patrolling games by steve alpern, alec morton, katerina papadaki presented by: yan t. yang

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

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

Introduction

Game theoretic formulation

[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

Advantage: how a patroller should randomize her patrols

[Paruchuri et al. 2007] heuristic models for Stackelberg formulation[Gordon 2007, Newsweek 2007] use Stackelberg formulation

Introduction



Search games Accumulation game Inspection game

Hider: Stationary

Searcher: Minimize his time

[Alpern and Gal 2003]

Photo credit: Hide and Seek (Tatiana Dery, 2007)

Searcher: several location

[Ruckle 2001 and Kikuta and Ruckle 2002]

Photo credit: Crackberry

Hider: distribute

Searcher: prevent

[Avenhaus et al. 2002]

Photo credit: HEXUS

Hider: infiltrate

Introduction



Search games Accumulation game Stackelberg Game

Patrolling Game

Hider: any time

Win-loss (zero sum game)



Searcher: mobile

Different Same

Formulation



Patrolling Game G(Q,T,m)

Q

Go(Q,T,m)

Gp(Q,T,m)

one-off

periodic

Formulation



Patrolling Game G(Q,T,m)Go(Q,T,m)

Gp(Q,T,m)

one-off

periodic

Time: 0

Formulation




Gp(Q,T,m)

one-off

periodic

Time: 1

Formulation




Gp(Q,T,m)

one-off

periodic

Time: 2

Formulation




Gp(Q,T,m)

one-off

periodic

Time: 0

Time: 1

Time: 2

Time: T-1

Formulation




Gp(Q,T,m)

one-off

periodic

Time: 0

Attacker = [i, I]

Node

Consecutive attacking intervalof the length m

Formulation




Gp(Q,T,m)

one-off

periodic

Time: 1

Formulation




Gp(Q,T,m)

one-off

periodic

Time: 2

Formulation




Gp(Q,T,m)

one-off

periodic

m

Formulation




Gp(Q,T,m)

one-off

periodic

Patroller = W: {0…T-1} => Q

Time: 0

Formulation




Gp(Q,T,m)

one-off

periodic

Time: 1

Formulation




Gp(Q,T,m)

one-off

periodic

Time: 2

Formulation




Gp(Q,T,m)

one-off

periodic

Patroller = W: {0…T-1} => Q

Formulation




Gp(Q,T,m)

one-off

periodic

Time: 0

i ϵ w(I)

Attacked node

Attacking interval

Patrolling route

If patrolling route during the attacking interval includesthe attacked nodePatroller wins:

Formulation




Gp(Q,T,m)

one-off

periodic

Time:1

Formulation




Gp(Q,T,m)

one-off

periodic

Time: 2

Formulation




Gp(Q,T,m)

one-off

periodic

Time: 0

Time: 1

Time: 2

Formulation




Gp(Q,T,m)

one-off

periodic

Time: 0

Time: 1

Time: 2

V the probability that the patroller wins (attack is intercepted)

Formulation




Gp(Q,T,m)

one-off

periodic

Time: 0

Time: 1

Time: T-1

Time: 0

Formulation




Gp(Q,T,m)

one-off

periodic

Time: 0

Time: 1

Time: T-1

Time: 0

The same starting point

Formulation

Patrolling Game

Assumptions:Zero sum game

Node values are equal

Distances between the nodes are the same

Patrolling Game

Lemma 11) V is non-decreasing in m (attacking interval)2) V is non-decreasing with more edges3) Vp ≤ Vo

4) V(Q’) ≥ V(Q)

Q

Patrolling Game

Lemma 11) V is non-decreasing in m (attacking interval)2) V is non-decreasing with more edges3) Vp ≤ Vo

4) V(Q’) ≥ V(Q)

Q’

Patrolling Game

Lemma 2 1/n ≤ V ≤ m/n

Number of nodes

Patrolling Game

Lemma 2 1/n ≤ V ≤ m/n

Required attacking interval length

Patrolling Game

Lemma 2 1/n ≤ V ≤ w/n

The maximum number of nodes that any patrol can cover

Patrolling Game


Patroller: pick a random node and wait there

Patrolling Game


Attacker: attack a random node during interval I

|w(I)| ≤ |I| = m

|w(I)| ≤ w

Patrolling Game

Proposition 3 1) Vo(T+1) ≤ Vo(T)2) Vp(kT) ≥ Vp(T)

k any non-zero integer

Patrolling Game

Proposition 3 1) Vo(T+1) ≤ Vo(T)

