1. Computing equilibria of security games using linear integer programming Dima Korzhyk dima@cs.duke.edu

2. Problem Description Targets t Utilities Resources n d, n a

3. Mixed strategies Defender strategy: probability of defending target t is d t, s.t. Attacker strategy: similarly, the probability of attacking t is a t, s.t.

4. Stackelberg strategy The defender commits to a mixed strategy d The attacker observes d and plays a best response a Note that the attacker’s best response doesn’t need to be randomized, so we can assume a is binary

5. Stackelberg – first formulation

6. Trick – fixing an implication with a binary right-hand side Suppose b is binary and we have implication if g(x) > 0 then b=1 Fix this as follows: b >= g(x) / M Here M is a sufficiently large number

7. Fixed conditions We can rewrite the implication As

8. Fixing the objective Note that the objective contains a product of two variables a t and d t

9. Trick – fixing a product of a binary and a continuous variable in the objective Suppose the objective is Maximize bx Where b is binary, and x>=0 is a continuous variable. Replace this with Maximize v v <= x v <= bM Where M is a sufficiently large number

10. MIP for computing a Stackelberg strategy

11. Nash equilibrium Find a pair of mixed strategies d,a such that both players are best-responding. d,a are not binary.

12. Trick – fixing the implications [aimms.com] Suppose we have an implication IF [g(x) > 0] THEN [h(x) > 0] This is equivalent to [g(x) 0] Now, we can use the either-or trick (next slide)

13. Either-Or Trick Suppose we have a constraint [g(x)>0] OR [h(x)>0] We need to change this to AND Replace with the following: g(x) + M b > 0 h(x) + M (1-b) > 0 where M is a sufficiently large number b is binary

14. Fixing the implications in the NE program

15. MIP for computing a Nash equilibrium

16. Scaling of the NE MIP in the number of targets/variables

17. Conclusion NE MIP is simpler than a specialized algorithm, but may take longer to execute A benefit is that we can combine the two MIPs in this presentation to find a defender strategy that is both a Stackelberg and a Nash strategy

18. Thank you! I thank Vince Conitzer, Ron Parr for detailed discussions.

19. References Security games (with a single attacker resource) were introduced in: Christopher Kiekintveld, Manish Jain, Jason Tsai, James Pita, Fernando Ordóñez, and Milind Tambe. Computing Optimal Randomized Resource Allocations for Massive Security Games. AAMAS-2009. Computing a Stackelberg strategy in a security game with multiple attacker resources s is NP-hard: Zhengyu Yin, Dmytro Korzhyk, Christopher Kiekintveld, Vincent Conitzer, and Milind Tambe. Stackelberg vs. Nash in security games: interchangeability, equivalence, and uniqueness. AAMAS-2010. Computing a Nash equilibrium in security games with multiple attacker resources can be done in polynomial time. The algorithm by Dmytro Korzhyk, Vincent Conitzer, Ronald Parr is currently unpublished and is under submission. Some of the integer programming tricks in this presentation were taken from AIMMS Modeling Guide http://www.aimms.com/aimms/download/manuals/aimms3om_integerprogrammingtricks.pdf