Counterclockwise Dissipativity, Potential Games, and Evolutionary Nash Equilibrium Learning

We use system-theoretic passivity methods to study evolutionary Nash equilibrium learning in large populations of agents engaged in strategic, non-cooperative interactions. The agents follow learning rules that specify their strategic preferences, while a payoff mechanism ascribes payoffs to the available strategies. The population's aggregate strategic profile is the state of an associated evolutionary dynamical system. Evolutionary Nash equilibrium learning refers to the convergence of this state to the set of Nash equilibria of the payoff mechanism. Most approaches consider memoryless payoff mechanisms, such as potential games. Methods using <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula>-passivity and equilibrium independent passivity (EIP) have introduced dynamic payoff mechanisms. However, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula>-passivity does not hold when agents follow rules exhibiting “imitation” behavior, such as in replicator dynamics. Conversely, EIP applies to the replicator dynamics but not to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula>-passive rules. We address this gap using counterclockwise dissipativity (CCW). First, we prove that any Lipschitz continuous memoryless payoff mechanism is CCW if and only if it is a potential game. Second, under any (possibly dynamic) CCW payoff mechanism, we establish evolutionary Nash equilibrium learning for all rules in a convex cone that includes most rules studied to date, such as, in particular, imitation and continuous <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula>-passive rules.