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 Email: wang.7691@osu.edu Feb. 5 2017

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

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 1 2 3 n 1 2 3 n

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.)

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?

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

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

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!

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

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

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}.

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

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

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

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

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

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

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