Skip to main content
SpringerLink
Log in

Safe navigation in adversarial environments

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

This work deals with the problem of navigation while avoiding detection by a mobile adversary, featuring adversarial modeling. In this problem, an evading agent is placed on a graph, where one or more nodes are defined as safehouses. The agent’s goal is to find a path from its current location to a safehouse, while minimizing the probability of meeting a mobile adversarial agent at a node along its path (i.e., being captured). We examine several models of this problem, where each one has different assumptions on what the agents know about their opponent, all using a framework for computing node utility, introduced herein. Using risk attitudes for computing the utility values, their impact on the constructed strategies is analyzed both theoretically and empirically. Furthermore, we allow the agents to use information gained along their movement, in order to efficiently update their motion strategies on-the-fly. Theoretical and empirical analysis shows the importance of using this information and these on-the-fly strategy updates.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adler, M., Räcke, H., Sivadasan, N., Sohler, C., Vöcking, B.: Randomized pursuit-evasion in graphs. Comb. Probab. Comput. 12(03), 225–244 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aigner, M., Fromme, M.: A game of cops and robbers. Discret. Appl. Math. 8(1), 1–12 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alexopoulos, A., Schmidt, T., Badreddin, E.: Cooperative pursue in pursuit-evasion games with unmanned aerial vehicles. In: Proceedings of the International Conference on Intelligent Robots and Systems (IROS). IEEE (2015)

  4. Alpern, S., Fokkink, R., Gal, S., Timmer, M.: On search games that include ambush. SIAM J. Control. Optim. 51(6), 4544–4556 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alspach, B.: Searching and sweeping graphs: a brief survey. Le matematiche 59 (1, 2), 5–37 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Barrett, S., Stone, P., Kraus, S.: Empirical evaluation of ad hoc teamwork in the pursuit domain. In: Proceedings of the International Conference on Autonomous Agents and Multiagent Systems (AAMAS), vol. 2, pp. 567–574. International Foundation for Autonomous Agents and Multiagent Systems (2011)

  7. Basilico, N., De Nittis, G., Gatti, N.: A security game combining patrolling and alarm–triggered responses under spatial and detection uncertainties. In: Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, pp. 397–403. AAAI Press (2016)

  8. Basilico, N., Gatti, N., Amigoni, F.: Leader-follower strategies for robotic patrolling in environments with arbitrary topologies. In: Proceedings of the International Conference on Autonomous Agents and Multiagent Systems(AAMAS), vol. 1, pp. 57–64. International Foundation for Autonomous Agents and Multiagent Systems (2009)

  9. Bhadauria, D., Klein, K., Isler, V., Suri, S.: Capturing an evader in polygonal environments with obstacles: The full visibility case. The International Journal of Robotics Research 31(10), 1176–1189 (2012)

    Article  Google Scholar 

  10. Bishop, C.: Pattern Recognition and Machine Learning (Information Science and Statistics). Springer, New York (2007)

    Google Scholar 

  11. Boidot, E., Marzuoli, A., Feron, E.: Optimal navigation policy for an autonomous agent operating in adversarial environments. In: IEEE International Conference on Robotics and Automation (ICRA), pp. 3154–3160. IEEE (2016)

  12. Borie, R.B., Tovey, C.A., Koenig, S.: Algorithms and complexity results for pursuit-evasion problems. In: Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI), vol. 9, pp. 59–66 (2009)

  13. Brémaud, P.: Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues, vol. 31. Springer, New Yoork (2013)

    MATH  Google Scholar 

  14. Chen, H., Zhang, F.: The expected hitting times for finite markov chains. Linear Algebra Appl. 428(11-12), 2730–2749 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chung, T.H., Hollinger, G.A., Isler, V.: Search and pursuit-evasion in mobile robotics. Auton. Robot. 31(4), 299–316 (2011)

