The breadth-depth dilemma in a finite capacity model of decision-making

Abstract

23 24 25 26 Acknowledgements: This work is supported by the Howard Hughes Medical Institute 27 (HHMI, ref 55008742), MINECO (Spain; BFU2017-85936-P) and ICREA Academia 28 (2016) to R.M.-B.; by NIH (DA037229) to B.Y.H.; by a scholar award from the James 29 S. McDonnell Foundation (grant# 220020462) to J.D.Abstract 34 Decision-makers are often faced with limited information about the outcomes of 35 their choices. Current formalizations of uncertain choice, such as the explore-exploit 36 dilemma, do not apply well to decisions in which search capacity can be allocated to each 37 option in variable amounts. Such choices confront decision-makers with the need to 38 tradeoff between breadth -allocating a small amount of capacity to each of many options 39 -and depth -focusing capacity on a few options. We formalize the breadth-depth 40 dilemma through a finite sample capacity model. We find that, if capacity is smaller than 41 4-7 samples, it is optimal to draw one sample per alternative, favoring breadth. However, 42 for larger capacities, a sharp transition is observed, and it becomes best to deeply sample 43 a very small fraction of alternatives, that decreases with the square root of capacity. Thus, 44 ignoring most options, even when capacity is large enough to shallowly sample all of 45 them, reflects a signature of optimal behavior. Our results also provide a rich casuistic 46 for metareasoning in multi-alternative decisions with bounded capacity. 47 48 49 50 51 52 53 54 55 56 57The breadth-depth (BD) dilemma is a ubiquitous problem in decision-making. 58Consider the example of going to graduate school, where one can enroll in many courses 59 in many topics. Let us assume that the goal is to determine the single one topic that is 60 most relevant, the one that will grant us a job. Should I enroll in few courses in many 61 topics -breadth search-at the risk of not learning enough about any topic to tell which 62 one is the best? Or should I enroll in many courses in very few topics -depth search-at 63 the risk of missing the really exciting topic for the future? One crucial element of this 64 type of decision is that the allocation of resources (time, in this case) needs to be done in 65 advance, before feedback is received (before classes start). Also, once decided, the 66 strategy cannot be changed on the fly, as doing so would be very costly. The BD dilemma 67 is popular in tree search algorithms (Horowitz and Sahni, 1978;Korf, 1985) and in 68 optimizing menu designs (Miller, 1981). It is also one faced by humans and other 69 foragers in many situations, as when we plan, schedule, or invest with finite resources. It 70 is remarkable that the bulk of research on the BD has been in fields outside of psychology 71 (e.g. (Halpert, 1958;Schwartz et al., 2009;Turner et al., 2002). We believe that one 72 reason is the lack of standard sets of tools for thinking about the problem and separating 73 it from other dilemmas. 74Many features of the BD dilemma warrant its study in isolation. First, BD 75 decisions are...