Annals of Operations Research

, Volume 275, Issue 2, pp 259–279 | Cite as

Approximate solutions for expanding search games on general networks

  • Steve Alpern
  • Thomas LidbetterEmail author
Original Research


We study the classical problem introduced by R. Isaacs and S. Gal of minimizing the time to find a hidden point H on a network Q moving from a known starting point. Rather than adopting the traditional continuous unit speed path paradigm, we use the dynamic “expanding search” paradigm recently introduced by the authors. Here the regions S(t) that have been searched by time t are increasing from the starting point and have total length t. Roughly speaking the search follows a sequence of arcs \(a_i\) such that each one starts at some point of an earlier one. This type of search is often carried out by real life search teams in the hunt for missing persons, escaped convicts, terrorists or lost airplanes. The paper which introduced this type of search solved the adversarial problem (where H is hidden to maximize the time to be found) for the cases where Q is a tree or is 2-arc-connected. This paper’s main contribution is to give two strategy classes which can be used on any network and have expected search times which are within a factor close to 1 of the value of the game (minimax search time). These strategies classes are respectively optimal for trees and 2-arc-connected networks. We also solve the game for circle-and-spike networks, which can be considered as the simplest class of networks for which a solution was previously unknown.


Search games Zero-sum games Networks Defense and security 



Steve Alpern wishes to acknowledge support from the Air Force Office of Scientific Research [Grant FA9550-14-1-0049].


  1. Alpern, S. (2010). Search games on trees with asymmetric travel times. SIAM Journal on Control and Optimization, 48(8), 5547–5563.Google Scholar
  2. Alpern, S. (2011a). Find-and-fetch search on a tree. Operations Research, 59(5), 1258–1268.Google Scholar
  3. Alpern, S. (2011b). A new approach to Gal’s theory of search games on weakly Eulerian networks. Dynamic Games and Applications, 1(2), 209–219.Google Scholar
  4. Alpern, S., & Gal, S. (2003). The theory of search games and rendezvous (p. 319)., Kluwer international series in operations research and management science Boston: Kluwer.Google Scholar
  5. Alpern, S., & Lidbetter, T. (2013). Mining coal or finding terrorists: The expanding search paradigm. Operations Research, 61(2), 265–279.Google Scholar
  6. Alpern, S., & Lidbetter, T. (2014). Searching a variable speed network. Mathematics of Operations Research, 39(3), 697–711.Google Scholar
  7. Alpern, S., & Lidbetter, T. (2015). Optimal trade-off between speed and acuity when searching for a small object. Operations Research, 63(1), 122–133.Google Scholar
  8. Alpern, S., Morton, A., & Papadaki, K. (2011). Patrolling games. Operations Research, 59(5), 1246–1257.Google Scholar
  9. Anderson, E. J., & Aramendia, M. (1990). The search game on a network with immobile hider. Networks, 20(7), 817–844.Google Scholar
  10. Angelopoulos, S. (2015). Connections between contract-scheduling and ray-searching problems. In Proceedings of the 24th international joint conference on artificial intelligence (IJCAI).Google Scholar
  11. Baston, V., & Kikuta, K. (2015). Search games on a network with travelling and search costs. International Journal of Game Theory, 44(2), 347–365.Google Scholar
  12. Chapman, M., Tyson, G., McBurney, P., Luck, M., & Parsons, S. (2014). Playing hide-and-seek: An abstract game for cyber security. In Proceedings of the 1st international workshop on agents and cybersecurity (p. 3). ACM.Google Scholar
  13. Eckman, D. J., Maillart, L. M., & Schaefer, A. J. (2016). Optimal pinging frequencies in the search for an immobile beacon. IIE Transactions, 48(6), 489–500.Google Scholar
  14. Fokkink, R., Kikuta, K., & Ramsey, D. (2016). The search value of a set. Annals of Operations Research, 256(1), 63–73.Google Scholar
  15. Gal, S. (1979). Search games with mobile and immobile hider. SIAM Journal on Control and Optimization, 17(1), 99–122.Google Scholar
  16. Gal, S. (2000). On the optimality of a simple strategy for searching graphs. International Journal of Game Theory, 29(4), 533–542.Google Scholar
  17. Gal, S. (2005). Strategies for searching graphs. In M. Golumbic & I. Hartman (Eds.), Graph theory, combinatorics and algorithms (Vol. 34, pp. 189–214)., Operations research/computer science interfaces series New York: Springer.Google Scholar
  18. Garnaev, A. (2000). Search games and other applications of game theory (Vol. 485)., Lecture notes in economics and mathematical systems New York: Springer.Google Scholar
  19. Hohzaki, R. (2016). Search games: Literature and survey. Journal of the Operations Research Society of Japan, 59(1), 1–34.Google Scholar
  20. Isaacs, R. (1965). Differential games. New York: Wiley.Google Scholar
  21. Kolokoltsov, V. (2014). The evolutionary game of pressure (or interference), resistance and collaboration. arXiv preprint arXiv:1412.1269.
  22. Lidbetter, T. (2013a). Search games with multiple hidden objects. SIAM Journal on Control and Optimization, 51(4), 3056–3074.Google Scholar
  23. Lidbetter, T. (2013b). Search games for an immobile hider. In S. Alpern, et al. (Eds.), Search theory: A game theoretic perspective (pp. 17–27). New York: Springer.Google Scholar
  24. Lin, K. Y., Atkinson, M. P., Chung, T. H., & Glazebrook, K. D. (2013). A graph patrol problem with random attack times. Operations Research, 61(3), 694–710.Google Scholar
  25. Liu, D., Xiao, X., Li, H., & Wang, W. (2015). Historical evolution and benefit-cost explanation of periodical fluctuation in coal mine safety supervision: An evolutionary game analysis framework. European Journal of Operational Research, 243(3), 974–984.Google Scholar
  26. Morice, S., Pincebourde, S., Darboux, F., Kaiser, W., & Casas, J. (2013). Predator-prey pursuit-evasion games in structurally complex environments. Integrative and Comparative Biology, 53(5), 767–779.Google Scholar
  27. Pavlovic, L. (1993). Search game on an odd number of arcs with immobile hider. Yugoslav Journal of Operations Research, 3(1), 11–19.Google Scholar
  28. Reijnierse, J. H., & Potter, J. A. M. (1993). Search games with immobile hider. International Journal of Game Theory, 21(4), 385–394.Google Scholar
  29. Shechter, S. M., Ghassemi, F., Gocgun, Y., & Puterman, M. L. (2015). Technical note-trading off quick versus slow actions in optimal search. Operations Research, 63(2), 353–362.Google Scholar
  30. Wenk, C. J. (2015). Control sequencing in a game of identity pursuit–evasion. SIAM Journal on Control and Optimization, 53(4), 1815–1841.Google Scholar
  31. Zoroa, N., Fernández-Sáez, M. J., & Zoroa, P. (2013). Tools to manage search games on lattices. In S. Alpern, et al. (Eds.), Search theory: A game theoretic perspective (pp. 29–58). New York: Springer.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Operations Research and Management Science Group, Warwick Business SchoolUniversity of WarwickCoventryUK
  2. 2.Department of Management Science and Information SystemsRutgers Business SchoolNewarkUSA

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