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Effects of neighbourhood size and connectivity on spatial Continuous Prisoner's Dilemma

The Prisoner's Dilemma, a 2-person game in which the players can either cooperate or defect, is a common paradigm for studying the evolution of cooperation, when individuals exhibit variable degrees of cooperation. It is known that in the presence of spatial structure, when individuals ``play against'' their neighbours, and ``compare to'' them, cooperative investments can evolve to considerable levels. Here we examine the effect of increasing the neighbourhood size: we find that the mean-field limit of no cooperation is reached for a critical neighbourhood size of about five neighbours. We also find the related result that in a network of players, the critical average degree (number of neighbours) of nodes for which defection is the final state depends only on the network topology. This critical average degree is considerably higher for clustered networks, than for distributed random networks. This result strengthens the argument that clustering is the mechanism which makes the development and maintenance of the cooperation possible.