the theory of search games and rendezvous - steve alpern

[...]... dimension, of Q and the boundary of Q.) In order to simplify the presentation of the results, we shall generally consider the case in which both the maximal velocity of the searcher and the radius of detection are constants However, we shall also extend the results to the case where the maximal velocity of the searcher depends on his location and the radius of detection depends on the location of the hider... game theoretic aspects of search problems was of course Gal (1980), but these are also considered in Ruckle (1983a) and form the basis of the recent stimulating book of Garnaev (2000) This volume is the first to cover the new field of rendezvous search theory Frequently Used Notations Search Games Lebesgue measure of the search space Minimal length of a tour that covers the search space Rate of discovery... is the 4 BOOK I SEARCH GAMES point occupied by the hider at time t The next step in describing the search game is to present a cost function (the game-theoretic payoff to the maximizing hider) c(S, H), where S is a pure search strategy and H is a pure hiding strategy The cost c(S, H) has to represent the loss of the searcher (or the effort spent in searching) if the searcher uses strategy S and the. .. updated edition of his book appeared in 1980.) The problem of finding the optimal distribution of effort spent in search is the main subject of the classic work of Stone (1989, 2nd ed.), Theory of Optimal Search , which was awarded the 1975 Lanchester Prize by the Operations Research Society of America Much of the early work on search theory surveyed by Dobbie (1968) was concerned with aspects other than... for all Define a search game in with an origin a detection radius and the same maximal velocitiesfor the searcher and the hider as in Then the value of satisfies and the optimal strategies of are obtained by applying the mapping to the trajectories in Q and changing the time scale by afactor of The proof is based on the simple observation that for any pair of trajectories S and H, in Q, the capture time... minimize T.) Of the two problems dealt with in the book, the area of Search Games (Book I) is the older These games stem in part from the The Princess and the Monster” games proposed by Rufus Isaacs (1965) in his well known book on Differential Games Beginning with the first search game with mobile hider to be solved (that on the circle, by Alpern (1974), Foreman (1974), and Zelinkin (1972)), and the subsequent... value and be the search by allowing the searcher to choose his starting point Then the The extension of the preceding discussion to search games with more than one searcher is presented in the following result Corollary 3.6 Consider a search game with one immobile hider and J searchers, with the j -th searcher having a maximal velocity Assume that all the searchers cooperate in order to discover the. .. BOOK I SEARCH GAMES The unrestricted game is similar to a discrete search game in which Q consists of n cells of equal size We now formulate and solve a more general discrete version of the search game In the game to be considered, Q consists of n cells of sizes and the measure of Q is defined as It is assumed that the maximal rate of discovery of the searcher is so that it takes him units of time... A pure search strategy (a search trajectory) The set of all admissible search trajectories A mixed search strategy Capture time Time parameter Value of the hiding strategy Value of the search strategy Minimal value obtained by a pure search strategy (the “pure value” Value of the search game Maximal velocity of the hider A point in the search space Integer part xiv FREQUENTLY USED NOTATIONS Rendezvous. .. Thomas, and others Particular interest has been paid to some discrete time rendezvous models, which have a separate history going back to the original papers of Crawford and Haller (1990) on coordination games in the economics literature, and Anderson and Weber (1990) in a search theory context Much of this work is surveyed in the paper of Alpern (2002a) An extensive introduction to the field of Rendezvous . wishes to minimize T.) Of the two problems dealt with in the book, the area of Search Games (Book I) is the older. These games stem in part from the The Princess and the Monster” games proposed by. rendezvous search theory. xii PREFACE Frequently Used Notations Search Games Lebesgue measure of the search space Minimal length of a tour that covers the search space Rate of discovery of the searcher Cost. classic work of Stone (1989, 2nd ed.), Theory of Optimal Search , which was awarded the 1975 Lanchester Prize by the Operations Research Society of America. Much of the early work on search theory