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Technical note: Find a hidden “treasure”

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  • A. Garnaev

Abstract

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 ∈ [1,n] is characterized by a detection parameter λi (μi) for Player 1 (Player 2) such that pi(1 − exp(−λixi)) (pi(1 − exp(−μiyi))) is the probability that Player 1 (Player 2) discovers the hidden object with amount of search effort xi (yi) applied at point i where pi ∈ (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 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 q1 and Player 2 gets q2 where q1 + q2 ≤ 1. The Nash equilibrium of this game is found and numerical examples are given. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007

Suggested Citation

  • A. Garnaev, 2007. "Technical note: Find a hidden “treasure”," Naval Research Logistics (NRL), John Wiley & Sons, vol. 54(1), pages 109-114, February.
    • Handle: RePEc:wly:navres:v:54:y:2007:i:1:p:109-114
    DOI: 10.1002/nav.20196
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    1. John S. Croucher, 1975. "Application of the fundamental theorem of games to an example concerning antiballistic missile defense," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 22(1), pages 197-203, March.
    2. Fershtman, Chaim & Rubinstein, Ariel, 1997. "A Simple Model of Equilibrium in Search Procedures," Journal of Economic Theory, Elsevier, vol. 72(2), pages 432-441, February.
    3. V.J. Baston & A.Y. Garnaev, 2000. "A search game with a protector," Naval Research Logistics (NRL), John Wiley & Sons, vol. 47(2), pages 85-96, March.
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