Find-and-Fetch Search on a Tree

We introduce a new type of search game called the “find-and-fetch” game F ( Q , O ). The Hider simply picks any point H in the network Q . The Searcher starts at time zero at a given point O of Q , moving at unit speed until he reaches H (finds the Hider). Then he returns at a given speed (rho) along the shortest path back to O , arriving at time R , the payoff. This models the problem faced in many types of search, including search-and-rescue problems and foraging problems of animals (where food must be found and returned to the lair). When Q is a binary tree, we derive optimal probabilities for the Searcher to branch at the nodes. These probabilities give a positive bias towards searching longer branches first. We show that the minimax value of the return time R (the game value of F ( Q , O )) is (mu) + D /(rho), where (mu) is the total length of Q and D is the mean distance from the root O to the leaves (terminal nodes) of Q , where the mean is taken with respect to what is known as the equal branch density distribution. As (rho) goes to infinity, our problem reduces to the search game model where the payoff is simply the time to reach the Hider, and our results tend to those obtained by Gal [Gal, S. 1979. Search games with mobile and immobile hider. SIAM J. Control Optim. 17 (1) 99--122] and Anderson and Gal [Anderson, E. J., S. Gal. 1990. Search in a maze. Probab. Engrg. Inform. Sci. 4 (3) 311--318] for that model. We also apply our return time formula (mu) + D /(rho) to determine the ideal location for the root (lair or rescue center) O , assuming it can be moved. In the traditional “find only” model, the location of O does not matter.