Markov Game Analysis for Attack and Defense of Power Networks Chris Y. T. Ma, David K. Y. Yau, Xin Lou, and Nageswara S. V. Rao

Power Networks are Important Infrastructures (And Vulnerable to Attacks) Growing reliance on electricity Aging infrastructure Introduced more connected digital sensing and control devices (and attract attacks on cyber space) Hard and expensive to protect Limited budget How to allocate the limited resources? – Optimal deployment to maximize long-term payoff

Modeling the Interactions – Game Theoretic Approaches Static game – Each player has a set of actions available – Players act simultaneously – Outcome and payoff determined by action of all players

Static Game Example Defend & Attack Defend & No Attack No defend & Attack No defend & No Attack

Modeling the Interactions – Game Theoretic Approaches Leader-follower game (Stackelberg game) – Defender as the leader – Adversary as the follower – Bi-level optimization – minimax operation Inner level: follower maximizes its payoff given a leader’s strategy Outer level: leader maximizes its payoff subject to the follower’s solution of the inner problem

Stackelberg Game Example Defend No defend Attack No Attack Attack No Attack Only model one-time interactions

Modeling the Interactions – Markov Decision Process Markov Decision Process (MDP) – System modeled as set of states with Markov transitions between them – Transition depends on action of one player and some passive disruptors of known probabilistic behaviors (acts of nature)

Markov Decision Process (MDP) Example (2 states, each has 2 actions available) updown Defend No defend Recover No recover Only models one intelligent player

Our Approach – Markov Game Generalization of MDP to an adversarial setting – Models the continual interactions between multiple players Players interact in the new state with different payoffs – Models probabilistic state transition because of inherent uncertainty in the system (e.g., random acts of nature)

Problem Formulation Defender and adversary of a power network Game formulation: – Adversary Actions: which link to attack Payoff: cost of load shedding by the defender because of the attack – Defender Actions: which (up) link to reinforce or which (down) link to recover Payoff: cost of load shedding because of the attack – Two-player zero-sum game

Markov Game – Reward Overview Assume five links; link 4 both attacked and defended (u,u,u,u,u) (u,u,u,d,u) (u,u,u,u,u) (u,u,u,d,u) p1p1 1-p 1 Immediate reward of such actions is the weighted sum of successful attack and successful defense Assume at state (u,u,u,d,u), link 4 both attacked and defended again p2p2 1-p 2 Immediate reward at state (u,u,u,d,u) is then the weighted sum of successful recovery and failed recovery This immediate reward is further “propagated” back to the original state (u,u,u,u,u) with a discount factor Hence, actions taken in a state will accrue a long-term reward

Solving the Markov Game – Definitions

Finding the Optimal Strategy – Solving a Linear Program

Solving the Markov Game – Value Iteration Dynamic program (value iteration) to solve the Markov game

Experiment Results Link diagram State {u,u,u,u,u} Links 4 and 5 both connect to generator, and generator at bus 4 has higher output

Experiment Results Payoff Matrix of state {u,u,u,u,u} for the static game. Payoff Matrix of state {u,u,u,u,u} for the Markov game. (ϒ = 0.3)

Conclusions Using Markov game to model the attack and defense of a power network between two players Results show the action of players depends not only on current state, but also later states – To obtain the optimal long term benefit