Markov Game Analysis for Attack and Defense of

Power NetworksChris 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– Outcome and payoff determined by action of all

players– Players act simultaneously

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 NoAttack 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)

up down

Defend

No defend

Recover

No recover

0.9

0.6

0.1

0.9

0.1

0.4

0.9

0.1

Only models one intelligent player

Our Approach – Markov Game

• Generalizations 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 underlying physical system (e.g., random acts of nature)

Problem Formulation

• Defender and adversary of a power network– Two-player zero-sum game

• 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

Markov Game – Reward Overview• Assume five links; link 4 both attacked and defended

(u,u,u,u,u) (u,u,u,u,u)

(u,u,u,d,u)

(u,u,u,u,u)

(u,u,u,d,u)

p1

1-p1

• 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

p2

1-p2

• 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