Birmingham mathematician Dr Hong Duong has joined forces with a computer scientist from Teesside University and a data analyst working in Norway to bring together two different concepts to further our understanding of the distribution of equilibrium points in dynamical systems involving multiple participants.

Evolutionary game theory (EGT) and random polynomial theory (RPT) are important theories in their own right. Crucially, they are also seemingly unrelated.

Professor Hong Duong’s ground-breaking work with colleagues Drs The Anh Han and Hoang Minh Tran is detailed in a research paper entitled ‘On the distribution of the number of internal equilibria in random evolutionary games’. Published in the Journal of Mathematical Biology, it was recently awarded the College’s Paper of the Month accolade.

EGT, which dates back nearly 50 years, is the application of game theory to evolving populations in biology – defining a framework of contests and strategies into which Darwinian competition can be modelled. It has helped to explain the basis of altruistic in animals – such as certain species sacrificing themselves for the good of the group.

In traditional game theory, the ‘equilibrium point’ refers to the Nash equilibrium – named after John Nash, who won the 1994 Nobel Prize in Economics for his work on game theory – which is a stable state of a system involving the interaction of different participants, where no participant can make a better choice for themselves if other people don’t change their strategies.

‘The Nash equilibrium is a foundational concept in game theory that has had a pervasive impact in economics and social sciences,’ explains Hong.

Bringing this concept to EGT creates an evolutionarily stable strategy (ESS), an equilibrium refinement of the Nash equilibrium: a Nash equilibrium that is ‘evolutionarily’ stable, meaning that once it is fixed in a population, natural selection alone is enough to stop alternative, or mutant, strategies from successfully invading.

‘So that is the motivation for our work – to study the equilibrium properties of evolutionary game theory’ explains Hong, a Lecturer in Mathematical Statistics.

‘We use the replicator equation, which is a fundamental equation in EGT that describes the change of frequencies of different strategies in the population. The change of each strategy depends on its current value, its fitness and the average fitness of the whole population, which are calculated using game theory. If the strategy’s fitness is better than the average of the population, then it increases. Conversely, if it’s not as good as the average, then it decreases. It’s Darwin’s theory of natural selection.’

‘So, we can see that the change depends on the interaction of strategies or players. But there are many situations we can’t have all the information for, or that the environment changes so rapidly that we can’t describe fully the interaction. In these scenarios, it’s necessary to introduce randomness into the model. And this is the situation we have considered in our work.’

The question Hong and his colleagues asked – and the subject of a previous paper – was: on average, how many equilibrium points does the system have?

The more challenging question is to find out the distribution of the equilibria, because that gives much more information than the answer to the first question.

‘It’s like asking a student what their average grade is, and then asking what grade they got in each subject. So that’s exactly what we did: we characterised the distribution of equilibrium points in multi-player evolutionary game theory.’

To do that, Hong and his collaborators brought together the two theories: EGT and RPT.

‘Mathematically, to find an equilibrium point you need to solve a polynomial equation, which is a classical mathematical problem. Abel’s impossibility theorem tells us that in general you can’t solve a polynomial equation of a degree bigger or equal to five. Because of this, studies were limited to games with a small number of players – two or three. So what we did was to use techniques from the RPT to answer our question from the EGT. The result was that we found a new class of random polynomial from EGT, which enabled us to answer the two questions – the simpler and the more challenging one – for any number of players.’

This has opened up a new avenue of research, such as to explore the connection to more complex systems and to study universality phenomena, which doesn’t depend as much on randomness.

‘I’m looking at this from a mathematical point of view, but it’s important also for biology, because it depicts biological diversity and the maintenance of polymorphisms.’