  We consider games of strategic substitutes and strategic complements on
networks. We introduce two different evolutionary dynamics in order to refine
their multiplicity of equilibria, and we analyse the system through a mean
field approach. We find that for the best-shot game, taken as a model for
substitutes, a replicator-like dynamics does not lead to Nash equilibria,
whereas it leads to unique equilibria (full cooperation or full defection,
depending on the initial condition and the game parameter) for complements,
represented by a coordination game. On the other hand, when the dynamics
becomes more cognitively demanding in the form of a best response evolution,
predictions are always Nash equilibria (at least when individuals are fully
rational): For the best-shot game we find equilibria with a definite value of
the fraction of contributors, whereas for the coordination game symmetric
equilibria arise only for low or high initial fractions of cooperators. We also
extend our study by considering complex heterogeneous topologies, and show that
the nature of the selected equilibria does not change for the best-shot game.
However for coordination games we reveal an important difference, namely that
on infinitely large scale-free networks cooperation arises for any value of the
incentive to cooperate.
