  Cultures around the world show varying levels of conservatism. While
maintaining traditional ideas prevents wrong ones from being embraced, it also
slows or prevents adaptation to new times. Without exploration there can be no
improvement, but often this effort is wasted as it fails to produce better
results, making it better to exploit the best known option. This tension is
known as the exploration/exploitation issue, and it occurs at the individual
and group levels, whenever decisions are made. As such, it is has been
investigated across many disciplines. In this work, we investigate the balance
between exploration and exploitation in changing environments by thinking of
exploration as mutation in a trait space with a varying fitness function.
Specifically, we study how exploration rates evolves by applying adaptive
dynamics to the replicator-mutator equation, under two types of fitness
functions. For the first, payoffs are accrued from playing a two-player,
two-action symmetric game, we consider representatives of all games in this
class and find exploration rates often evolve downwards, but can also undergo
neutral selection as well. Second, we study time dependent fitness with a
function having a single oscillating peak. By increasing the period, we see a
jump in the optimal exploration rate, which then decreases towards zero. These
results establish several possible evolutionary scenarios for exploration
rates, providing insight into many applications, including why we can see such
diversity in rates of cultural change.