Attacker has more strategies – bad for patroller

Patrolling Game

Proposition 3

2) Vp(kT) ≥ Vp(T)

Patroller: pick identical strategies as in T case repeat it k timesBut more strategies for patroller

Patrolling Game


One-off game: increasing time T helps the attacker

Periodic game: increasing T by a multiplicative factor helps the patroller.

Patrolling Game


Corollary 4 0 ≤ Vo(kT) - Vp(kT) ≤ Vo(T) - Vp(T)

Lemma 13) Vp ≤ Vo

Gap between on-off game and periodic game decreases

Strategy Reduction

In general: these problems are quite hardLarge number of strategies

Reduction

Symmetrization

DominanceDecomposition

Strategy Reduction

Symmetrization


Both attacker and patroller

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.

Strategy Reduction

Symmetrization


Strategy s1 dominates strategy s2, if s1 wins more than s2

Leaf nodePenultimate node

Strategy Reduction

Symmetrization


Lemma 5Assume Q is connected, T ≥ 31) m ≥ 2, patrolling staying for 3 consecutive

period are dominated 2) m ≥ 3, attacking on penultimate nodes

are dominated

Strategy Reduction

Symmetrization


Lemma 5Assume Q is connected, T ≥ 31) m ≥ 2, patrolling staying for 3 consecutive

period are dominated 2) m ≥ 3, attacking on penultimate nodes

are dominated

Proof: 1) Patrolling w1 = {t-1, t, t+1} staying at node i

Patrolling w2(t) = i’ adjacent to iw2 intercepts every attack w1 intercepts

+ attack on i’ 2) Denote i’ penultimate node i is the leaf node

If w wins against the attack [i,l], w(t) = I for some t ϵ Im ≥ 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 aroundsince we need to reach i via i’ then w also wins against [i’,I]

Strategy Reduction

Symmetrization


Strategy Reduction

Symmetrization


Proof:

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)

[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.

Generic Strategies

Attackers Strategies

Uniform strategy: a random node is attacked at a random time


Lemma 8If T is odd and Q is bipartite, the bound of Lemma 2

Vp ≤ ((T-1)m + 1)/nTIf attackers adopt uniform strategy

Periodic game: nT possible attacks.

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.

Case 1 đ large with respect to m and T, wait at one pointCase 2 đ small, go back and forth

Corresponding patroller’s strategy:

Case 1Case 2

Generic Strategies

Patroller’s Strategies

Definition: w is called intercepting, if it interceptsEvery attack on a node that it contains.

Definition: covering set, a set of intercepting patrols that contains all nodes.

Covering number, minimum cardinalityffcc

Generic Strategies


Definition: any two nodes are independent setIf any patrol intercepting node i cannot for j

Definition: independent number is the cardinality of a maximal independent set. ffii

Generic Strategies


ffi i ≤ f≤ fcc

Lemma 121/fc ≤ V ≤ 1/f≤ V ≤ 1/fii

Patrolling on Special Classes of Graphs

Graph with a Hamiltonian Cycle

Bipartite

Line graphs

Patroller follows the Hamiltonian cycle

Attacker: choose a node from the larger halfset.

Patroller: randomize from strategies visit the larger halfset every second time period

Complicated already

General graph

Difficult

Conclusion

The model can be used to Plan patrol route Design facility: Hamiltonian

Possible Improvement Multiple patroller

Superficially similar games Continuous time Knowledge set Memory

Superficialy similar games

Continuous time Attacker

• Any point (node, arcs)• Continuous time interval

Patroller• Unit speed path

Resemble more to search game


Knowledge setPatroller:

• Alerted by noise

Attacker:• Confederate


MemoryNo-memory

• Markov

Memory• Cycle vs. line

patrolling games by steve alpern, alec morton, katerina papadaki presented by: yan t. yang

Documents

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oneoff periodic time