  Evolution is based on the assumption that competing players update their
strategies to increase their individual payoffs. However, while the applied
updating method can be different, most of previous works proposed uniform
models where players use identical way to revise their strategies. In this work
we explore how imitation-based or learning attitude and innovation-based or
myopic best response attitude compete for space in a complex model where both
attitudes are available. In the absence of additional cost the best response
trait practically dominates the whole snow-drift game parameter space which is
in agreement with the average payoff difference of basic models. When
additional cost is involved then the imitation attitude can gradually invade
the whole parameter space but this transition happens in a highly nontrivial
way. However, the role of competing attitudes is reversed in the stag-hunt
parameter space where imitation is more successful in general. Interestingly, a
four-state solution can be observed for the latter game which is a consequence
of an emerging cyclic dominance between possible states. These phenomena can be
understood by analyzing the microscopic invasion processes, which reveals the
unequal propagation velocities of strategies and attitudes.
