The search value of a network

Abstract

Let Q be a connected network with a distinguished (starting) point qo, whose arc lengths sum to one. We associate with Q a "search value" V ( Q ) representing the expected time needed for a searcher, starting at qo and moving at unit speed, to find a moving hider. We assume neither sees the other until they meet. We demonstrate that the "figure-eight" network, consisting of two equal loops joined at a central starting point, has a search value not exceeding #. This contradicts a conjecture of Gal that the search value of any network is at least 1. In the other direction, we show that V ( Q ) S 6kD for a network with k edges and diameter D.

On Search Games That Include Ambush

“…For instance, the game that we consider here is similar to the hunter and rabbit game on a graph that is studied in [1]. In our paper, we study the expected capture time, or search value [3] …”

“…This game on a graph has remained largely unsolved, contrary to the game in a domain, which was solved by Gal [17]. In the 1980s Alpern and Asic [3,4] solved the game on a graph with three unit length arcs connecting two nodes. This is the only nontrivial graph for which the game has been solved up to now.…”

“…For instance, the game that we consider here is similar to the hunter and rabbit game on a graph that is studied in [1]. In our paper, we study the expected capture time, or search value [3], which jump to a different location. This type of run-or-hide tactic of the prey animal has also been observed in lizards [25].…”

“…The noisy ambush game is a search game in which we ignore the geometry of the search space. In the introduction we mentioned that our ambush game is motivated by a search game on a star with k rays (or a k-leafed clover) with k going to infinity, which is a game that was proposed in [3] and in [7,Remark 4.15] and that has been studied in [1]. The noisy search game on a star with k rays has been studied by Timmer in [30].…”

The “Princess and Monster” Game on an Interval

“…Recall that a strategy is ε-optimal if the expected payoff is at least V − ε against any strategy of the opponent. Upper bounds on V = V (Q) in terms of the structure of Q (and hence the finiteness of V ) are derived in [3]. For general networks Q, it is sometimes advantageous for the Searcher to wait for a while at a node, a so-called ambush strategy, and these games are known to be difficult -none have been solved.…”