  On rugged fitness landscapes where sign epistasis is common, adaptation can
often involve either individually beneficial "uphill" mutations or more complex
mutational trajectories involving fitness valleys or plateaus. The dynamics of
the evolutionary process determine the probability that evolution will take any
specific path among a variety of competing possible trajectories. Understanding
this evolutionary choice is essential if we are to understand the outcomes and
predictability of adaptation on rugged landscapes. We present a simple model to
analyze the probability that evolution will eschew immediately uphill paths in
favor of crossing fitness valleys or plateaus that lead to higher fitness but
less accessible genotypes. We calculate how this probability depends on the
population size, mutation rates, and relevant selection pressures, and compare
our analytical results to Wright-Fisher simulations. We find that the
probability of valley crossing depends nonmonotonically on population size:
intermediate size populations are most likely to follow a "greedy" strategy of
acquiring immediately beneficial mutations even if they lead to evolutionary
dead ends, while larger and smaller populations are more likely to cross
fitness valleys to reach distant advantageous genotypes. We explicitly identify
the boundaries between these different regimes in terms of the relevant
evolutionary parameters. Above a certain threshold population size, we show
that the degree of evolutionary "foresight" depends only on a single simple
combination of the relevant parameters.
