Technical note: Find a hidden "treasure"

This paper deals with a two searchers game and it investigates the problem of how the possibility of finding a hidden object simultaneously by players influences their behavior. Namely, we consider the following two-sided allocation non-zero-sum game on an integer interval [1, n]. Two teams (Player 1 and 2) want to find an immobile object (say, a treasure) hidden at one of n points. Each point i is an element of [1, n] is characterized by a detection parameter; lambda(i) (mu(i)) for Player 1 (Player 2) such that p(i) (1 - exp(-lambda x(i))) (p(i) (1 - exp(--mu(i)y(i)))) is the probability that Player 1 (Player 2) discovers the hidden object with amount of search effort x(i) (y(i)) applied at point i where p(i) is an element of (0, 1) is the probability that the object is hidden at point i. Player 1 (Player 2) undertakes the search by allocating the total amount of effort X(Y). The payoff for Player I (Player 2) is I if he detects the object but his opponent does not. If both players detect the object they can share it proportionally and even can pay some share to an umpire who takes care that the players do not cheat each other, namely Player 1 gets q(1) and Player 2 gets q(2) where q(1) + q(2) <= 1. The Nash equilibrium of this game is found and numerical examples are given.