August 19, 2019

With my sister Josefine Bohr Brask, we a have a new paper out on the arXiv today: https://arxiv.org/abs/1908.05923

It deals with the question of how evolution can lead to cooperation. According to Darwin, individuals tend to behave in ways that maximise their own gain and chance of survival – ‘survival of the fittest’. Cooperative behaviour seems to contradict this, if we understand cooperation to mean that individuals help others without directly getting anything out of it. But we do see cooperation in nature, in many different species, from insects to humans. This is a major puzzle for science. In fact, explaining the evolution of cooperation has been called one of the biggest challenges facing scientist today.

This topic doesn’t have a lot to do with my usual business in quantum physics, but it is a very exciting research field in which Josefine is active, and she got me onboard for this project :). In the paper, we study certain models for evolution of cooperation based on combining game theory and social networks.

Game theory provides a simplified but promising approach to understanding the evolution of cooperation. In evolutionary game theory, the tension between selfishly focusing on one’s own gain or working for the common good is captured by simple games between two players. A famous example is the co-called Prisoner’s Dilemma:

Two prisoners are facing jail time, but their sentence depends on whether they are each willing to rat out the other. They are offered the following bargain:

• If they both stay silent, they each get 1 year in jail.
• If only one of them tells on the other, the one who tells goes free and the other gets 3 years.
• If they both tell on each other, they both get 2 years.

Seen from the perspective of one prisoner, it is always better to rat the other out. If the other doesn’t say anything you go free, and if the other tells on you, you get 2 years instead of 3 by also telling on them. So selfish prisoners would end up both telling and so getting 2 years each. But that is worse that if they would have cooperated! If they both stay silent, each only gets 1 year.

What does that have to do with evolution?

We can use the game to model evolution in the following way. Imagine a large population of individuals. Each individual has a fixed strategy – either always cooperate (i.e. stay silent) or always defect (i.e. rat the opponent out). We let the individuals in the population play against each other and register how many games they win or loose. Then we allow them to adapt their strategies. For example by copying the strategy of more successful individuals. And then we do it again. And again. After many rounds, one strategy may start to dominate in the population. For example, the cooperators die out and only defectors are left. By varying the parameters and rules of the game, we can try to figure out, under what conditions cooperation can survive and spread.

It turns out that if everyone in the population just plays against everyone else, cooperation doesn’t stand a very good chance. The same is true if individuals are just randomly paired up in every round. When there is no structure in the population, cooperation generally cannot survive. Something more is needed.

A ‘something more’ which can make cooperation survive, is social network structure. In nature – and in human contexts – an individual does not usually interact with every other individual in the population, but mostly with a particular bunch of individuals. Each of these in turn are connected to certain other individuals, and so on. Just like you are connected to you friends on Facebook (or in the real world), and they each have their own circles of friends. Evolution on a network can be modelled by having each individual play only against its neighbours in the network and adapting its strategy based on the strategies and performance of its neighbours.

A number of studies have found that social network structure can in fact stabilise cooperation, even for games with a strong incentive for selfishness, such as Prisoner’s Dilemma. So network structure is a strong candidate for explaining, how it is possible for cooperation to evolve.

Of course, the actual dynamics of interactions, adaption, and survival in nature are very much more complicated than simple two-player games with just two strategies. However, simple models can also be powerful exactly because they potentially allow us to cut through the noise and identify the key underlying mechanisms. But one must be careful to check how general conclusions can really be drawn from them. Evolutionary game theory models for cooperation are often studied using numerical simulations as they are not easily solved analytically. In that case, one needs to be sure that the technical details of the simulations do not affect the general conclusions (about whether cooperation survives) too much.

In our paper, we study the effect of initially placing cooperators and defectors in different types of positions in the network. Most simulations start by distributing equal numbers of cooperators and defectors in the network at random. We wanted to know, if changing this initial distribution affects the outcome, and how. For example, if we initially place cooperators in positions with many neighbours and defectors in positions with few neighbours that might give the cooperators an advantage (more neighbours might copy their strategy thus becoming cooperators too).

We find that in certain cases, the conclusions about the evolution of cooperation are robust. But for some commonly studied kinds of networks, correlating the initial positions of cooperators with the number of neighbours strongly affects whether cooperation survives or dies out. So one does need to be careful!