish and I’m starting with a paper called Measuring the price of anarchy in critical care unit interactions. In this paper, we used game theory and Markov models to model interactions between two critical care units.

The underlying idea here is that when you have uncoordinated behaviour you get inefficiency. A good example of this is the Prisoner’s Dilemma. If two prisoner’s act in a coordinated way, where there was some form of central control that wanted to reduce time in Prison: they would cooperate with each other and go to prison for 2 years each. If however we remove that central control and let both players do exactly what is rational for themselves they would go to prison for 4 years each.

This paper is not the first that’s looked at inefficiencies in healthcare caused by uncoordinated behaviour:

In the paper I’m discussing here we took a look at modelling interactions between critical care units. We noticed in another piece of research: a 2015 paper about combining two critical care units, that behavioural models of waiting lines seemed to model two critical care units better than standard models. This gave us the idea to look at how two critical care units would interact when one accepts slightly less patients (sending them to the other) if its occupancy is passed a certain point.

The first thing we had to do was build a Markov chain model of the occupancy of the hospital: we do this by considering the state of the system as a two dimensional vector. For example: \((2, 15)\) would indicate that there were 2 people in the first unit and 15 in the second. We can then build up linear relationships between any two states using the rate at which a patient arrives and how long they stay in service. This does make an assumption about the fact that the length of stay follows a so called exponential distribution. An image of the entire chain is shown here:

The red lines shows the “strategies” of both hospital: ie the point of occupancy at which they take in less patients. We considered two possible scenarios:

  1. If both hospitals are past that point (ie they both stop taking patients) then those patients are lost to the system;
  2. If both hospitals are past that point then they do take the patients that were originally intended for them.

From the point of a central controller, we would hope to ensure that as many patient as possible are seen. This does not necessarily imply that both units should never send patients to the other. Indeed our Markov model will capture the variability in the system and always working at full capacity will imply that new patients often find the system full.

From the point of view of the individual unit they will try to be as efficient as possible. Often, with a hope to align the social good with the goals of the individual agents in a system, targets will be imposed. For example, perhaps some guideline about the target occupancy rate of a unit

Once we have this we can build a normal form game representation, of the number of total patients seen in total. So that would look like this:

Where \(A_{ij}\) is the “utility” to the first unit when the first unit’s threshold is at \(i\) patients and the second is at \(j\) patients.

Following this, in the paper we proved a theoretic result about the location of the Nash equilibrium: the point at which both hospitals will end up when aiming to get as close as possible to the target occupancy \(t\).

Then finally we carried out a number of numerical experiments. All using sagemath which was my tool of choice at the time, if we were to do this again we would probably use Python.

Some of the insight we gained from these numerical experiments:

  • Inefficiency (ie lost patients) can be relatively high in some scenarios (10% loss of patients for example).
  • It is possible to find a target that ensures that there is no inefficiency.

This was a paper I very much enjoyed working on as it was a direct application of Game Theory and demonstrates the importance of understanding the effect of behaviour on complex systems.