Heuristics and optimal solutions to the breadth–depth dilemma

Significance

From choosing among the many courses offered in graduate school to dividing budget into research programs, the breadth–depth is a commonplace dilemma that arises when finite resources (e.g., time, money, cognitive capabilities) need to be allocated among a large range of alternatives. For such problems, decision makers need to trade off breadth—allocating little capacity to each of many alternatives—and depth—focusing capacity on a few options. We found that little available capacity (less than 10 samples for search) promotes allocating resources broadly, and thus breadth search is favored. Increased capacity results in an abrupt transition toward favoring a balance between breadth and depth. We finally describe a rich casuistic and heuristics for metareasoning with finite resources.

Abstract

In multialternative risky choice, we are often faced with the opportunity to allocate our limited information-gathering capacity between several options before receiving feedback. In such cases, we face a natural trade-off between breadth—spreading our capacity across many options—and depth —gaining more information about a smaller number of options. Despite its broad relevance to daily life, including in many naturalistic foraging situations, the optimal strategy in the breadth–depth trade-off has not been delineated. Here, we formalize the breadth–depth dilemma through a finite-sample capacity model. We find that, if capacity is small (∼10 samples), it is optimal to draw one sample per alternative, favoring breadth. However, for larger capacities, a sharp transition is observed, and it becomes best to deeply sample a very small fraction of alternatives, which roughly decreases with the square root of capacity. Thus, ignoring most options, even when capacity is large enough to shallowly sample all of them, is a signature of optimal behavior. Our results also provide a rich casuistic for metareasoning in multialternative decisions with bounded capacity using close-to-optimal heuristics.