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Patrolling Games

Patrolling Games. By Steve Alpern, Alec Morton, Katerina Papadaki Presented by: Yan T. Yang. Agenda. Introduction The Patrolling Game Reduction Strategies Generic Strategies Patrolling on Special Classes of Graphs Conclusions. Agenda. Introduction The Patrolling Game

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Patrolling Games

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  1. Patrolling Games By Steve Alpern, Alec Morton, Katerina Papadaki Presented by: Yan T. Yang

  2. Agenda • Introduction • The Patrolling Game • Reduction Strategies • Generic Strategies • Patrolling on Special Classes of Graphs • Conclusions

  3. Agenda • Introduction • The Patrolling Game • Reduction Strategies • Generic Strategies • Patrolling on Special Classes of Graphs • Conclusions

  4. Introduction Scheduling and deployment of patrols Art gallery Airport or shopping mall • City containing a number of targets • Airline network • Cargo warehouse Past research [Urrutia 2000] Computational geometry: art gallery security guards [Larson 1972] Operations research: the scheduling of police patrols [Sherman and Eck 1972] Randomized patrols [Chelst 1978] Maximize the probability of intercepting a crime in progress Not game theoretic

  5. Introduction Game theoretic formulation Advantage: how a patroller should randomize her patrols [Isaacs 1999, Feichtinger 1983] dynamic adjustment rather than planning [Brown et al. 2006, Bier and Azaiez 2009, Lindelauf et al. 2009] homeland security and counter terrorism [Paruchuri et al. 2007] heuristic models for Stackelberg formulation [Gordon 2007, Newsweek 2007] use Stackelberg formulation

  6. Introduction Game theoretic formulation Advantage: how a patroller should randomize her patrols Search games Accumulation game Inspection game Hider: Stationary Hider: distribute Hider: infiltrate Searcher: Minimize his time Searcher: several location Searcher: prevent [Alpern and Gal 2003] [Ruckle 2001 and Kikuta and Ruckle 2002] [Avenhaus et al. 2002] Photo credit: Hide and Seek (Tatiana Dery, 2007) Photo credit: Crackberry Photo credit: HEXUS

  7. Introduction Game theoretic formulation Advantage: how a patroller should randomize her patrols Patrolling Game Search games Accumulation game Stackelberg Game Hider: any time Win-loss (zero sum game) Searcher: mobile Win-loss (zero sum game) Win-loss (zero sum game) Different Same

  8. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Q

  9. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 0

  10. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 1

  11. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 2

  12. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 0 Time: 1 Time: 2 Time: T-1

  13. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 0 Attacker = [i, I] Node Consecutive attacking interval of the length m

  14. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 1

  15. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 2

  16. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) m

  17. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 0 Patroller = W: {0…T-1} => Q

  18. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 1

  19. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 2

  20. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Patroller = W: {0…T-1} => Q

  21. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 0 Attacking interval If patrolling route during the attacking interval includes the attacked node Patroller wins: i ϵ w(I) Patrolling route Attacked node

  22. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time:1

  23. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 2

  24. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 0 Time: 1 Time: 2

  25. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 0 Time: 1 Time: 2 V the probability that the patroller wins (attack is intercepted)

  26. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 0 Time: 1 Time: T-1 Time: 0

  27. Formulation Game theoretic formulation Advantage: how a patroller should randomize her patrols Go(Q,T,m) one-off G(Q,T,m) Patrolling Game periodic Gp(Q,T,m) Time: 0 Time: 1 The same starting point Time: T-1 Time: 0

  28. Formulation Patrolling Game Assumptions: Zero sum game Node values are equal Distances between the nodes are the same

  29. Patrolling Game Lemma 1 1) V is non-decreasing in m (attacking interval) 2) V is non-decreasing with more edges 3) Vp ≤ Vo 4) V(Q’) ≥ V(Q) Q

  30. Patrolling Game Lemma 1 1) V is non-decreasing in m (attacking interval) 2) V is non-decreasing with more edges 3) Vp ≤ Vo 4) V(Q’) ≥ V(Q) Q’

