Steven Alpern

Rendezvous Search: A Personal Perspective

The rendezvous-search problem was posed by the author 25years ago. In its basic form, it asks how two unit speed players can find each other in least expected time, when randomly placed in a known dark region. The problem received little attention until about 10 years ago. This article surveys the rapid progress that has been made since then, and also presents some new results.

Spatial dispersion as a dynamic coordination problem

Following Schelling (1960), coordination problems have mainly been considered in a context where agents can achieve a common goal (e.g., rendezvous) only by taking common actions. Dynamic versions of this problem have been studied by Crawford and Haller (1990), Ponssard (1994), and Kramarz (1996). This paper considers an alternative dynamic formulation in which the common goal (dispersion) can only be achieved by agents taking distinct actions. The goal of spatial dispersion has been studied in static models of habitat selection, location or congestion games, and network analysis. Our results show how this goal can be achieved gradually, by indistinguishable non-communicating agents, in a dynamic setting.

Ambush frequency should increase over time during optimal predator search for prey

We advance and apply the mathematical theory of search games to model the problem faced by a predator searching for prey. Two search modes are available: ambush and cruising search. Some species can adopt either mode, with their choice at a given time traditionally explained in terms of varying habitat and physiological conditions. We present an additional explanation of the observed predator alternation between these search modes, which is based on the dynamical nature of the search game they are playing: the possibility of ambush decreases the propensity of the prey to frequently change locations and thereby renders it more susceptible to the systematic cruising search portion of the strategy. This heuristic explanation is supported by showing that in a new idealized search game where the predator is allowed to ambush or search at any time, and the prey can change locations at intermittent times, optimal predator play requires an alternation (or mixture) over time of ambush and cruise search. Thus, our game is an extension of the well-studied ‘Princess and Monster’ search game. Search games are zero sum games, where the pay-off is the capture time and neither the Searcher nor the Hider knows the location of the other. We are able to determine the optimal mixture of the search modes when the predator uses a mixture which is constant over time, and also to determine how the mode mixture changes over time when dynamic strategies are allowed (the ambush probability increases over time). In particular, we establish the ‘square root law of search predation’: the optimal proportion of active search equals the square root of the fraction of the region that has not yet been explored.

Search Games on Trees with Asymmetric Travel Times

A point

The “Princess and Monster” Game on an Interval

H

Find-and-Fetch Search on a Tree

is hidden in a rooted tree

Pure Strategy Asymmetric Rendezvous on the Line with an Unknown Initial Distance

Q

Rendezvous on a Planar Lattice

which is endowed with asymmetric distances (travel times) between nodes. We determine the randomized search strategy, starting from the root, which minimizes the expected time to reach

Asymmetric rendezvous search on the circle

H

On Ruckle's Conjecture on Accumulation Games

, in the worst case. This is equivalent to a zero-sum search game