    Article  Google Scholar 

  16. Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. De Alfaro, L., Henzinger, T.A., Kupferman, O.: Concurrent reachability games. Theor. Comput. Sci. 386(3), 188–217 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dereniowski, D., Dyer, D., Tifenbach, R.M., Yang, B.: Zero-visibility cops and robber game on a graph. In: Frontiers in Algorithmics and Algorithmic Aspects in Information and Management, pp. 175–186. Springer (2013)

  19. Foderaro, G., Raju, V., Ferrari, S.: A cell decomposition approach to online evasive path planning and the video game Ms. Pac-Man. In: IEEE International Symposium on Intelligent Control (ISIC), pp. 191–197 (2011)

  20. Fomin, F.V., Thilikos, D.M.: An annotated bibliography on guaranteed graph searching. Theor. Comput. Sci. 399(3), 236–245 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Geraerts, R., Schager, E.: Stealth-based path planning using corridor maps. In: Proceedings of Computer Animation and Social Agents (CASA) (2010)

  22. Gibbs, A., Su, F.: On choosing and bounding probability metrics. Int. Stat. Rev. 70(3), 419–435 (2002)

    Article  MATH  Google Scholar 

  23. Gmytrasiewicz, P.J., Doshi, P.: A framework for sequential planning in multi-agent settings. J. Artif. Intell. Res. (JAIR) 24, 49–79 (2005)

    MATH  Google Scholar 

  24. Göbel, F., Jagers, A.: Random walks on graphs. Stochastic Processes and their Applications 2(4), 311–336 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  25. Guibas, L.J., Latombe, J.C., LaValle, S.M., Lin, D., Motwani, R.: A visibility-based pursuit-evasion problem. Int. J. Comput. Geom. Appl. 9(4 & 5), 471–493 (1999)

    Article  MathSciNet  Google Scholar 

  26. Huang, H., Ding, J., Zhang, W., Tomlin, C.J.: Automation-assisted capture-the-flag: a differential game approach. IEEE Trans. Control Syst. Technol. 23 (3), 1014–1028 (2015)

    Article  Google Scholar 

  27. Jain, M., Conitzer, V., Tambe, M.: Security scheduling for real-world networks. In: Proceedings of the 2013 International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), pp. 215–222 (2013)

  28. Karlin, S.: A First Course in Stochastic Processes. Academic Press, New York (2014)

    Google Scholar 

  29. Latombe, J.C.: Robot Motion Planning. Kluwer Academic Publishers, Boston (1991)

    Book  MATH  Google Scholar 

  30. Littman, M.L.: Markov games as a framework for multi-agent reinforcement learning. In: Proceedings of the Eleventh International Conference on Machine Learning, vol. 157, pp. 157–163 (1994)

  31. Lubiw, A., Vosoughpour, H.: Visibility graphs, dismantlability, and the cops and robbers game. In: 26th Canadian conference on computational geometry (2014)

  32. Marzouqi, M., Jarvis, R.A.: Covert path planning for autonomous robot navigation in known environments. In: Proceedings of the Australasian Conference on Robotics and Automation, Citeseer, Brisbane (2003)

  33. Marzouqi, M., Jarvis, R.A.: Covert robotics: covert path planning in unknown environments. In: Proceedings of the Australasian Conference on Robotics and Automation, Citeseer, Brisbane (2003)

  34. Maskin, E., Tirole, J.: Markov perfect equilibrium, i: Observable actions. J. Econ. Theory 100(2), 191–219 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  35. McCarthy, S., Tambe, M., Kiekintveld, C., Gore, M.L., Killion, A.: Preventing illegal logging: simultaneous optimization of resource teams and tactics for security. In: Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, pp. 3880–3886 (2016)

  36. Nearchou, A.C.: Path planning of a mobile robot using genetic heuristics. Robotica 16(05), 575–588 (1998)

    Article  Google Scholar 

  37. Nowakowski, R., Winkler, P.: Vertex-to-vertex pursuit in a graph. Discret. Math. 43(2), 235–239 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  38. Örgen, P., Winstrand, M.: Minimizing mission risk in fuel constrained uav path planning. AIAA Journal of Guidance Control, and Dynamics 31(5), 1497–1500 (2008)