  31. Patrolling Game Lemma 2 1/n ≤ V ≤ m/n Number of nodes

  32. Patrolling Game Lemma 2 1/n ≤ V ≤ m/n Required attacking interval length

  33. Patrolling Game Lemma 2 1/n ≤ V ≤ w/n The maximum number of nodes that any patrol can cover

  34. Patrolling Game Lemma 2 1/n ≤ V ≤ w/n Patroller: pick a random node and wait there

  35. Patrolling Game Lemma 2 1/n ≤ V ≤ w/n Attacker: attack a random node during interval I |w(I)| ≤ |I| = m |w(I)| ≤ w

  36. Patrolling Game • Proposition 3 • Vo(T+1) ≤ Vo(T) • Vp(kT) ≥ Vp(T) k any non-zero integer

  37. Patrolling Game • Proposition 3 • Vo(T+1) ≤ Vo(T) Attacker has more strategies – bad for patroller

  38. Patrolling Game Proposition 3 2) Vp(kT) ≥ Vp(T) Patroller: pick identical strategies as in T case repeat it k times But more strategies for patroller

  39. Patrolling Game One-off game: increasing time T helps the attacker • Proposition 3 • Vo(T+1) ≤ Vo(T) • Vp(kT) ≥ Vp(T) Periodic game: increasing T by a multiplicative factor helps the patroller.

  40. Patrolling Game • Proposition 3 • Vo(T+1) ≤ Vo(T) • Vp(kT) ≥ Vp(T) Lemma 1 3) Vp ≤ Vo Corollary 4 0 ≤ Vo(kT) - Vp(kT) ≤ Vo(T) - Vp(T) Gap between on-off game and periodic game decreases

  41. Strategy Reduction In general: these problems are quite hard Large number of strategies Reduction Symmetrization Dominance Decomposition

  42. Strategy Reduction Symmetrization Dominance Both attacker and patroller Decomposition Theorem [Alpern and Asic 1985, Zoroa and Zoroa 1993] There is an optimal mixed attacker strategy with the property that for any attack interval I, these two symmetric nodes are attacked with equal probability.

  43. Strategy Reduction Penultimate node Leaf node Symmetrization Dominance Strategy s1dominates strategy s2, if s1 wins more than s2 Decomposition

  44. Strategy Reduction Lemma 5 Assume Q is connected, T ≥ 3 1) m ≥ 2, patrolling staying for 3 consecutive period are dominated 2) m ≥ 3, attacking on penultimate nodes are dominated Symmetrization Dominance Decomposition

  45. Strategy Reduction Lemma 5 Assume Q is connected, T ≥ 3 1) m ≥ 2, patrolling staying for 3 consecutive period are dominated 2) m ≥ 3, attacking on penultimate nodes are dominated Symmetrization Proof: Dominance • Patrolling w1 = {t-1, t, t+1} staying at node i • Patrolling w2(t) = i’ adjacent to i • w2 intercepts every attack w1 intercepts + attack on i’ • Denote i’ penultimate node i is the leaf node • If w wins against the attack [i,l], w(t) = I for some t ϵ I • m ≥ 3, I contains at least 3 consecutive periods • {t-2, t-1, t} or {t-1,t,t+1} or {t,t+1,t+2} in I • by 1) patrolling will move around • since we need to reach i via i’ then w also wins against [i’,I] Decomposition

  46. Strategy Reduction Symmetrization Dominance Decomposition

  47. Strategy Reduction Proof: Symmetrization Dominance Restrict Patroller Sk an optimal mixed strategy for game G(Qk) Pick Sk with probability qk such that qkVk = c is constant 1 = ∑ qk = c ∑1/Vk or c = 1/(∑1/Vk) Decomposition [i,I], the node i belongs to the node set of some graph Qk With qk: optimally patrol Qk; interept the attacker with probability at least Vk Win with probability at least qkVk = c.

  48. Generic Strategies Attackers Strategies Uniform strategy: a random node is attacked at a random time Periodic game: nT possible attacks. Lemma 2 1/n ≤ V ≤ w/n Lemma 8 If T is odd and Q is bipartite, the bound of Lemma 2 Vp ≤ ((T-1)m + 1)/nT If attackers adopt uniform strategy

  49. Generic Strategies Attackers Strategies Diametrical strategy: Diameter: đ = maxi,i’d(i,i’) Diametrical nodes: i, i’ Attack these nodes equi-probably during a random time interval I. Corresponding patroller’s strategy: Case 1 đ large with respect to m and T, wait at one point Case 2 đ small, go back and forth Case 2 Case 1

  50. Generic Strategies Patroller’s Strategies Definition: w is called intercepting, if it intercepts Every attack on a node that it contains. Definition: covering set, a set of intercepting patrols that contains all nodes. Covering number, minimum cardinality fc

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