Scenario-Based Risk-Sensitive Computations of Equilibria for Two-Person Zero-Sum Games

A scenario-based risk-sensitive optimization framework is presented to approximate minimax solutions with high confidence. The approach involves first drawing several random samples from the maximizing variable, then solving a sample-based risk-sensitive optimization problem. This letter derives the sample complexity and the required risk-sensitivity level to ensure a specified tolerance and confidence in approximating the minimax solution. The derived sample complexity highlights the impact of the underlying probability distribution of the random samples. The framework is demonstrated through applications to zero-sum games and model predictive control for linear dynamical systems with bounded disturbances.