  We analyse the strategy equilibrium of dilemma games considering a payoff
matrix affected by small and random perturbations on the off-diagonal.
  Notably, a recent work [1] reported that, while cooperation is sustained by
perturbations acting on the main diagonal, a less clear scenario emerges when
perturbations act on the off-diagonal. Thus, the second case represents the
core of this investigation, aimed at completing the description of the effects
that payoff perturbations have on the dynamics of evolutionary games. Our
results, achieved by analysing the proposed model under a variety of
configurations, as different update rules, suggest that off-diagonal
perturbations actually constitute a non-trivial form of noise. In particular,
the most interesting effects are detected near the phase transition, as
perturbations tend to move the strategy distribution towards non-ordered states
of equilibrium, supporting cooperation when defection is pervading the
population, and supporting defection in the opposite case. To conclude, we
identified a form of noise that, under controlled conditions, could be used to
enhance cooperation, and greatly delay its extinction.
