The princess and Monster differential game

The Princess and the Monster is a search game posed by R. Isaacs. It is a game with a mobile hider and incomplete information. The search is in a set D with the payoff being the time until capture. For a wide class of sets D in the plane, the value of the game is determined for sufficiently small detection radius. The value of the game is approximately the time necessary for the monster to cover an area equal that of D. This result extends work of S. Gal and proves one of his conjectures. The game is also considered where D is a graph. It is shown that the expected time to capture may be arbitrarily many times the period required for the monster to search the length of the graph. This result settles another conjecture of Gal.