Hide-and-seek games on a tree to which Eulerian networks are attached
We analyze the hide-and-seek game Gamma(G) on certain networks G. The hider picks a hiding point y in G and the searcher picks a unit speed path S(t) in G, starting at any point S(0). The payoff in this zero-sum game is the capture time T = T(S, y) = mini t : S(t) = y). Such games have been studied before, but mainly with the simplifying assumption that the searcher's starting point S(0) is specified and known to the hider. We call a network partly Eulerian if it consists of a tree (of length a and radius r) to which a finite number of disjoint Eulerian networks (of total length b) are attached, each at a single point. We show that for such networks, a strategy consisting equiprobably of a minimal (Chinese Postman) covering path and its reverse path is optimal for the searcher, while the optimal hider strategy is to assume that the searcher must start at the center of the tree, and to optimize in that (known) game. The value of the game Gamma(G) is a + b/2 - r. This simplifies and extends a similar result of Dagan and Gal for search games on trees. (C) 2008 Wiley Periodicals, Inc.
network zero-sum search game