Nash Equilibrium and Belief Evolution in Differential Games

Differential games [4, 6] involve multiple players controlling a dynamical system through their actions, which are described by differential state equations. These games evolve over a continuous-time horizon, where each player seeks to optimize an objective function that depends on the system's state, their own actions, and potentially the actions of others. In this study, we extend the classic differential game model to scenarios involving motion-payoff uncertainty, where players face uncertainties in both the dynamic equations and the payoff functions, and are unaware of certain parameters in the environment or in their opponents' payoff structures. In dynamic games, optimal control techniques are generalized to accommodate multiple players with both shared and conflicting interests. As shown in [9], if a set of interconnected partial differential equations--commonly referred to as the Hamilton-Jacobi-Bellman (HJB) equations--has solutions, then a Nash equilibrium can be achieved. At this equilibrium, no player can improve their outcome by unilaterally changing their strategy. However, traditional dynamic game models often assume that all players possess complete knowledge of the game. In many real-world scenarios, players face rapidly changing and uncertain environments, leading to incomplete information about the system's dynamics and payoffs [22, 3, 15, 1]. To address this uncertainty, we apply Bayesian updating methods, where players update their beliefs about unknown parameters as new information becomes available.