Co-operation adds up, the math says

I’ve been reading Jonathan Haidt and Edward O. Wilson this week, so it was serendipitous to run across a very different take on one of their favourite topics — the dynamics of co-operation.

“Does it pay to be nice? – the maths of altruism ” by Rachel Thomas was published in two parts by +Plus magazine on April 23rd. The article highlights the work of Harvard biologist and mathematician Martin Nowak, who has long applied mathematical analysis to such classic co-operation exercises as the “Prisoner’s Dilemma.”

Most of the material in Thomas’s article isn’t new to readers who are up on game theory or social psychology, but its importance for the claims of Haidt, Wilson, and others — whose works will be reviewed here soon — makes it relevant as a general background for those later articles.

In the Prisoner’s Dilemma, participants are put into a situation similar to that faced by two criminal accomplices who are interviewed separately by the authorities. Each criminal is offered a deal: rat out your partner, and you’ll go free. If neither partner turns on his partner (“defects”), they’ll both go free. But what happens if you keep your mouth shut and your partner defects? The best individual strategy for each criminal is to defect, which results in both partners being convicted. This losing choice is the best choice for each criminal, but it’s the worst strategy for both of them. Hence the “dilemma” in the name.

Nowak tweaked the test by changing it from a one time choice to a repeatable procedure. Now, the dynamic includes not only this choice but the choices made in previous exercises. This change makes all the difference. What you should do depends on the interaction of two choices in each run, plus the choices from the run before (and the one before that …).

Instead of “always defect,” the best strategy is now “direct reciprocity.” If you co-operated last time, I co-operate this time. If you defected last time, I defect this time. Always Defect, the best/worst strategy, is superseded.

Nowak and his team converted the social dynamics of this version of the test into mathematical terms, then inserted a semi-random feature (e.g., heads on a coin flip means that this is the last test in the set, while tails means that you’ll play again with the same partner).

Nowak created a computer model of the test as now constituted and let it run. The computer players began with Always Defect, but they soon switched to a Tit-for-Tat reciprocity. The surprise came when the simulation continued to evolve. Soon, the silicon criminals had developed what Nowak calls “Generous Tit-for-Tat,” in which once in a while (optimally, one time in three) a player “forgives” its partner’s defection and co-operates on the next round anyway.

But the simulation wasn’t through evolving. As time passed, the system came to be dominated by players who always co-operated. The computer participants became more and more lenient, and the number of defections plummeted.

The inevitable outcome was that defection suddenly became more profitable again, and the “nice” society of players collapsed. The system defaulted to the starting point, and the entire evolution began again. Nowak says:

“It is very beautiful because you have these cycles of cooperation and defection.” Nowak’s first observations in the field have since been confirmed by many other studies over the years: cooperation is never fully stable. “So we have a simple mathematical version of oscillations in human history, where you have cooperation for some time, then it is destroyed, then it is rebuilt, and so on.”

But of course this simulation does not run the way most co-operation works in the real world, where our interactions are typically in groups, not pairs. So the conditions of the test were changed again, to “Indirect Reciprocity.”

Indirect Reciprocity involves a number of games in each round, with players paired randomly, with one player as donor and the other as recipient. The donor decides whether or not to co-operate with the recipient. Even though the donor will not play with that recipient again in the round, it’s in the donor’s interest to co-operate in order to create a better reputation with the other players, with whom the present donor might later be a recipient: “Strategies for playing indirect reciprocity games consist of a social norm and an action rule. The social norm gives players a way to judge other players’ reputations and interpret their actions.”

Computer tournaments based on indirect reciprocity produced the same kind of “mutation” cycle seen earlier, “but the most successful, in terms of how long they remained dominant in the population, were those strategies that behaved cooperatively and discriminated on the basis of their opponents’ reputation.” Nowak believes that “indirect reciprocity can explain many altruistic behaviours, such as acts of charity.”
While other animals that reciprocate have to observe the behaviour of others directly, thanks to language (“gossip”) we can practice full indirect reciprocity. “No other organism has the full blown indirect reciprocity because of the lack of human language. They can have indirect reciprocity by observation, but not augmented by communication.”
Nowak believes that “the cooperative force of indirect reciprocity is not only responsible for the evolution of human language, but that it also drove the development of our understanding of social complexity.”

[Previously] there wasn’t a good mathematical approach to group selection, it was only a verbal discussion. But when you look at the mathematics it is clear when [group selection] is valid and when it isn’t. The discussion disappears and mathematics decides the argument.

Directly related to the “eusociality” ideas of Wilson, Nowak and his colleagues raised a firestorm when “they published a paper criticising the standard mathematical argument for kin selection, called inclusive fitness theory.”

They showed that inclusive fitness theory had much greater limitations than people had realised. In particular, inclusive fitness could not be used to explain the evolution of the highly complex cooperative insect societies, such as leaf cutter ant colonies, where millions of ants work and die so that only one individual, the queen ant, can reproduce.

Nowak claims that science that lacks a clear mathematical description isn’t fully understood. “Real understanding in science in terms of mathematics is ultimately elegant, it’s ultimately simple, so we understand the situation well if we have a simple description. If we don’t, then I’m not quite satisfied.”

Perhaps. But no matter what you think of the mathematician’s assertion that science without maths isn’t really science, there’s something to ponder in results like the ones Nowak’s team produced on the Prisoner’s Dilemma.

More on the notions surrounding co-operation and “group selection” when Haidt’s and Wilson’s books come up for review.

One thought on “Co-operation adds up, the math says

  1. With the numbers added this turns further towards hard science which seems to lead to being a description of how things are. Is a law now created for human populations or are there going tobe exceptions?

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