23rd June 2020

An international research project based at the University of Leicester recently provided the first detailed mathematical proof that antibiotic over-prescribing is a social dilemma of ‘the tragedy of the commons type’. Andrew Colman outlines how this was achieved with the help of game theory.

Researchers and commentators have frequently suggested that antibiotic over-prescribing may be a tragedy of the commons – a social dilemma in which rational decision-making by every participant leads to a collective outcome that is worse for all of them than different behaviour. A recent survey confirmed that 96% of a sample of 1,530 infectious disease professionals in the US believed that antibiotic over-prescribing is a tragedy of the commons, but no one had ever proved this until we did. Several colleagues and I recently received funding from the Economic and Social Research Council to study antimicrobial resistance internationally. One of the products of our research was the first rigorous, mathematical proof of the tragedy of the commons claim, published in the journal PLoS ONE in 2019.

Modelling antimicrobial prescribing as an evolutionary game turned out to be quite tricky. We started with the assumption that doctors typically have to treat patients without being certain whether or not they have bacterial infections. We defined a population game as a set N of prescribing doctors in a defined population, each choosing repeatedly (whenever confronted by a patient with symptoms of possible bacterial infection) either to prescribe or not to prescribe antibiotics. After every decision, each doctor receives a payoff π. We assumed that doctors are motivated to do the best for their patients, hence the lower the morbidity (from life-threatening to mild illness) among their patients, the higher their payoffs (π).

The first step was to work out exactly how the π depend on the relative risk of antibiotics to patients with and without bacterial infection, the proportion of patients in the population currently being treated with antibiotics, and the probability that a given symptomatic patient does indeed have a bacterial infection. We then proved that the unique Nash equilibrium of this game occurs when all doctors prescribe antibiotics for all symptomatic patients. A Nash equilibrium is an outcome in any game in which every player receives the best possible payoff from a chosen strategy, given the strategies chosen by the others. Next, we proved that this outcome yields worse payoffs for every doctor – and, incidentally, hence also for their patients – than the outcome that would have occurred if some or all of them had behaved differently by restraining their antibiotic prescribing strategies. We then used evolutionary game theory to show that the Nash equilibrium outcome when all doctors prescribe antibiotics at every opportunity is an evolutionarily stable strategy – a strategy that, if all doctors choose it, it cannot be driven out over time by a different strategy yielding better payoffs.

Finally and importantly, we used replicator dynamics, a mathematical technique based on ordinary differential equations, to prove that the Nash equilibrium is a global attractor. This means that maximal antibiotic prescribing by all doctors is not only an evolutionarily stable Nash equilibrium, but there is also a dynamic attraction to this state from virtually all other possible states. In other words, doctors will increase their prescribing of antibiotics irrespective of the level of antibiotic resistance in the population and the behaviour of other doctors, although they would all be better-off if they restrained their prescribing behaviour.

We concluded from all this that ever-increasing antibiotic resistance may therefore be inevitable unless the payoffs of the game are changed. In a subsequent (non-mathematical) publication, we made recommendations about how this change might be achieved, by establishing consensus-based rules around limiting antibiotic use, enabling the community of prescribers to self-monitor through sharing of data on prescribing levels, and using social and reputational rewards and sanctions may enhance current approaches to conserving antibiotics. Alongside this, misaligned goals and incentives need to be addressed, and local mechanisms for conflict resolution established. Developing a better understanding of the nature of the trade-offs involved in the dilemma will inform decisions about how far stewardship should go in restricting current antibiotic use.

Andrew M. Colman, BA, MA, PhD, FBPsS, Professor of Psychology, University of Leicester

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