# Paper: Cutting ties leads to cooperation

One field of study that greatly interests me is evolutionary game theory. The central question of the field is: how does altruism evolve? In a naive analysis, it would seem that uncooperative individuals can get ahead of the rest of their species, and uncooperative offspring will come to dominate the population. Nonetheless, in the real world we observe mixtures of cooperative and non-cooperative behavior.

In evolutionary game theory, “cooperation” is understood as a strategy in a two-player game. Most commonly, we consider the prisoner’s dilemma game, where two players each have the choice to cooperate or defect. “Defecting” is a strategy that benefits yourself, but hurts your opponent even more. And so if both players defect, then they’re both worse off than if they had cooperated.

The key to the evolution of cooperation is the ability to react to defectors. In particular, one needs to punish defectors, such that defection is no longer beneficial.

Here I’ll talk about one model that allows for such punishment, based on a paper titled, “Cooperation prevails when individuals adjust their social ties“. As suggested by the title, the mechanism for punishment is to cut off ties with defectors.

Figure 1 from the paper illustrates a network of cooperators and defectors

### A model of network evolution

The basic model is to start with a population of individuals in a network. Each individual has a single strategy: either cooperate, or defect. They play a Prisoner’s Dilemma game with each of their neighbors, as determined by the network. Their fitness is calculated from the sum of outcomes of all those games.

Now we allow the population to “evolve”, using two processes: strategic evolution, and network evolution. Strategic evolution follows this procedure:

1. Consider a random pair of connected individuals.
2. Calculate the fitness of each of those individuals.
3. Randomly select one of the two individuals, with greater weight given to the fitter one. Call the selected individual A, and call the other one B.
4. B copies the strategy of A.

Network evolution follows a similar procedure, only the last step is replaced with the following:

4. If B is a defector, A cuts ties with B, and randomly creates a new connection with one of B’s neighbors.

Strategic evolution and network evolution operate simultaneously, although not necessarily at the same rates. If network evolution is fast enough compared to strategic evolution, then cooperators dominate; otherwise, defectors dominate. That’s the main result of this paper. The more power we have to cut ties, the more this leads to cooperation.

### Why does cooperation prevail?

Now I’ll say a bit why the main result isn’t surprising. The way the model is designed, punishment of defectors is hardwired into individual behavior. Whenever someone defects, all their neighbors want to cut ties with them. Having ties is good! Overall, an individual with more ties is more fit (unless they’re unlucky enough to only be tied with defectors).

It may be helpful to draw comparisons to another evolutionary model. This paper uses a model based on the iterated prisoner’s dilemma (I blogged about this one in 2013). Rather than having a network, individuals simply play games with other individuals selected at random. And rather than playing one game of prisoner’s dilemma, they play many games of prisoner’s dilemma, and are allowed to react to the outcomes of the previous games.

The iterated prisoner’s dilemma model leads to cooperation, but this result is highly nontrivial! Defectors may be punished, but punishment comes in the form of “tit for tat”–that is, when someone defects against you, you defect against them in the next game. It is far from clear that “tit for tat” is a strategy that would be evolutionarily successful.

Contrast with the network model, where punishment is hardwired in, and there is no cost to the punisher. I would be interested in seeing an alternate model where cutting ties hurts both sides. (Perhaps when A cuts ties with B, A does not get to form a new connection with one of B’s neighbors.)

So to me, the main result of the paper is not very surprising. Of course cutting ties will lead to cooperation, at least in the limit where network evolution is faster than strategic evolution. The only question is how fast does network evolution need to be? And how is the result affected by the parameters of the model?

In fact, most of the paper is devoted to these lesser questions. For example, they find that the more connected the network is, the more defection is favored. The more we weight probabilities towards fitter individuals, the more defection is favored. And they also investigate games besides the Prisoner’s dilemma. I invite you to read details in the article, which is open access.

### Relation to human behavior

Although it’s an abstract mathematical model, it’s easy to think of how it might apply to human relationships. When someone is being uncooperative or hurtful, there are two things you can do to promote cooperation. You could follow “tit for tat”, and be uncooperative right back. Or you could cut ties. Either of these strategies may be individually successful, while also benefiting the population as a whole in the long run. So cut ties, cut ties!