  According to the standard imitation protocol, a less successful player adopts
the strategy of the more successful one faithfully for future success. This is
the cornerstone of evolutionary game theory that explores the vitality of
competing strategies in different social dilemma situations. In our present
work we explore the possible consequences of two slightly modified imitation
protocols that are exaggerated and gradual learning rules. In the former case a
learner does not only adopt, but also enlarges the strategy change for the hope
of a higher income. Similarly, in the latter case a cautious learner does not
adopt the alternative behavior precisely, but takes only a smaller step towards
the other's strategy during the updating process. Evidently, both scenarios
assume that the players' propensity to cooperate may vary gradually between
zero (always defect) and one (always cooperate) where these extreme states
represent the traditional two-strategy social dilemma. We have observed that
while the usage of exaggerated learning has no particular consequence on the
final state, but gradual learning can support cooperation significantly. The
latter protocol mitigates the invasion speeds of both main strategies, but the
decline of successful defector invasion is more significant, hence the biased
impact of the modified microscopic rule on invasion processes explains our
observations.
