1. Game-theoretic Resource Allocation for Protecting Large Public Events Yue Yin 1, Bo An 2, Manish Jain 3 1 Institute of Computing Technology, Chinese Academy of Sciences, China 2 Nanyang Technological University, Singapore 3 Armorway, U.S.A. July 30, 2014 1

2. Boston Marathon Bombings 2 On April 15, 2013, two bombs exploded near the finish line, killing 3 people and injuring an estimated 264 others.

3. Security in Public Events 3 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 reaction Research Question: When and How to transfer resources?

4. 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  4

5. 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 5

6. 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 : 6 Attacker utility 0 r - # of resources Target value 014523 t - time ∞

7. Model: Strategies and Equilibrium Defender’s pure strategy: Initial assignment & All transfers Example: 2 targets (T1, T2),2 resources, transfer time 1 7 0 1 23 t - time (2, 0) (1, 0) (1, 1) Attacker’s pure strategy: Attack target i at time t Equilibrium: Minimizing the maximum attacker utility T1 T2T2

8. 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 v 1 (t) v 2 (t) t-time 014523 Target value 8 0 0 Attacker utilityResources Minimax assignment at time 0 T1 T2T2 Infeasible to find the minimax assignment at each time point since time is continuous

9. SCOUT-A:Negligible Transfer Time (2) 9 t-time 01 4 523 Attacker Utility ‘Minimax assignment’ does not change continuously 0 0 v 2 (t) v 1 (t) v 1 (t) / e λ v 1 (t) / e 2λ v 2 (t) / e λ v 2 (t) / e 2λ T1 T2T2 T2 T1 T2 SCOUT-A computes the time point at which a minimax assignment ‘expires’, then finds the ‘next’ minimax assignment

10. 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 10 The equilibrium of the constructed game is also an equilibrium of the initial game 0 tete  0 tete Transfer at any time Transfer at discretized points Initial Game Constructed Game

11. SCOUT-C: Nonzero Transfer Time (2) 11 target pair (i, j), assignment of resources (a i, a j ), compute Transfers can only begin at θ

12. Experimental Results: Solution Quality *more in the paper (a) Varying transfer time(b) Varying value of λ 12 Value of targets: [0, 50] Baseline  SDS: Optimal static defender strategy  DDS: Optimal dynamic defender strategy with arbitrarily discretized time

13. 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 13 Thank you! melody1235813@gmail.com