    Article  Google Scholar 

  39. Petres, C., Pailhas, Y., Patron, P., Petillot, Y., Evans, J., Lane, D.: Path planning for autonomous underwater vehicles. IEEE Trans. Robot. 23(2), 331–341 (2007)

    Article  Google Scholar 

  40. Raghavan, T.: Zero-sum two-person games. Handbook of Game Theory with Economic Applications 2, 735–768 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sgall, J.: Solution of david gale’s lion and man problem. Theor. Comput. Sci. 259(1-2), 663–670 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  42. Shoham, Y., Leyton-Brown, K.: Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. Cambridge University Press, Cambridge (2008)

    Book  MATH  Google Scholar 

  43. Stentz, A.: Optimal and efficient path planning for partially-known environments. In: Proceedings of IEEE International Conference on Robotics and Automation, pp. 3310–3317 (1994)

  44. Sutton, R.S., Barto, A.G.: Introduction to Reinforcement Learning. MIT Press, Cambridge (1998)

    Google Scholar 

  45. Tews, A., Matarić, M.J., Sukhatme, G.S.: Avoiding detection in a dynamic environment. In: Proceedings of the International Conference on Intelligent Robots and Systems (IROS), vol. 4, pp. 3773–3778 (2004)

  46. Tews, A., Sukhatme, G.S., Matarić, M.J.: A multi-robot approach to stealthy navigation in the presence of an observer. In: Proceedings of the International Conference on Robotics and Automation, pp. 2379–2385 (2004)

  47. Yehoshua, R., Agmon, N.: Adversarial modeling in the robotic coverage problem. In: Proceedings of International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS) (2015)

  48. Zhang, Y., An, B., Tran-Thanh, L., Wang, Z., Gan, J., Jennings, N.R.: Optimal escape interdiction on transportation networks. In: Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI) (2017)

Download references

Acknowledgments

This paper was supported in part by ISF grant 1337/15.

Author information

Authors and Affiliations

  1. Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel

    Ofri Keidar & Noa Agmon

Authors
  1. Ofri Keidar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Noa Agmon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Noa Agmon.

Appendix

Appendix

We shall now introduce the full proof of Theorem 4:

Theorem 4

Denote the actual probability distribution over C’s location at time t as \(P_{\mathrm {C}}^{(t)}\) (i.e., \(P_{\mathrm {C}}^{(t)}\) is 1 for C’s location and 0 for any other node). \(\tilde {P}_{\mathrm {C}}^{(t)}\) denotes R’s estimation for \(P_{\mathrm {C}}^{(t)}\) before updating it with the information received from \(V_{visible}(v_{\mathrm {R}}^t)\), while \(\hat {P}_{\mathrm {C}}^{(t)}\)denotes \(\tilde {P}_{\mathrm {C}}^{(t)}\) after being updated. Incorporating the information received from the visibility edges for updating \(\hat {P}_{\mathrm {C}}^{(t)}\), improves the accuracy of this estimation. Namely, \(H(\tilde {P}_{\mathrm {C}}^{(t)},P_{\mathrm {C}}^{(t)}) \geq H(\hat {P}_{\mathrm {C}}^{(t)},P_{\mathrm {C}}^{(t)})\).

Proof

As stated in Section 6.2, if \({\sum }_{v \in V_{visible}(v_{\mathrm {R}}^t)} \tilde {P}_{\mathrm {C}}^{(t)}[v] = 0\) then \(\hat {P}_{\mathrm {C}}^{(t)} = \tilde {P}_{\mathrm {C}}^{(t)}\). Hence, \(H(\tilde {P}_{\mathrm {C}}^{(t)},P_{\mathrm {C}}^{(t)}) = H(\hat {P}_{\mathrm {C}}^{(t)},P_{\mathrm {C}}^{(t)})\).

