1. Non-additive Security Games
The Thirty-First AAAI Conference on Artificial Intelligence
Non-additive Security Games
Sinong Wang, Fang Liu and Ness Shroff
Electrical and Computer Engineering
Feb

2. Classic Security Game Model
Two players: attacker and defender
n targets:
Defender’s resources:
Homogenous resources
Heterogeneous resources
Attacker choose only one target to attack
Defender will get benefit if if target i is covered, and loss if target i is uncovered;
The attacker’s utility is denoted similarly by
Resources
Targets
Defender
Attacker
Gap>0
covered
uncovered
Gap>0

3. Classic Security Game Model
Pure strategies
Defender’s pure strategy can be represented by a subset , the defender’s pure strategy space is
Attacker’s pure strategy is , the attacker’s pure strategy space is simply
Toy example, defender can assign at most two resources
D={1,3} D={1,n}
Resources
Resources
Targets
Targets n n

4. Related Works Challenges to solve the security game
Exponential large defender pure strategy space
Research directions
Efficient algorithm design (heuristic algorithm, might be exponential time in the worst case)
Examining the complexity of the security game (polynomial time algorithm design or NP-hard)
Extending to other settings (uncertain attacker behavior, etc.)

5. Compact representation (Kiekintveld, AAMAS 09’)
Related Works
Examining the complexity of the security game
Compact representation (Kiekintveld, AAMAS 09’)
Recover mixed strategy and complexity of several security games (Korzhyk, AAAI 11’)
Security game with multiple attacker resources (Korzhyk, AAAI 13’)
Complexity of security game with single attacker resource (Yin 09’;Xu, EC 16’)
Complexity of security game with multiple attacker resources and non-additive utility functions?

6. Non-additive Security Game Model
Complete pure strategy space:
Attacker and defender’s pure strategy space is the power set of [n]:
Non-additive utility function:
Benefit function:
Attacker cost function:
Defender cost function:
The expected utility is given by
Bilinear form
Defender’s pure strategy is is a feasible assignment of resources to targets

7. Challenges and open questions
Both exponential large attacker’s and defender’s pure strategy space
Exponential number of utility functions
Main results:
Condition of compactly representing the NASG.
An oracle-based algorithm to efficiently compute the mixed strategies of NASG.
The complexity of computing the mixed strategies of NASG.
Proposition 1. The set of Nash equilibriums of NASG is equivalent to the set of Nash equilibriums of zero-sum game with payoff matrix

8. Compact Representation
Based on von Neumann’s minimax theorem, computing the NE of zero- sum game can be formulated as the following minimax problem,
Challenges: exponential large optimization problem
Conceptual derivation:
r is the rank of payoff matrix, if we define affine transformation:
We can obtain the following equivalent optimization problem
We can further simplify it as
size dependent on rank!

9. Compact Representation
Decomposable property of payoff matrix
Q is binary matrix, D is diagonal matrix, V and L are two sparse matrices with only one non-zero column and row.
Close-from expression of each elements:
Common utilities:
Theorem 1. They payoff matrix can be decomposed as

10. Compact Representation
Formal description
Based on above decomposition results we can let elementary matrices
Affine transformation:
Compact minimax problem
S is defined as the support set of the NASG:
transformation
projection

11. Compact Representation
Sufficient condition of compact representation
Theorem 2. The rank of payoff matrix is equal to
is NE of NASG
is optimal solution of compact problem
Implication of compact representation
where and
Additive case:
Common utilities satisfy
Support set S = {1,2,3,…,n}.

12. Oracle-based Algorithmic Framework
Our results
Main technical development:
A geometric approach
Ellipsoid method
Minimax Problem:
poly(n) reduction
Defender oracle problem: for any vector , compute

13. Oracle-based Algorithmic Framework
The main technical development
The poly(n) time reduction from minimax problem to compact NASG.
The poly(n) time from compact NASG to defender oracle problem.
poly(n) reduction
poly(n) reduction

14. Solving NASG is a Combinatorial Problem
Restriction:
Attacker attack at most c targets, defender defend at most k targets.
Defender’s cost function are additive.
Main results:
Main technique:
Partial matrix decomposition
Polytope transformation and projection.
Reduction between the separation oracles of two polytopes.
D - Minimax Problem:
poly(n) reduction
Defender oracle problem: for any vector , compute

15. What is the defender oracle problem
Defender oracle problem is indeed a combinatorial optimization problem.
When the utility function is additive, the defender oracle problem is a linear optimization problem.
Theorem 7. The defender oracle problem is equivalent to, for any vector
, maximize a pesudo - boolean function under a cardinality constraint

16. Application to the Network Security Domain
Definition. Network security game is given by a tuple (G,T,F,c)
G=(V,E) with node set V and edge set E
T is the network value function, F is the failure operator
c is the maximum number of targets attacker can choose
First solvable class: additive utility functions such as betweenness centrality.
Second solvable class: separable support set S with largest component equal to
Third solvable class: negative common utilities except for singleton set

17. Conclusions and Future Work
We extended the security game to the multiple attacker resources and non- additive utility functions
We proposed a completely new theoretical framework
We extend both the polynomial solvable and NP-hard classes
Future works:
The computation of Nash equilibrium of non-zero-sum and non-additive security game
The approximate version of our equivalence theorem
Uncertain attacker behavior

18. Q&A
Thanks