Analysis of Lyapunov Chaos in Game Dynamics

In game theory, taking zero-sum games as an example, there are many cutting-edge theoretical studies that believe the time-averaged behavior of agents will eventually lead to convergence to its Nash equilibrium, which shows economic stability to some extent. But in fact, from the behavioral perspective, it is very unreasonable to merely focus on the time-averaged agent behavior, and in this paper, we argue that in the context of zero-sum games, the day-to-day behavior in online learning dynamics is difficult to predict. Specifically, gradient ascent and descent algorithms will eventually lead to Lyapunov chaos in the payoff space. The small square area representing the initial condition value will expand exponentially in the payoff space over time, and this result is not only algorithmically robust, but also robust in the game-theoretical aspect. The topic of this paper will simulate and analyze the algorithms related to this chaotic phenomenon, and finally refute the conclusion that “the time-averaged behavior under online learning will eventually converge to the Nash equilibrium”.