A review of evolutionary game theory, focusing on the prisoner’s dilemma, in structured populations. Has some good stuff on pair approximations, including a straightforward introduction to the math in an appendix.
The pair approximation gives much closer results to simulations for voluntary, as opposed to compulsory participation in the PD game.
The pair approx does not take into account corrections arising from loops. For this reason, the pair approximation is unable to distinguish between square lattices, regular small world networks, and regular random graphs (RRG) with identical connectivities. (Fig. 14)
Generally, predictions by the pair approximation are less reliable near the extinction thresholds, because this approximation does not account for corrections arising from loops nor the long range correlations occurring in the vicinity of critical transitions. The accuracy of this technique can be improved by considering configuration probabilities of larger clusters. The improvement may not only be quantitative, but in some cases even qualitative.
Also, on positive assortment:
The most prominent mechanisms of cooperation are direct and indirect reciprocity, voluntary interactions, and spatial structure. All these mechanisms have one thing in common: they hinge on different forms of assortative (that is, nonrandom or conditional) interactions i int inttinfdn interactions. Such assortment can be actively implemented through discriminating strategic behavior of the interacting individuals or passively by imposing environmental constraints such as local interactions in spatially extended systems. The dynamics that results from constraining interactions to nearest neighbors suggests new and interesting and intriguing links to physics and, in particular, to statistical mechanics.
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