  When facing many options, we narrow down our focus to very few of them.
Although behaviors like this can be a sign of heuristics, they can actually be
optimal under limited cognitive resources. Here we study the problem of how to
optimally allocate limited sampling time to multiple options, modelled as
accumulators of noisy evidence, to determine the most profitable one. We show
that the effective sampling capacity of an agent increases with both available
time and the discriminability of the options, and optimal policies undergo a
sharp transition as a function of it. For small capacity, it is best to
allocate time evenly to exactly five options and to ignore all the others,
regardless of the prior distribution of rewards. For large capacities, the
optimal number of sampled accumulators grows sub-linearly, closely following a
power law for a wide variety of priors. We find that allocating equal times to
the sampled accumulators is better than using uneven time allocations. Our work
highlights that multi-alternative decisions are endowed with breadth-depth
tradeoffs, demonstrates how their optimal solutions depend on the amount of
limited resources and the variability of the environment, and shows that
narrowing down to a handful of options is always optimal for small capacities.
