A Generalization of the Multi-Stage Search Allocation Game

  • Ryusuke Hohzaki
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 11)


This chapter deals with a multistage two-person zero-sum game called multistage search allocation game (MSSAG), in which a searcher and a target participate. The searcher distributes his searching resource in a search space to detect the target and the target moves under an energy constraint to evade the searcher. At the beginning of each stage, the searcher is informed of the target’s position and of his moving energy, and the target knows the amount of the searcher’s resource used so far. The game ends when the searcher detects the target. The payoff of the game is the probability of detecting the target during the search. There have been few search games on the MSSAG. We started the discussion on the MSSAG in a previous paper, where we assumed an exponential function for the detection probability of the target. In this chapter, we introduce a general function for the detection probability, aiming at more applicability. We first formulate the problem as a dynamic program. Then, by convex analysis, we propose a general method to solve the problem, and elucidate some general properties of optimal solutions.


Equilibrium Point Optimal Strategy Detection Probability Residual Energy Stochastic Game 
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  1. 1.
    Baston, V.J., Bostock, F.A.: A one-dimensional helicopter-submarine game. Naval Research Logistics 36, 479–490 (1989)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baston, V.J., Garnaev, A.Y.: A search game with a protector. Naval Research Logistics 47, 85–96 (2000)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Danskin, J.M.: The Theory of Max-Min. Springer, N.Y. (1967)MATHGoogle Scholar
  4. 4.
    Danskin, J.M.: A helicopter versus submarine search game. Operations Research 16, 509-517 (1968)CrossRefGoogle Scholar
  5. 5.
    Eagle, J.N., Washburn, A.R.: Cumulative search-evasion games. Naval Research Logistics 38, 495–510 (1991)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gal, S.: Search Games. Academic, N.Y. (1980)MATHGoogle Scholar
  7. 7.
    Garnaev, A.Y.: On a Ruckle problem in discrete games of ambush. Naval Research Logistics 44, 353–364 (1997)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Garnaev, A.Y.: Search Games and Other Applications of Game Theory. Springer, Tokyo (2000)MATHGoogle Scholar
  9. 9.
    Hohzaki, R.: A search allocation game. European Journal of Operational Research 172, 101–119 (2006)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hohzaki, R.: A multi-stage searh allocation game with the payoff of detection probability. Journal of the Operations Research Society of Japan 50, 178–200 (2007)MATHMathSciNetGoogle Scholar
  11. 11.
    Hohzaki, R., Iida, K.: A search game with reward criterion. Journal of the Operations Research Society of Japan 41, 629–642 (1998)MATHMathSciNetGoogle Scholar
  12. 12.
    Hohzaki, R., Iida, K.: A solution for a two-person zero-sum game with a concave payoff function. In: Takahashi, W., Tanaka, T. (eds.) Nonlinear Analysis and Convex Analysis, pp. 157–166. World Science Publishing, London (1999)Google Scholar
  13. 13.
    Hohzaki, R., Iida, K.: A search game when a search path is given. European Journal of Operational Research 124, 114–124 (2000)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hohzaki, R., Iida, K., Komiya, T.: Discrete search allocation game with energy constraints. Journal of the Operations Research Society of Japan 45, 93–108 (2002)MATHMathSciNetGoogle Scholar
  15. 15.
    Iida, K., Hohzaki, R., Furui, S.: A search game for a mobile target with the conditionally deterministic motion defined by paths. Journal of the Operations Research Society of Japan 39, 501–511 (1996)MATHMathSciNetGoogle Scholar
  16. 16.
    Iida, K., Hohzaki, R., Sato, K.: Hide-and-search game with the risk criterion. Journal of the Operations Research Society of Japan 37, 287–296 (1994)MATHMathSciNetGoogle Scholar
  17. 17.
    Koopman, B.O.: Search and Screening. pp. 221–227. Pergamon, (1980)Google Scholar
  18. 18.
    Meinardi, J.J.: A sequentially compounded search game. In: Theory of Games: Techniquea and Applications, pp. 285–299. The English Universities Press, London (1964)Google Scholar
  19. 19.
    Nakai, T.: A sequential evasion-search game with a goal. Journal of the Operations Research Society of Japan 29, 113–122 (1986)MATHMathSciNetGoogle Scholar
  20. 20.
    Nakai, T.: Search models with continuous effort under various criteria. Journal of the Operations Research Society of Japan 31, 335–351 (1988)MATHMathSciNetGoogle Scholar
  21. 21.
    Owen, G.: Game Theory. Academic, N.Y. (1995)Google Scholar
  22. 22.
    Ruckle, W.H.: Ambushing random walk II: continuous models. Operations Research 29, 108–120 (1981)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Shapley, L.S.: Stochastic games. Proceedings of the National Academy of Sciences of the U.S.A. 39, 1095–1100 (1953)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Stone, L.D.: Theory of Optimal Search. pp. 136–178. Academic, N.Y. (1975)Google Scholar
  25. 25.
    Washburn, A.R.: Search-evasion game in a fixed region. Operations Research 28, 1290–1298 (1980)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Washburn, A.R., Hohzaki, R.: The diesel submarine flaming datum problem. Military Operations Research 6, 19–33 (2001)Google Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Computer ScienceNational Defense AcademyYokosukaJapan

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