A condition of cooperation. Games on network

Tomohiko Konno

doi:10.1038/npre.2009.3028.1

Abstract

AbstractNatural selection is often regarded as a result of severe competition. Defect seems beneficial for a single individual in many cases.However, cooperation is observed in many levels of biological systems ranging from single cells to animals, including human society. We have yet known that in unstructured populations, evolution favors defectors over cooperators. On the other hand, there have been much interest on evolutionary games^1,2^ on structured population and on graphs^3-16^. Structures of biological systems and societies of animals can be taken as networks. They discover that network structures determine results of the games. Together with the recent interest of complex networks^17,18^, many researchers investigate real network structures. Recently even economists study firms&apos; transactions structure^19^. Seminal work^11^ derives the condition of favoring cooperation for evolutionary games on networks, that is, benefit divided by cost, b/c, exceeds average degree, (k). Although this condition has been believed so far^20^, we find the condition is b/c (k~nm~) instead. k~nm~ is the mean nearest neighbor degree. Our condition enables us to compare how network structure enhances cooperation across different kinds of networks. Regular network favors most, scale free network least. On ideal scale free networks, cooperation is unfeasible. We could say that (k) is the degree of itself, while k~nm~ is that of others. One of the most interesting points in network theory is that results depend not only on itself but also on others. In evolutionary games on network, we find the same characteristic.

Highlights

There are only two kinds of agent: cooperator and defector
In an unstructured population, where all the agents interact each other, defectors tend to have higher average gain than cooperators, so that cooperators would extinct as a result of natural selection
When the payoff of the game is relatively smaller than the baseline fitness, we call this as weak selection

Summary

Introduction

There are only two kinds of agent: cooperator and defector. A cooperator pays a cost, c, for each neighbor to receive a benefit, b. If you look at vertices linked to the randomly chosen one, the mean degree of which is not k but knn . If selection neither favors nor opposes cooperation the probability is 1/N , N is the size of networks. If the fixation probability is larger than 1/N , we say that the network structure favors cooperation.

Results

Conclusion