Now, assume \({\sum }_{v \in V_{visible}(v_{\mathrm {R}}^t)} \tilde {P}_{\mathrm {C}}^{(t)}[v] > 0\) (a sum of probabilities is either 0 or positive). If C has been observed (\(v_{\mathrm {C}}^{t} \in V_{visible}(v_{\mathrm {R}}^t)\)), then \(\hat {P}_{\mathrm {C}}^{(t)}\) = \(P_{\mathrm {C}}^{(t)}\).

Otherwise, \(v_{\mathrm {C}}^{t} \notin V_{visible}(v_{\mathrm {R}}^t)\). Let us compute the Hellinger distance between the actual distribution over C’s location to R’ estimated distribution prior being updated, \(H\left (\tilde {P}_{\mathrm {C}}^{(t)},P_{\mathrm {C}}^{(t)}\right )\):

$$\begin{array}{@{}rcl@{}} H\left( \tilde{P}_{\mathrm{C}}^{(t)},P_{\mathrm{C}}^{(t)}\right)\!\! &=&\!\! \sqrt{ \sum\limits_{v \in V} \left( \sqrt{\tilde{P}_{\mathrm{C}}^{(t)}[v]} - \sqrt{P_{\mathrm{C}}^{(t)}[v]}\right)^{2} } \\ &=&\!\left( \sum\limits_{v \in V \setminus \left( V_{visible}(v_{\mathrm{R}}^t) \cup \{v_{\mathrm{C}}^{t}\} \right)}\left( \sqrt{\tilde{P}_{\mathrm{C}}^{(t)}[v]} - \sqrt{P_{\mathrm{C}}^{(t)}[v]} \right)^{2} \right. \\ &&\!\! \,+\,\!\!\, \left. \sum\limits_{v \in V_{visible}(v_{\mathrm{R}}^t)}\left( \sqrt{\tilde{P}_{\mathrm{C}}^{(t)}[v]} \,-\, \sqrt{P_{\mathrm{C}}^{(t)}[v]}\right)^{2} \,+\, \left( \sqrt{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}]} \,-\, \sqrt{P_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}]} \right)^{2}\right)^{{1}/{2}} \end{array} $$

C resides at \(v_{\mathrm {C}}^{t}\), hence \(\forall v \in V \setminus \{v_{\mathrm {C}}^{t}\}\), \(P_{\mathrm {C}}^{t}[v] = 0\), \(P_{\mathrm {C}}^{t}[v_{\mathrm {C}}^{t}] = 1\). Therefore:

$$\begin{array}{@{}rcl@{}} H\!\left( \!\tilde{P}_{\mathrm{C}}^{(t)},P_{\mathrm{C}}^{(t)}\right)\! &=& \left( \sum\limits_{v \in V \setminus \left( V_{visible}(v_{\mathrm{R}}^t) \cup \{v_{\mathrm{C}}^{t}\} \right)}\left( \sqrt{\tilde{P}_{\mathrm{C}}^{(t)}[v]}\right)^{2} \right. \\ &&+\,\left.\sum\limits_{v \in V_{visible}(v_{\mathrm{R}}^t)}\left( \sqrt{\tilde{P}_{\mathrm{C}}^{(t)}[v]}\right)^{2} + \left( \sqrt{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}]} - 1 \right)^{2} \right)^{{1}/{2}} \\ &=&\left( \sum\limits_{v \in V \setminus \left( V_{visible}(v_{\mathrm{R}}^t) \cup \{v_{\mathrm{C}}^{t}\} \right)}\tilde{P}_{\mathrm{C}}^{(t)}[v] + \sum\limits_{v \in V_{visible}(v_{\mathrm{R}}^t)}\tilde{P}_{\mathrm{C}}^{(t)}[v] \right. \,+\, \left. \left( \sqrt{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}]} \,-\, 1 \right)^{2} \right)^{{1}/{2}} \\ &=& \left( \sum\limits_{v \in V \setminus \left( V_{visible}(v_{\mathrm{R}}^t) \cup \{v_{\mathrm{C}}^{t}\} \right)}\tilde{P}_{\mathrm{C}}^{(t)}[v] + \sum\limits_{v \in V_{visible}(v_{\mathrm{R}}^t)}\tilde{P}_{\mathrm{C}}^{(t)}[v] \right. \,+\, \left. \tilde{P}_{\mathrm{C}}^{(t)} \,-\,2\sqrt{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}]} + 1 \right)^{{1}/{2}} \end{array} $$

