Combining Graph Contraction and Strategy Generation for Green Security Games 1 Anjon Basak, Fei Fang, Thanh Hong Nguyen, Christopher Kiekintveld
Environmental Crimes 2 Illegal fishing Poaching Illegal logging
Consequences 3 Major threat to biodiversity Global warming Financial loss
Stackelberg Security Game(SSG) 4
Green Security Game(GSG) 5 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
Patrolling Path in GSG 6 Base station
Problem 7 Huge area Exponential number of paths LP optimizes over all of the paths. Largest problem solved : 25 targets(approximately)
Motivation 8 Mean numbers of elephants/0.16km^2 in Queen Elizabeth National Park, Uganda
Solution Idea 9 Automated + Strategy generation contraction ACSG
Abstraction Using Graph 10 Contraction Original Contract Contracted Game Game solve Use solution in Reverse Solution map original game
Contraction 11 Removes unnecessary nodes one by one Introduces edges Evaluates edges whether to keep or not
Contraction 12 1 1 2 3 2 3 4 4 2 2 4 4 2 2 4 1 2 4 2 5 4 5 4 2 2
Instant Contraction 13 Removes unnecessary nodes altogether Finds shortest path through unnecessary nodes
Instant Contraction 14 Base station 4 1 5 2 1 2 2 9 1 1 3 1 5 0 0 12 7 10 2 1 3 9
Instant Contraction 15 Base station 4 1 5 2 2 1 2 9 4 1 1 3 1 5 0 0 12 7 10 2 1 3 9
Instant Contraction 16 Base station 4 1 5 2 2 1 2 9 4 1 1 3 1 5 0 0 9 12 7 10 2 1 3 9
Instant Contraction 17 Base station 4 9 4 3 9 Which nodes to contract ? 3
Single Oracle 18 Restrict attacker’s strategy space Incrementally add targets Consider full defender strategy space
Single Oracle 19 Phase n Phase 2 Add Restricted set of … targets targets T’’ to T’ Phase 1 Contract game Restricted set of targets T’ Solve using LP Contract game Compute Best response for attacker Solve using LP … Compute Best response for Stop attacker BR in T’
Single Oracle 20 Phase n Phase 2 Add Restricted set of … targets targets T’’ to T’ Phase 1 Contract game Restricted set of targets T’ Solve using LP Contract game Compute Best response for attacker Solve using LP … Compute Best response for Stop attacker BR in T’
Automated Contraction Using Double Oracle 21 Phase n Phase 2 Add Restricted set of … targets targets T’’ to T’ Phase 1 Contract game Restricted set of targets T’ DO Contract game Compute Best response for attacker DO … Compute Best response for Stop attacker BR in T’
Automated Contraction With 22 Double Oracle Restrict attacker’s strategy space Restrict defender’s strategy space
Double Oracle 23 … Restrict set of defender strategy S’ Solve using LP Add paths to S’ Compute best response for defender BR already in S’ …
Single Oracle 24 Base station 1 4 5 2 2 1 2 1 1 3 1 5 0 0 12 7 10 2 1 3 9
Single Oracle 25 Base station 4 .5 9 .5 3
Single Oracle 26 Compute Best Base station response of .5 1 4 attacker 5 2 2 1 9 2 .5 1 1 3 1 5 0 0 12 7 10 2 1 3 9 .5
Single Oracle 27 Base station 1 4 5 2 2 1 2 1 1 3 1 5 0 0 12 7 10 2 1 3 9
Single Oracle 28 Base station 4 .33 9 4 3 .33 9 3 .33
Single Oracle 29 Can be improved Compute Best Base station .33 response of 1 4 5 attacker 2 2 1 9 2 .33 1 1 3 1 5 0 0 .33 .33 12 7 10 2 1 3 9 .33
Automated Contraction With 30 Double Oracle Restrict attacker’s strategy space Restrict defender’s strategy space
Automated Contraction With 31 Double Oracle Base station 1 4 5 2 2 1 2 1 1 3 1 5 0 0 12 7 10 2 1 3 9
Automated Contraction With 32 Double Oracle Base station 4 .5 Restrict set of defender strategy S’ Solve using LP 9 Add paths to S’ Compute best response for defender BR already in S’ .5 3
Automated Contraction With 33 Double Oracle Base station Compute Best .5 1 4 response of 5 attacker 2 2 1 9 2 .5 1 1 3 1 5 0 0 12 7 10 2 1 3 9 .5
Automated Contraction With 34 Double Oracle Base station 1 4 5 2 2 1 2 1 1 3 1 5 0 0 12 7 10 2 1 3 9
Automated Contraction Using 35 Double Oracle Base station 4 .33 Restrict set of defender strategy S’ 9 4 Solve using LP 3 .33 Add paths Compute best 9 to S’ response for defender BR already in S’ 3 .33
Automated Contraction Using 36 Double Oracle Compute Best Base station .33 response of 1 4 5 attacker 2 2 1 9 2 .33 1 1 3 1 5 0 0 .33 .33 12 7 10 2 1 3 9 .33
Experiments 37 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.
Heuristics 38 For initializing attacker's strategy space and strategy generation: GreedyCover1(GC1) GreedyCoverR(GCR) For initializing defender's strategy space: GreedyPath3(GP3) Base station Taget t Base station Base station Insert targets greedily
Results 39 DO = Double Oracle with heuristics SO = Single Oracle Baseline = No contraction
Results 40 DO = Double Oracle with heuristics SO = Single Oracle Baseline = No contraction
41 OP = Orienteering Problem lexicoOP = Lexicographic solution for OP with multiple resources TOP = Team orienteering problem with multiple visitations DO = Double Oracle with heuristics
42 OP = Orienteering Problem lexicoOP = Lexicographic solution for OP with multiple resources TOP = Team orienteering problem with multiple visitations DO = Double Oracle with heuristics
Conclusion 43 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 optimal/sub-optimal solvers
Thanks! 44
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