game-theoretic resource allocation for protecting large public events yue yin 1, bo an 2, manish...
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Game-theoretic Resource Allocation for Protecting Large Public Events
Yue Yin1, Bo An2, Manish Jain3
1Institute of Computing Technology, Chinese Academy of Sciences, China2Nanyang Technological University, Singapore
3Armorway, U.S.A.
July 30, 2014
Boston Marathon Bombings
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On April 15, 2013, two bombs exploded near the finish line, killing 3 people and injuring an estimated 264 others.
Security in Public Events
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Potential attack targets in marathon
Varying target value
• Target value changes over time• Dynamically allocate security resources
- Transfer resources at any time- A resource in transfer is not protecting any target
• Attacker’s reactionResearch Question: When and How to transfer resources?
Related work
• Applying game theory to security domains - ARMOR (Los Angeles International Airport), PROTECT
(United States Coast Guard),IRIS (Federal Air Marshals) et al.
• Static target value
• Discretized time
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Our Contributions
• New security game model – Varying value of targets & Continuous strategy space
• Algorithms computing the equilibrium– SCOUT-A: Negligible transfer time– SCOUT-C: Non-zero transfer time
• Evaluation
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Model: Target Value and Utilities
• Value of target i: – Continuous function w.r.t time t
(piecewise linear or others)
• Attacker utility of attacking
target i at time t : – r: # of resources – Decreasing marginal effect
• Zero-sum Game:
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Atta
cker
ut
ility
0r - # of resources
Targ
et v
alue
0 1 4 52 3t - time
∞
Model: Strategies and Equilibrium
• Defender’s pure strategy: Initial assignment & All transfers
Example: 2 targets (T1, T2),2 resources, transfer time 1
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0 1 2 3t - time
(2, 0) (1, 0) (1, 1)
• Attacker’s pure strategy: Attack target i at time t
• Equilibrium: Minimizing the maximum attacker utility
T1 T2
SCOUT-A:Negligible Transfer Time
• Context: Resources can be transferred quickly• Find the minimax assignment of resources at each time point
Example: 2 targets (T1, T2),2 resources
v1(t) v2(t)
t-time0 1 4 52 3
Target value
8
0
0
Attacker utilityResources
Minimax assignment at time 0
T1
T2
Infeasible to find the minimax assignment at each time point since time is continuous
SCOUT-A:Negligible Transfer Time (2)
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t-time0 1 4 52 3
Attacker Utility
• ‘Minimax assignment’ does not change continuously
0
0
v2(t)
v1(t)
v1(t) / eλ
v1(t) / e2λ
v2(t) / eλ
v2(t) / e2λ
T1
T2
T1 T2
T1 T2
SCOUT-A computes the time point at which a minimax assignment ‘expires’, then finds the ‘next’ minimax assignment
SCOUT-C: Nonzero Transfer Time
• Key Result– For any game with continuous defender strategy space, we can
construct an equivalent game with discrete defender strategy space
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The equilibrium of the constructed game is also an equilibrium of the initial game
0 te
0 te
Transfer at any time Transfer at discretized points
Initial Game Constructed Game
SCOUT-C: Nonzero Transfer Time (2)
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target pair (i, j), assignment of resources (ai, aj), compute
Transfers can only begin at θ
Experimental Results: Solution Quality*more in the paper
(a) Varying transfer time (b) Varying value of λ
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• Value of targets: [0, 50]
• Baseline - SDS: Optimal static defender strategy- DDS: Optimal dynamic defender strategy with arbitrarily
discretized time
Conclusions
• Contributions– Security game model considering varying value of targets and
continuous strategy space– Algorithms to compute optimal defender strategy– Evaluation
• Future work– Scale up the algorithm– Consider uncertainty
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Thank [email protected]