  Spatial games are crucial for understanding patterns of cooperation in nature
(and to some extent society). They are known to be more sensitive to local
symmetries than e.g. spin models. This paper concerns the evolution of the
prisoner's dilemma game on regular lattices with three different types of
neighborhoods -- the von Neumann-, Moore-, and kagome types. We investigate two
kinds of dynamics for the players to update their strategies (that can be
unconditional cooperator or defector). Depending on the payoff difference, an
individual can adopt the strategy of a random neighbor (a voter-model-like
dynamics, VMLD), or impose its strategy on a random neighbor, i.e.,
invasion-like dynamics (IPLD). In particular, we focus on the effects of noise,
in combination with the strategy dynamics, on the evolution of cooperation. We
find that VMLD, compared to IPLD, better supports the spreading and sustaining
of cooperation. We see that noise has nontrivial effects on the evolution of
cooperation: maximum cooperation density can be realized either at a medium
noise level, in the limit of zero noise, or in both these regions. The
temptation to defect and the local interaction structure determine the outcome.
Especially, in the low noise limit, the local interaction plays a crucial role
in determining the fate of cooperators. We elucidate these both by numerical
simulations and mean-field cluster approximation methods.
