#### THE MATHEMATICS OF COOPERATION

MN: So here we have a sea of co-operators in blue and a single defector.

David: Tell me what you mean by co-operator and defector.

MN: A co-operator is somebody who pays a cost to help somebody else. So, for example, I pay a certain cost and you have a benefit. That’s cooperation.

David: Okay.

MN: And a defector does not do that. A defector does nothing: does not pay the cost, refuses to pay the cost, but also does not distribute benefit.

Ard: The defector’s selfish.

David: Is it selfish?

MN: Defectors are selfish and the co-operator does something very strange. The co-operator…

David: More generous?

MN: They’re generous, but kind of generous to a competitor, because everybody here is a competitor to everybody else. And the co-operator helps a competitor. It reduces its own potential in order to augment the potential of somebody else.

David: Which you wouldn’t expect natural selection to allow.

MN: Yes, natural selection should basically make sure those co-operators who do this strange thing, they get wiped out.

David: Yeah.

MN: We would expect them to get wiped out. Okay.

David: So what happens?

MN: Yeah, let us start the game. And we see, yes, these defectors, they are spreading, you know. So the colours that we are observing here are: red are defectors and blue are co-operators, and yellow and green are changing sides.

So yellow is now a new defector: it was a co-operator before. And green is a new co-operator: it was a defector before. We see the pattern is spreading. A kind of symmetry is being maintained here because they are deterministic of the rules, and the initial symmetry is never broken. But the amazing thing is these co-operators, they are persistent. They just refuse to get wiped out and they survive in those clusters. And if you watch very carefully, you see the clusters, they are actually growing until they collide with other clusters, and then they’re shrinking, and then they grow.

David: But they never go out of business.

MN: But they never get out of business. And the amazing thing is the average abundance of co-operators in this pattern is very close to 12log2-8.

David: What does that mean?

Ard: It’s close to constant.

MN: It converges to a constant which is approximately 31%, and this is a mathematical curiosity.

Ard: Natural selection would say the co-operators would get wiped out, because they are paying a cost to help their competitors. And yet it’s not happening. So why?

MN: The reason why it’s not happening is because the co-operators form clusters and in those clusters of co-operators, they actually get a high payoff. They have a high fitness.

David: So they do better?

MN: They do better than the defectors that are surrounded by other defectors. So we always have to ask, on the edge between a co-operator and defector cluster, who is actually winning?

MN: Because the co-operator, even though sitting on the edge, is still getting all the help from other co-operators inside, but a defector is, sort of, getting no help from his defectors. And therefore the co-operators form these clusters that can persist and can even grow in the presence of defectors.

Ard: So this is a bit like, if I’m with my neighbours and we help each other, then we’ll, in the end, be better off than the neighbours one block down who don’t help one another.

MN: Yes, that’s right. So the neighbours that help each other, they form a community that is cooperative, and the neighbours who don’t help each other, they form a community that is defective. And the first can prosper and the other one will kind of perish.

David: Is this, sort of, an addition to the rule of competition in natural selection. Is this a natural law of cooperation?

MN: I think the very interesting observation is the following: going back to first principles, natural selection favours defectors over co-operators. Yet we have cooperation in nature, and we need to find a reason why there is cooperation in nature. And thousands of papers have been written on that topic, actually, and I have been trying to classify all those different propositions into five mechanisms.

And what we’re seeing here is one of those five mechanisms, for the evolution of cooperation, that I call spatial selection.

The other four are… The most important mechanisms for humans, in my opinion, are indirect reciprocity and direct reciprocity. Direct reciprocity is the fundamental idea that we have repeated interaction between two people: I help you, you help me.

And indirect reciprocity is where there’s a repeated interaction in the group: I help you, somebody will help me. I help you, I gain the reputation of a helpful person and I receive help.

Spatial selection, that’s the third mechanism.

The fourth mechanism is group selection. Groups of co-operators out-compete other groups.

And the last one is kin selection: you help relatives. So J.B.S. Haldane said, ‘I will jump into the river to save two brothers or eight cousins.’

Ard: And these five, you call them laws or principles? What would you call them?

MN: They are mechanisms for the evolution of cooperation. So a mechanism is an interaction structure.

David: And this is a different part of nature from… We’re always taught in school of nature red in tooth and claw. This isn’t that, is it?

MN: I would say nature red in tooth and claw is the first, simple, interpretation of Darwinian selection, of Darwinian evolution, and this is now the add-on. I think this is the modern view. And the modern view is somehow also the one that I’ve wanted to promote over the last twenty years: that cooperation is a fundamental principal of the living world, of evolution. And cooperation is necessary whenever evolution makes a step to a higher level of organisation. So whenever evolutions leads from a simple level of organisation to a higher level of organisation, cooperation is essentially involved. And this is the emergence of the first cell; the emergence of eukaryotic cells that have actually organised inside; the emergence of multi-cell organisms, of social insects and of human society.

Ard: So these principles are what explain how very complex societies or organisms arose from evolution?

MN: I think that’s the fundamental step. So I call cooperation an architect, a master architect of complexity.

Ard: And without cooperation, this wouldn’t happen?

MN: I would expect that without cooperation, it is very likely that you would be stuck on a certain level of organisation. So, for example, just a world of single cell bacteria.

David: So, it’s really a very creative…?

MN: I think this is what makes evolution creative.

David: Yes. If you just had competition, then you would just repeat versions of whatever simple thing you started with.

MN: Yes, I think so. And also, competition would lead to the most selfish, to the most successful type dominating.

David: Ah.

MN: Everything would also be more impoverished.

David: So this is going to give you a richer kind of…?

MN: Yes, that is what I definitely, what I…

Ard: And more beautiful?

MN: Yes, more beautiful.

David: When you first had the idea, did you realise then what effect it would have on your life, or how powerful an idea it has become? Was it clear, or was it something you felt like you were just, sort of, exploring?

MN: As soon as I saw the pattern, I thought there’s something great here, but I don’t fully understand it and I don’t fully understand the implications. But the sheer beauty of it made it clear there was something amazing.

Ard: Something true?

MN: Something true.

Ard: You think the beauty of it told you there was something true about it?

MN: Something true. Something extremely relevant, yes, because this beauty emerged from that game in a completely simple way.

Ard: And was that a surprising emergence?

MN: Yeah, once we had it, you know, one could explain it, but when I set out I wouldn’t have expected something like this.