On salesmen and tourists: Two-step optimization in deterministic foragers

Abstract

We explore a two-step optimization problem in random environments, the so-called restaurant-coffee shop problem, where a walker aims at visiting the nearest and better restaurant in an area and then move to the nearest and better coffee-shop. This is an extension of the Tourist Problem, a one-step optimization dynamics that can be viewed as a deterministic walk in a random medium. A certain amount of heterogeneity in the values of the resources to be visited causes the emergence of power-laws distributions for the steps performed by the walker, similarly to a Lévy flight. The fluctuations of the step lengths tend to decrease as a consequence of multiple-step planning, thus reducing the foraging uncertainty. We find that the first and second steps of each planned movement play very different roles in heterogeneous environments. The two-step process improves only slightly the foraging efficiency compared to the one-step optimization, at a much higher computational cost. We discuss the implications of these findings for animal and human mobility, in particular in relation to the computational effort that informed agents should deploy to solve search problems.