Computing Stackelberg Equilibria in Discounted Stochastic Games

Stackelberg games increasingly influence security policies deployed in real-world settings. Much of the work to date focuses on devising a fixed randomized strategy for the defender, accounting for an attacker who optimally responds to it. In practice, defense policies are often subject to constraints and vary over time, allowing an attacker to infer characteristics of future policies based on current observations. A defender must therefore account for an attacker's observation capabilities in devising a security policy. We show that this general modeling framework can be captured using stochastic Stackelberg games (SSGs), where a defender commits to a dynamic policy to which the attacker devises an optimal dynamic response. We then offer the following contributions. 1) We show that Markov stationary policies suffice in SSGs, 2) present a finite-time mixed-integer non-linear program for computing a Stackelberg equilibrium in SSGs, and 3) present a mixed-integer linear program to approximate it. 4) We illustrate our algorithms on a simple SSG representing an adversarial patrolling scenario, where we study the impact of attacker patience and risk aversion on optimal defense policies.