Online Performative Gradient Descent for Learning Nash Equilibria in Decision-Dependent Games

We study multi-agent games within the innovative framework of decision-dependent games, which establishes a feedback mechanism that population data reacts to agents' actions and further characterizes the strategic interactions among agents. We focus on finding the Nash equilibrium of decision-dependent games in the bandit feedback setting. However, since agents are strategically coupled, classical gradient-based methods are infeasible without the gradient oracle. To overcome this challenge, we model the strategic interactions by a general parametric model and propose a novel online algorithm, Online Performative Gradient Descent (OPGD), which leverages the ideas of online stochastic approximation and projected gradient descent to learn the Nash equilibrium in the context of function approximation for the unknown gradient. In particular, under mild assumptions on the function classes defined in the parametric model, we prove that the OPGD algorithm finds the Nash equilibrium efficiently for strongly monotone decision-dependent games. Synthetic numerical experiments validate our theory.