Combining Graph Contraction and Strategy Generation for Green - - PowerPoint PPT Presentation

Combining Graph Contraction and Strategy Generation for Green Security Games

Anjon Basak, Fei Fang, Thanh Hong Nguyen, Christopher Kiekintveld

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Environmental Crimes

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Illegal fishing Poaching Illegal logging

Consequences

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 Major threat to biodiversity  Global warming  Financial loss

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Stackelberg Security Game(SSG)

 Graph based representation of terrain (e.g. national park)  Node represents a small portion of the terrain(1kmx1km)  Attacker: poacher Defender: patroller  Solution: Optimized patrolling strategy

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Green Security Game(GSG)

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Patrolling Path in GSG

Base station

 Huge area  Exponential number of paths  LP optimizes over all of the paths.  Largest problem solved : 25 targets(approximately)

Problem

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Motivation

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Mean numbers of elephants/0.16km^2 in Queen Elizabeth National Park, Uganda

Solution Idea

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Automated contraction Strategy generation

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ACSG

Abstraction Using Graph Contraction

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Original Game Contracted Game Solution

solve Contract

Use solution in

• riginal game

Reverse map

Contraction

 Removes unnecessary nodes one by one  Introduces edges  Evaluates edges whether to keep or not

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5 4 3 2 1 1 2 2 2 2 2 2 5 4 3 2 1 2 4 2 4 4 4 4 4

Contraction

Instant Contraction

 Removes unnecessary nodes altogether  Finds shortest path through unnecessary nodes

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Instant Contraction

Base station

4 3 3

2 5 1 1 1 9 12 7 10

1 1 1

2 2 5 2 9

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Instant Contraction

Base station

4 3 3

2 5 1 1 1 9 12 7 10

1 1

2 2 5 2 9 4 1

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Instant Contraction

Base station

4 3 3

2 5 1 1 1 9 12 7 10

1 1 1

2 2 5 2 9 4 9

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Instant Contraction

Base station

4 3 3

9 4 9

Which nodes to contract ?

Single Oracle

 Restrict attacker’s strategy space  Incrementally add targets  Consider full defender strategy space

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Restricted set of targets T’ Contract game Solve using LP Compute Best response for attacker Restricted set of targets T’’ Contract game Solve using LP Compute Best response for attacker

Phase 1 Phase 2

Stop

… Phase n …

Add targets to T’ BR in T’

Single Oracle

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Restricted set of targets T’ Contract game Solve using LP Compute Best response for attacker Restricted set of targets T’’ Contract game Solve using LP Compute Best response for attacker

Phase 1 Phase 2

Stop

… Phase n …

BR in T’

Single Oracle

Add targets to T’

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Automated Contraction Using Double Oracle

Restricted set of targets T’ Contract game DO Compute Best response for attacker Restricted set of targets T’’ Contract game Compute Best response for attacker

Phase 1 Phase 2

Stop

… Phase n …

DO BR in T’ Add targets to T’

Automated Contraction With Double Oracle

 Restrict attacker’s strategy space  Restrict defender’s strategy space

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Double Oracle

Solve using LP Compute best response for defender Restrict set of defender strategy S’

… …

Add paths to S’ BR already in S’

Single Oracle

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Base station

4 3 3

2 5 1 1 1 9 12 7 10

1 1 1

2 2 5 2

25

Base station

4 3

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Single Oracle

.5 .5

Single Oracle

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Base station

4 3 3

2 5 1 1 1 9 12 7 10

1 1 1

2 2 5 2

.5 .5 .5

9 Compute Best response of attacker

Single Oracle

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Base station

4 3 3

2 5 1 1 1 9 12 7 10

1 1 1

2 2 5 2

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Single Oracle

.33

Base station

4 3 3

9 4 9

.33 .33

Single Oracle

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Base station

4 3 3

2 5 1 1 1 9 12 7 10

1 1 1

2 2 5 2

.33 .33

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.33 .33 .33 Can be improved

Compute Best response of attacker

Automated Contraction With Double Oracle

 Restrict attacker’s strategy space  Restrict defender’s strategy space

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Base station

4 3 3

2 5 1 1 1 9 12 7 10

1 1 1

2 2 5 2

Automated Contraction With Double Oracle

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Base station

4 3

9

.5 .5

Automated Contraction With Double Oracle

Solve using LP Compute best response for defender Restrict set of defender strategy S’ Add paths to S’ BR already in S’

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Base station

4 3 3

2 5 1 1 1 9 12 7 10

1 1 1

2 2 5 2

.5 .5 .5

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Automated Contraction With Double Oracle

Compute Best response of attacker

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Base station

4 3 3

2 5 1 1 1 9 12 7 10

1 1 1

2 2 5 2

Automated Contraction With Double Oracle

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Automated Contraction Using Double Oracle

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Base station

4 3 3

9 4 9

.33 .33

Solve using LP Compute best response for defender Restrict set of defender strategy S’ Add paths to S’ BR already in S’

Automated Contraction Using Double Oracle

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Base station

4 3 3

2 5 1 1 1 9 12 7 10

1 1 1

2 2 5 2

.33 .33

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.33 .33 .33

Compute Best response of attacker

Experiments

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20 random 2 player zero-sum games



size {25, 50, 100, 200}



Payoffs are randomly chosen from [0, 4] and [8,10] range



Payoff ranges maintain 90% and 10% frequency respectively.

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Heuristics



For initializing attacker's strategy space and strategy generation:



GreedyCover1(GC1)



GreedyCoverR(GCR)

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For initializing defender's strategy space:



GreedyPath3(GP3)

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Base station Base station Insert targets greedily Base station Taget t

Results

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DO = Double Oracle with heuristics SO = Single Oracle Baseline = No contraction

Results

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DO = Double Oracle with heuristics SO = Single Oracle Baseline = No contraction

OP = Orienteering Problem lexicoOP = Lexicographic solution for OP with multiple resources TOP = Team orienteering problem with multiple visitations DO = Double Oracle with heuristics

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OP = Orienteering Problem lexicoOP = Lexicographic solution for OP with multiple resources TOP = Team orienteering problem with multiple visitations DO = Double Oracle with heuristics

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Conclusion

 First algorithm to combine automated contraction with strategy generation  Scalable enough to solve GSG having 200 targets within seconds  Heuristics good and fast enough compared with

• ptimal/sub-optimal solvers

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Thanks!

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