Now, we shall compute the Hellinger distance between the actual distribution over C’s location to R’ estimated distribution after being updated, \(H\left (\hat {P}_{\mathrm {C}}^{(t)},P_{\mathrm {C}}^{(t)}\right )\):

$$\begin{array}{@{}rcl@{}} H\left( \hat{P}_{\mathrm{C}}^{(t)},P_{\mathrm{C}}^{(t)}\right) &=& \left( \sum\limits_{v \in V \setminus \left( V_{visible}(v_{\mathrm{R}}^t) \cup \{v_{\mathrm{C}}^{t}\} \right)}\left( \sqrt{\hat{P}_{\mathrm{C}}^{(t)}[v]} - \sqrt{P_{\mathrm{C}}^{(t)}[v]} \right)^{2} \right. \\ &&+\left. \sum\limits_{v \in V_{visible}(v_{\mathrm{R}}^t)}\left( \sqrt{\hat{P}_{\mathrm{C}}^{(t)}[v]} - \sqrt{P_{\mathrm{C}}^{(t)}[v]}\right)^{2} + \left( \sqrt{\hat{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}]} - \sqrt{P_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}]} \right)^{2}\right)^{{1}/{2}} \\ &=&\left( \sum\limits_{v \in V \setminus \left( V_{visible}(v_{\mathrm{R}}^t) \cup \{v_{\mathrm{C}}^{t}\} \right)}\hat{P}_{\mathrm{C}}^{(t)}[v] + \sum\limits_{v \in V_{visible}(v_{\mathrm{R}}^t)}\hat{P}_{\mathrm{C}}^{(t)}[v] \right. + \left. \left( \sqrt{\hat{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}]} - 1 \right)^{2} \right)^{{1}/{2}} \end{array} $$

C resides at \(v_{\mathrm {C}}^{t}\) and \(v_{\mathrm {C}}^{t} \notin V_{visible}(v_{\mathrm {R}}^t)\), hence:

$$\begin{array}{@{}rcl@{}} &&\forall v \in V_{visible}(v_{\mathrm{R}}^t): \hat{P}_{\mathrm{C}}^{(t)}[v] = 0 \\ &&\forall v \in V \setminus V_{visible}(v_{\mathrm{R}}^t): \hat{P}_{\mathrm{C}}^{(t)}[v] = \tilde{P}_{\mathrm{C}}^{(t)}[v] + \frac{\tilde{P}_{\mathrm{C}}^{(t)}[v] \cdot {\sum}_{v \in V_{visible}(v_{\mathrm{R}}^t)}\tilde{P}_{\mathrm{C}}^{(t)}[v]}{ {\sum}_{v \in V \setminus V_{visible}(v_{\mathrm{R}}^t)} \hat{P}_{\mathrm{C}}^{(t)}[v]} \end{array} $$

