Mining Coal or Finding Terrorists: The Expanding Search Paradigm

Abstract

We show how to optimize the search for a hidden object, terrorist, or simply Hider, located at a point H according to a known or unknown distribution on a rooted network Q. We modify the traditional 'pathwise search' approach to a more general notion of 'expanding search'. When the Hider is restricted to the nodes of Q; an expanding search S consists of an ordering (a 1 ; a 2 ; : : : ) of the arcs of a spanning subtree such that the root node is in a 1 and every arc a i is adjacent to a previous arc a j ; j < i: If a k contains H; the search time T is (a 1 ) + + (a k ) ; where is length measure on Q: For more general distributions ; an expanding search S is described by the nested family of connected sets S (t) which specify the area of Q that has been covered by time t: S (0) is the root, (S (t)) = t; and T = min ft : H 2 S (t)g : For a known Hider distribution on a tree Q; the expected time minimizing strategy S begins with the rooted subtree Q 0 maximizing the 'density' (Q 0 ) = (Q 0 ) : (For arbitrary networks, we use this criterion on all spanning subtrees.) The search S can be interpreted as the optimal method of mining known coal seams, when the time to move miners or machines is negligible compared to digging time. When the Hider distribution is unknown, we consider the zero-sum search game where the Hider picks H; the Searcher S and the payo¤ is T: For trees Q; the value is V = ( (Q) + D) =2, where D is a mean distance from root to leaf nodes. If Q is 2 arc connected, V = (Q) =2. Applications and interpretations of the expanding search paradigm are given, particularly to multiple agent search.

Search Games with Multiple Hidden Objects

“…This bound for the expanding search game with one hidden object can be found in [8], and our results therefore both generalize this bound. Downloaded 11/25/15 to 158.143.197.114.…”

“…For k = 1 the solution of the game Λ(Q, k) can be found in [8] for general tree networks Q, and a formula is given for the value. This formula is V = (μ + D)/2, where μ is the total length of the network and D is an average of the distances of the root to the leaf notes, weighted with respect to the optimal Hider distribution on the leaf nodes.…”

“…The optimal Hider strategy given in [8] is also the same, but the optimal Searcher strategy is different. It mixes between 2 n−1 pure strategies, whereas our optimal strategy mixes between all n!…”

The search value of a set

“…The expanding search game on edge-weighted networks was introduced by Alpern and Lidbetter in Alpern and Lidbetter (2013). The game on a weighted tree was completely solved in that paper.…”

“…In this paper we want to study search games that are not necessarily placed on a network, and therefore we follow an axiomatic approach, imposing general conditions only. Our results are motivated by a recent study of Alpern and Lidbetter on expanding search Alpern and Lidbetter (2013).…”