Dissipativity Theory in Game Theory: On the Role of Dissipativity and Passivity in Nash Equilibrium Seeking

I n this article, we show how dissipativity and passivity theory impacts game theory, in particular, learning in multiplayer games. Over the years, a plethora of algorithms/dynamics have been proposed in the game theoretic literature for learning (or seeking) a Nash equilibrium. From the best-response play, proximal dynamics and (projected) gradient-play to fictitious-play, payoff-based play or Q-learning (reinforcement-learning), the list is long. Herein, we consider some these popular game-theoretic algorithms and show how the principle of balancing passivity can explain their operation, as well as the trade-off between game properties and learning dynamics properties. We discuss how passivity and basic properties of interconnected systems lead to simplified proofs of convergence of such algorithms, and furthermore, how by leveraging them, novel algorithms and game dynamics with better properties can be generated.