Alexander, J. McKenzie (2002) *Random Boolean Networks and Evolutionary Game Theory.* [Preprint]

## Abstract

Recent years have seen increased interest in the question of whether it is possible to provide an evolutionary game theoretic explanation for certain kinds of social norms. These explanatory approaches often rely on the fact that, in certain evolutionary models, the basin of attraction of "fair" or "just" strategies occupies a certain percentage of the state space. I sketch a proof of a general representation theorem for a large class of evolutionary game theoretic models played on a social network, in the hope that this will contribute to a greater understanding of the basins of attraction of such models -- and hence the evolution of social norms. More precisely, I show how many kinds of social networks can be translated into random boolean networks. The interesting and useful part of this result is that, for many social networks, one can find a bijection $f$ between the state space of the social network and the state space of the random boolean network, such that the state $S`$ follows the state $S$ under the dynamical laws of the social network if and only if $f(S`)$ follows the state $f(S)$ under the dynamics of the random boolean network. In some cases, it is not possible to find such a bijection; in these cases, one can find an injection $f$ with the property that if $S`$ follows $S$ under the dynamics of the social network, then $f(S`)$ follows $f(S)$ under the dynamics of the random boolean network. I then use this method to catalog all the basins of attraction for some simple two-strategy games (the prisoner`s dilemma and the stag hunt) played on a ring, drawing on the work of Wuensche and Lesser (1992).

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