Therefore, we obtain:

$$\begin{array}{@{}rcl@{}} &&H\left( \hat{P}_{\mathrm{C}}^{(t)},P_{\mathrm{C}}^{(t)}\right)\\&=&\left( \sum\limits_{v \in V \setminus \left( V_{visible}(v_{\mathrm{R}}^t) \cup \{v_{\mathrm{C}}^{t}\} \right)}\hat{P}_{\mathrm{C}}^{(t)}[v] + \left( \sqrt{\hat{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}]} - 1 \right)^{2} \right)^{{1}/{2}} \\ &=&\left( \sum\limits_{v \in V \setminus \left( V_{visible}(v_{\mathrm{R}}^t) \cup \{v_{\mathrm{C}}^{t}\} \right)}\left( \tilde{P}_{\mathrm{C}}^{(t)}[v] + \frac{\tilde{P}_{\mathrm{C}}^{(t)}[v] \cdot {\sum}_{v \in V_{visible}(v_{\mathrm{R}}^t)}\tilde{P}_{\mathrm{C}}^{(t)}[v]}{ {\sum}_{v \in V \setminus V_{visible}(v_{\mathrm{R}}^t)} \tilde{P}_{\mathrm{C}}^{(t)}[v]} \right) \right. \\ &&+\left. \left( \sqrt{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] + \frac{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] \cdot {\sum}_{v \in V_{visible}(v_{\mathrm{R}}^t)}\tilde{P}_{\mathrm{C}}^{(t)}[v]}{ {\sum}_{v \in V \setminus V_{visible}(v_{\mathrm{R}}^t)} \tilde{P}_{\mathrm{C}}^{(t)}[v]}} - 1 \right)^{2} \right)^{{1}/{2}} \\ &=&\left( \sum\limits_{v \in V \setminus \left( V_{visible}(v_{\mathrm{R}}^t) \cup \{v_{\mathrm{C}}^{t}\} \right)}\left( \tilde{P}_{\mathrm{C}}^{(t)}[v] + \frac{\tilde{P}_{\mathrm{C}}^{(t)}[v] \cdot {\sum}_{v \in V_{visible}(v_{\mathrm{R}}^t)}\tilde{P}_{\mathrm{C}}^{(t)}[v]}{ {\sum}_{v \in V \setminus V_{visible}(v_{\mathrm{R}}^t)} \tilde{P}_{\mathrm{C}}^{(t)}[v]} \right) \right. \\ &&+\left. \tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] + \frac{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] \cdot {\sum}_{v \in V_{visible}(v_{\mathrm{R}}^t)}\tilde{P}_{\mathrm{C}}^{(t)}[v]}{ {\sum}_{v \in V \setminus V_{visible}(v_{\mathrm{R}}^t)} \tilde{P}_{\mathrm{C}}^{(t)}[v]} -2 \sqrt{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] + \frac{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] \cdot {\sum}_{v \in V_{visible}(v_{\mathrm{R}}^t)}\tilde{P}_{\mathrm{C}}^{(t)}[v]}{ {\sum}_{v \in V \setminus V_{visible}(v_{\mathrm{R}}^t)} \tilde{P}_{\mathrm{C}}^{(t)}[v]}} + 1 \right)^{{1}/{2}} \\ &=&\left( \sum\limits_{v \in V \setminus \left( V_{visible}(v_{\mathrm{R}}^t) \cup \{v_{\mathrm{C}}^{t}\} \right)}\tilde{P}_{\mathrm{C}}^{(t)}[v] + \sum\limits_{v \in V \setminus \left( V_{visible}(v_{\mathrm{R}}^t) \cup \{v_{\mathrm{C}}^{t}\} \right)} \frac{\tilde{P}_{\mathrm{C}}^{(t)}[v] \cdot {\sum}_{v \in V_{visible}(v_{\mathrm{R}}^t)}\tilde{P}_{\mathrm{C}}^{(t)}[v]}{ {\sum}_{v \in V \setminus V_{visible}(v_{\mathrm{R}}^t)} \tilde{P}_{\mathrm{C}}^{(t)}[v]} \right. \\ &&+\left. \tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] + \frac{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] \cdot {\sum}_{v \in V_{visible}(v_{\mathrm{R}}^t)}\tilde{P}_{\mathrm{C}}^{(t)}[v]}{ {\sum}_{v \in V \setminus V_{visible}(v_{\mathrm{R}}^t)} \tilde{P}_{\mathrm{C}}^{(t)}[v]} -2 \sqrt{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] + \frac{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] \cdot {\sum}_{v \in V_{visible}(v_{\mathrm{R}}^t)}\tilde{P}_{\mathrm{C}}^{(t)}[v]}{ {\sum}_{v \in V \setminus V_{visible}(v_{\mathrm{R}}^t)} \tilde{P}_{\mathrm{C}}^{(t)}[v]}} + 1 \right)^{{1}/{2}} \\ &=&\left( \sum\limits_{v \in V \setminus \left( V_{visible}(v_{\mathrm{R}}^t) \cup \{v_{\mathrm{C}}^{t}\} \right)}\tilde{P}_{\mathrm{C}}^{(t)}[v] + \sum\limits_{v \in V_{visible}(v_{\mathrm{R}}^t)}\tilde{P}_{\mathrm{C}}^{(t)}[v] + \tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] \right. \\ &&\left. -2 \sqrt{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] + \frac{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] \cdot {\sum}_{v \in V_{visible}(v_{\mathrm{R}}^t)}\tilde{P}_{\mathrm{C}}^{(t)}[v]}{ {\sum}_{v \in V \setminus V_{visible}(v_{\mathrm{R}}^t)} \tilde{P}_{\mathrm{C}}^{(t)}[v]}} + 1 \right)^{{1}/{2}} \end{array} $$

