1. CPS 173 Security games Vincent Conitzer conitzer@cs.duke.edu

2. Recent deployments in security Tambe’s TEAMCORE group at USC Airport security Where should checkpoints, canine units, etc. be deployed? Deployed at LAX and another US airport, being evaluated for deployment at all US airports Federal Air Marshals Coast Guard …

3. Security example action Terminal ATerminal B

4. Security game 0, 0-1, 2 -1, 10, 0 A B AB

5. Some of the questions raised Equilibrium selection? How should we model temporal / information structure? What structure should utility functions have? Do our algorithms scale? 0, 0-1, 1 1, -1-5, -5 D S DS 2, 2-1, 0 -7, -80, 0

6. Observing the defender’s distribution in security Terminal A Terminal B Mo TuWeThFr Sa observe This model is not uncontroversial… [Pita, Jain, Tambe, Ordóñez, Kraus AIJ’10; Korzhyk, Yin, Kiekintveld, Conitzer, Tambe JAIR’11; Korzhyk, Conitzer, Parr AAMAS’11]

7. Other nice properties of commitment to mixed strategies Agrees w. Nash in zero-sum games No equilibrium selection problem Leader’s payoff at least as good as any Nash eq. or even correlated eq. ( von Stengel & Zamir [GEB ‘10]; see also Conitzer & Korzhyk [AAAI ‘11], Letchford, Korzhyk, Conitzer [draft] ) ≥ 0, 0-1, 1 0, 0 -1, 1 1, -1-5, -5

8. Discussion about appropriateness of leadership model in security applications Mixed strategy not actually communicated Observability of mixed strategies? –Imperfect observation? Does it matter much (close to zero-sum anyway)? Modeling follower payoffs? –Sensitivity to modeling mistakes Human players… [Pita et al. 2009] 2, 14, 0 1, 03, 1

9. Example security game 3 airport terminals to defend (A, B, C) Defender can place checkpoints at 2 of them Attacker can attack any 1 terminal 0, -1 -2, 3 0, -1-1, 10, 0 -1, 10, -10, 0 {A, B} {A, C} {B, C} ABC

10. Set of targets T Set of security resources aavailable to the defender (leader) Set of schedules Resource  can be assigned to one of the schedules in Attacker (follower) chooses one target to attack Utilities: if the attacked target is defended, otherwise Security resource allocation games [Kiekintveld, Jain, Tsai, Pita, Ordóñez, Tambe AAMAS’09]  1  2 s1s1 s2s2 s3s3 t5t5 t1t1 t2t2 t3t3 t4t4

11. Game-theoretic properties of security resource allocation games [Korzhyk, Yin, Kiekintveld, Conitzer, Tambe JAIR’11] For the defender: Stackelberg strategies are also Nash strategies –minor assumption needed –not true with multiple attacks Interchangeability property for Nash equilibria (“solvable”) no equilibrium selection problem still true with multiple attacks [Korzhyk, Conitzer, Parr IJCAI’11] 1, 21, 02, 2 1, 11, 02, 1 0, 10, 00, 1

12. Scalability in security games Techniques: basic model [Kiekintveld, Jain, Tsai, Pita, Ordóñez, Tambe AAMAS’09; Korzhyk, Conitzer, Parr, AAAI’10; Jain, Kardeş, Kiekintveld, Ordóñez, Tambe AAAI’10; Korzhyk, Conitzer, Parr, IJCAI’11] games on graphs (usually zero-sum) [Halvorson, Conitzer, Parr IJCAI’09; Tsai, Yin, Kwak, Kempe, Kiekintveld, Tambe AAAI’10; Jain, Korzhyk, Vaněk, Conitzer, Pěchouček, Tambe AAMAS’11]; ongoing work compact linear/integer programs strategy generation

13. Compact LP Cf. ERASER-C algorithm by Kiekintveld et al. [2009] Separate LP for every possible t* attacked: Defender utility Distributional constraints Attacker optimality Marginal probability of t* being defended (?) Slide 11

14. Counter-example to the compact LP LP suggests that we can cover every target with probability 1… … but in fact we can cover at most 3 targets at a time 11 22.5 Slide 12 tt t t

15. Will the compact LP work for homogeneous resources? Suppose that every resource can be assigned to any schedule. We can still find a counter-example for this case: t tt.5 t tt rrr 3 homogeneous resources

16. Birkhoff-von Neumann theorem Every doubly stochastic n x n matrix can be represented as a convex combination of n x n permutation matrices Decomposition can be found in polynomial time O(n 4.5 ), and the size is O(n 2 ) [Dulmage and Halperin, 1955] Can be extended to rectangular doubly substochastic matrices.1.4.5.3.5.2.6.1.3 100 001 010 =.1 010 001 100 +.1 001 010 100 +.5 010 100 001 +.3 Slide 14

17. Schedules of size 1 using BvN  1  2 t1t1 t2t2 t3t3.7.1.7.3.2 t1t1 t2t2 t3t3 11.7.2.1 22 0.3.7 001 010 010 001 100 010 100 001.1.2.5

18. Algorithms & complexity [Korzhyk, Conitzer, Parr AAAI’10] Homogeneous Resources Heterogeneous resources Schedules Size 1P P (BvN theorem) Size ≤2, bipartite Size ≤2 Size ≥3 P (BvN theorem) P (constraint generation) NP-hard (SAT) NP-hard (3-COVER) Slide 16

19. Placing checkpoints in a city [Tsai, Yin, Kwak, Kempe, Kiekintveld, Tambe AAAI’10; Jain, Korzhyk, Vaněk, Conitzer, Pěchouček, Tambe AAMAS’11]