Super Cooperators By Martin Nowak

I would safely say that the summer of 2012 has been the summer of reading and playing around with different ideas about evolution, cooperation and competition. I have not heard of Martin Nowak until I read his article in Scientific American's July issue. Then, a few weeks later, I was reading about him again in Chaitin's book Proving Darwin. Martin Nowak is a researcher in Mathematics and Biology. He spent most of his research career tackling problems about evolution and evolutionary games mathematically and through numerous computer simulations. His latest book, Super Cooperators, aims at showing that evolution (i.e. selection + mutation) needs cooperation as well to work. The book starts with a discussion of the famous Prisoner's Dilemma problem, it then describes 5 principles that allow cooperation to thrive in communities. The book also has two nice chapters about the evolution of language and the tragedy of the common goods.

The Prisoner's Dilemma

The Prisoner's Dilemma is a very well-konow problem in Game Theory. The problem comes in different flavors and has been extensively studied in the past decades. It has been first formalized by Alan Tucker in the fifties (You might remember Alan Tucker from A Beautiful Mind movie, he was the supervisor of John Nash and the head of the Mathematics Department at Princeton University).

The basic game is as follows. Two prisoners are being questioned by an investigator about a robbery. Each prisoner can either cooperate (deny the robbery) or defect (confess). The game has the following rules:

• If both prisoners cooperate, then they both get a 2 years sentence.
• If both prisoners defect, then they both get a 3 years sentence.
• If one prisoner cooperates while her partner defects, then the cooperator is sentenced to 4 years while the defector gets only a 1 year sentence.

This is a formalization of the problem using my own concepts and symbols. Let A (Alice) and B (Bob) denote both players. Let c_c(A) (resp. c_c(B)) be the cost function (in terms of prison years) that Alice (resp. Bob) incurs when she (resp. he) cooperates. We define,in a similar way c_d(A) and c_d(B) the costs to defect for each player. Let p_c(A) (resp. p_c(B)) be the probabiltiy that Alice (resp. Bob) cooperates. Then, we have the following:

and,

Equations (1) and (2) show that defection is a dominant strategy in the Prisoner's Dilemma game. This means, regardless of the opponent's probability to cooperate, defection always leads to the best pay-off (i.e. minimum cost).

Now, let us consider the case where Alice and Bob play N rounds of the game. If the decisions made by Alice do not influence future descisions made by Bob (and reciprocally), i.e. when cooperation propabilities remain static all along the game, then defection is still a dominant strategry. Indeed, in this case we are merlely multiplying the cost functions by N.

In an N-rounds game, the game space has a cardinality of 2^2N games. In this space, the best outcome of a single player is N and the worst outcome is 4N. The pay-off for the group is the sum of the outcomes of all the players in this group. The best outcome for the group is 4N. This value is reached when both players cooperate on each round. The worst outcome for the group is 6N. This value is reached when both players engage in vengful behavior on each round. Cooperation is better for the group at a little cost (1 year of prison) for the individuals and hence the dilemma

Strategies for this problem will typically try to throw a carrot to the opponent once in a while hoping that this will increase its p_c(.)

Game Generalizations

Martin Nowak and others have generalized the Prisoner's Dilemma game to reflect situations encountered in economy, biology, cancer cells propagation... First, the problem has been generalized to M players. Pairs of individuals play N rounds of the game. Moreover, this population of individuals undergoes evolution. Each individual plays the N-rounds game with 0 to N - 1 players. The pay-off of every individual is calculated at the end of each generation, that is the fitness of the individual. A selection operator will then select the individuals that will make up the next generation with a probability proportional to their fitness. Individuals may also mutate, unfortunately, the mutation operator has not been described in the book.

Nowak et al. have also introduced noise in the basic model. This means that Alice might defect on the next round with a very small probability even though the strategy she is following dictates cooperation. This noisification aims to take into account of some phenomena that make us sometimes go for the "wrong" decision like misinterpretations of the opponent's moves, emotions, noise in the communication channel between players...

Early models of the M-player extension assumed that every player plays with everybody else for every generation. In other words, individuals in these popultations were connected via complete graphs. Nowak and his group studied the impact of the topology of the population's graph on the evolution of cooperation. Edges in such graphs can model geographical proximity. This field is now known as Evolutionary Graph Theory.

Finally, Evolutionary Set Theory deals with the impact of having individuals belong to some common set on their probability to cooperate. A set may include people observing the same faith, people who are members of the same gym, the same political party...

The Cooperation Five Principles

The book describes 5 principles for cooperation to thrive in a population and hence counter defection which is the rational dominant strategy.

Direct reciprocity

This principle states: be nice who has been nice to you and be nasty to who has been nasty with you. The most well-know strategy that abides by this principle is the Tit-for-tat strategy.

Indirect reciprocity

This principle makes individuals cooperate with individuals that they have never met but who have the reputation to be nice. In this case, the player is paying some cost hoping to get a higher return in the future from these reputably-nice individuals. Language and more recently the Internet reinforce this principle and promote strategies that abide by it.

Spatial selection

Strategies that follow this principle will make individuals cooperate much more with their immediate neighbors in some population graph.

Multilevel selection

In this case, the probability that Alice will cooperate with Bob is also influenced by the reputation of the group to which Bob belongs, even she has no clue if Bob is a nice person or not. Some groups, for instance members of the same religion, will have an overall reputation of being nice people, and reciprocally.

Kin selection

Kin selection dictates to individuals to cooperate with others who share more genes with them. The purpose of this is to allow the genes to survive. For instance, following this principle you will more likely save a drowing brother than a drowining cousin since you have more genes in common with your brother than with your cousin

Overall Impressions

Super Cooperators is a nice and entertaining read. The book does not include a single mathematical equation. Many anecdotes about the author and his coworkers come to give the reader some breaks from the main topic of the book. A well-written book and highly recommended.