If \(\tilde {P}_{\mathrm {C}}^{(t)}[v_{\mathrm {C}}^{t}] = 0\) then \(\hat {P}_{\mathrm {C}}^{(t)}[v_{\mathrm {C}}^{t}] = \tilde {P}_{\mathrm {C}}^{(t)}[v_{\mathrm {C}}^{t}]\) and \(H(\tilde {P}_{\mathrm {C}}^{(t)},P_{\mathrm {C}}^{(t)}) = H(\hat {P}_{\mathrm {C}}^{(t)},P_{\mathrm {C}}^{(t)})\). Otherwise:

$$\begin{array}{@{}rcl@{}} &&\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] > 0 \Rightarrow \frac{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] \cdot {\sum}_{v \in V_{visible}(v_{\mathrm{R}}^t)}\tilde{P}_{\mathrm{C}}^{(t)}[v]}{ {\sum}_{v \in V \setminus V_{visible}(v_{\mathrm{R}}^t)} \tilde{P}_{\mathrm{C}}^{(t)}[v]} > 0 \\ &&\Rightarrow \sqrt{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] + \frac{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}] \cdot {\sum}_{v \in V_{visible}(v_{\mathrm{R}}^t)}\tilde{P}_{\mathrm{C}}^{(t)}[v]}{ {\sum}_{v \in V \setminus V_{visible}(v_{\mathrm{R}}^t)} \tilde{P}_{\mathrm{C}}^{(t)}[v]}} > \sqrt{\tilde{P}_{\mathrm{C}}^{(t)}[v_{\mathrm{C}}^{t}]} \\ &&\Rightarrow H(\tilde{P}_{\mathrm{C}}^{(t)},P_{\mathrm{C}}^{(t)}) > H(\hat{P}_{\mathrm{C}}^{(t)},P_{\mathrm{C}}^{(t)}) \end{array} $$

In conclusion, we obtain that \(H(\tilde {P}_{\mathrm {C}}^{(t)},P_{\mathrm {C}}^{(t)}) \geq H(\hat {P}_{\mathrm {C}}^{(t)},P_{\mathrm {C}}^{(t)})\), hence R’s estimated probability distribution over C’s location is closer to \(P_{\mathrm {C}}^{(t)}\) after incorporating the information gained from the visibility edges. □

Proving Theorem 4 for C is similar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Keidar, O., Agmon, N. Safe navigation in adversarial environments. Ann Math Artif Intell 83, 121–164 (2018). https://doi.org/10.1007/s10472-018-9591-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-018-9591-0

Keywords

Mathematics Subject Classification (2010)

Search

Navigation