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Lion and Man Game in Compact Spaces

  • Olga Yufereva
Article
  • 80 Downloads

Abstract

The pursuit-evasion game with two persons is considered. Both players are moving in a metric space, have equal maximum speeds and complete information about the location of each other. We study the sufficient conditions for a capture (with a positive capture radius). We assume that Lion wins if he manages the capture independently of the initial positions of the players and the evader’s strategy. We prove that the discrete-time simple pursuit strategy is a Lion’s winning strategy in a compact geodesic space satisfying the betweenness property. In particular, it means that Lion wins in compact CAT(0)-spaces, Ptolemy spaces, Buseman convex spaces, or any geodesic space with convex metric. We also do not need to use such properties as finite dimension, smoothness, boundary regularity, or contractibility of the loops.

Keywords

Pursuit-evasion game Lion-and-man game Simple pursuit Betweenness Convex metrics 

Mathematics Subject Classification

91A23 91A24 49N75 53C22 

References

  1. 1.
    Alexander S, Bishop R, Ghrist R (2006) Pursuit and evasion in non-convex domains of arbitrary dimensions. In: Proc. robotics: science and systems conferenceGoogle Scholar
  2. 2.
    Alexander S, Bishop R, Ghrist R (2010) Total curvature and simple pursuit on domains of curvature bounded above. Geom Dedicata 149(1):275–290MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alonso L, Goldstein AS, Reingold EM (1992) Lion and man: upper and lower bounds. ORSA J Comput 4(4):447–452MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bačák M (2012) Note on a compactness characterization via a pursuit game. Geom Dedicata 160(1):195–197MathSciNetCrossRefGoogle Scholar
  5. 5.
    Barmak JA (2017) Lion and man in non-metric spaces. arXiv preprint arXiv:1703.01480
  6. 6.
    Beveridge A, Cai Y (2015) Two-dimensional pursuit-evasion in a compact domain with piecewise analytic boundary. arXiv preprint arXiv:1505.00297
  7. 7.
    Bollobás B, Leader I, Walters M (2012) Lion and man—can both win? Isr J Math 189(1):267–286MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bramson M, Burdzy K, Kendall WS (2014) Rubber bands, pursuit games and shy couplings. Proc Lond Math Soc 109(1):121–160MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bridson MR, Haefliger A (2011) Metric spaces of non-positive curvature, vol 319. Springer Science & Business Media, BerlinzbMATHGoogle Scholar
  10. 10.
    Chernous’ko F (1976) A problem of evasion from many pursuers. J Appl Math Mech 40(1):11–20MathSciNetCrossRefGoogle Scholar
  11. 11.
    Isaacs R (1965) Differential games, a mathematical theory with applications to optimization, control and warfare. Wiley, New YorkzbMATHGoogle Scholar
  12. 12.
    Isler V, Karnad N (2009) Lion and man game in the presence of a circular obstacle. In: IEEE/RSJ international conference on intelligent robots and systems, pp 5045–5050Google Scholar
  13. 13.
    Isler V, Noori N (2015) The lion and man game on convex terrains. Algorithmic Found Robot XI:443–460MathSciNetGoogle Scholar
  14. 14.
    Ivanov R, Ledyaev YS (1983) Optimality of the pursuit time in a differential game with several pursuers under simple motion. Proc Steklov Inst Math 158:93–103zbMATHGoogle Scholar
  15. 15.
    Kumkov SS, Le Ménec S, Patsko VS (2017) Zero-sum pursuit-evasion differential games with many objects: survey of publications. Dyn Games Appl 7(4):609–633MathSciNetCrossRefGoogle Scholar
  16. 16.
    Littlewood JE (1953) A mathematicians miscellany. Methuen & Co., Ltd., LondonzbMATHGoogle Scholar
  17. 17.
    López-Acedo G, Nicolae A, Pia̧tek B (2017) ‘Lion-Man’ and the fixed point property. arXiv preprint arXiv:1712.04005
  18. 18.
    Nicolae A (2013) Asymptotic behavior of averaged and firmly nonexpansive mappings in geodesic spaces. Nonlinear Anal Theory Methods Appl 87:102–115MathSciNetCrossRefGoogle Scholar
  19. 19.
    O’Kane JM, Stiffler NM (2012) Shortest paths for visibility-based pursuit-evasion. In: 2012 IEEE international conference on robotics and automation (ICRA). IEEE, pp 3997–4002Google Scholar
  20. 20.
    Papadopoulos A (2005) Metric spaces, convexity and nonpositive curvature, vol 6. European Mathematical SocietyGoogle Scholar
  21. 21.
    Petrosjan LA (1977) Differential pursuit games (Russian). Izdat Leningrad Univ, LeningradGoogle Scholar
  22. 22.
    Pontryagin LS (1966) On the theory of differential games. Russ Math Surv 21(4):193–246MathSciNetCrossRefGoogle Scholar
  23. 23.
    Sgall J (2001) Solution of David gale’s lion and man problem. Theor Comput Sci 259(1):663–670MathSciNetCrossRefGoogle Scholar
  24. 24.
    Tovar B, LaValle SM (2008) Visibility-based pursuit-evasion with bounded speed. Int J Robot Res 27(11–12):1350–1360CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018
Corrected publication October/2018

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and Mechanics UrB RASEkaterinburgRussia
  2. 2.Chair of Applied Mathematics and Mechanics, Institute of Mathematics and Computer ScienceUral Federal UniversityEkaterinburgRussia

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