# Expanding the solutions in game theory

Game theory models the problems that arise when different interests are difficult to reconcile. While most applications assume that the players make decisions based only on the so-called payoff matrix, a more detailed modelling is necessary if we also want to consider the influence of correlations on the decisions of the players. Physicists Adriana Correia and Prof Henk Stoof from Utrecht University therefore studied the extension of the existing framework of correlated strategies. The results of their research were recently published in the journal Scientific Reports.

### Tossing a biased coin

Suppose you you are driving a car in a country without any traffic rules, and you arrive at an intersection. Also, the cars in that country don’t allow you to see what cars coming from the other directions will do. What can you do to ensure the best possible chance of crossing the intersection safely? This is an (almost) real-life situation where Game Theory can be used to help make a choice. This situation can be modelled using a game known as Snowdrift or Chicken, and it turns out that the best solution is to toss a biased coin that tells you whether or not to cross the intersection with a calculated probability. This is known as the mixed strategy Nash equilibrium.

### Correlation device

When you come back to the same country next year, you notice that they have dramatically improved their traffic system: now they have installed traffic lights at the intersections. However, these are not like the ones you’re accustomed to back home, where when you see a green light, the traffic from the perpendicular direction is certainly seeing a red light. These traffic lights have a certain defect that strips that certainty away, so that when you see a green light, you only know with a certain probability whether the other direction is showing red. The traffic light here acts as what is called in Game Theory a “correlation device”, and there is a correlated equilibrium if the probabilities are such that the best solution is still to follow the device, or otherwise discard the device altogether.

### New equilibria

Correia and Stoof explored the possibility that this is not the only solution. They considered whether the players could use the information from the traffic light to arrive at a new equilibrium, where the new strategies involve a probability to follow or not follow it, instead of going or not going. The researchers found that there is indeed a new equilibrium in areas where it is not stable to always follow the device, but it is stable to sometimes follow the device. Besides these new equilibria providing a better outcome than ignoring the device for certain parameters, sometimes they are the best solution even if always following is still a stable solution.

Applications in other disciplines Besides the existence of new equilibria being a new and interesting finding on its own, this result can have vast consequences for fields like evolutionary biology, where game theoretical models are used to explain why certain species make it instead of others. It shows that many systems might actually be in an equilibrium that is triggered by an underlying “device”, which can be interpreted as shared information between the agents, helping to explain emergent complexity. The researchers also expect that this result has an impact in the analysis of games played by several players on a network. By introducing the device and the probabilities of following it, the situation can be expressed as an Ising Model and one can derive statistical information about it at equilibrium.

### Publication

Correia, A.D., Stoof, H. T. C.: Nash Equilibria in the Response Strategy of Correlated Games, Scientific Reports, 9, 2352, 2019, https://doi.org/10.1038/s41598-018-36562-2