A search game with a protector

A classic problem in Search Theory is one in which a searcher allocates res ources to the points of the integer interval [1, n] in an attempt to find a n object which has been hidden in them using a known probability function. In this paper we consider a modification of this problem in which there is a protector who can also allocate resources to the points; allocating these resources makes it more difficult for the searcher to find an object. We m odel the situation as a two-person non-zero-sum game so that we can take in to account the fact that using resources can be costly. It is shown that th is game has a unique Nash equilibrium when the searcher's probability of fi nding an object located at point i is of the form (1 - exp(-lambda(i)x(i))) exp(-mu(i)y(i)) when the starcher and protector allocate resources x(i) an d y(i) respectively to point i. An algorithm to find this Nash equilibrium is given. (C) 2000 John Wiley & Sons, Inc.