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RUTCOR Colloquia - February 28, 2007
Speaker: Andrey Garnaev
Affiliation: Faculty of Applied Mathematics and Control Processes, St Petersburg State University
Title: Find a Hidden Treasure
Time: 1:30 - 2:30 PM
Location: RUTCOR Building - Room 139, Rutgers University, Busch Campus, Piscataway, NJ
Abstract:
In this report a game modeling the situation where two parties try to find a
hidden treasure or a sunk ship is studied. This scenario is modeled by a
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 \in [1,n] is characterized by a detection
parameter \lambda_i (\mu_i) for Player 1 (Player 2) such that p_i(1 -
exp(-\lambda_i 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 \in (0,1) is the probability that the object is
hidden at point i. Player 1 (Player 2) undertakes search by allocating the
total amount effort X (Y). The payoff for Player 1 (Player 2) is 1 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 \leq 1. The Nash equilibrium of this game
is found and numerical examples are given.
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