Game theory has a long history, but computing Nash equilibria in dynamic non-cooperative games remains highly challenging. In this talk, we introduce a new framework for dynamic N-player games, called α-potential games, which avoids the homogeneity assumptions and infinite-population limits commonly imposed in mean field games. In an α-potential game, the change in a player’s objective under a unilateral deviation coincides with the change in an α-potential function up to an error of order α. This reduces the computation of approximate Nash equilibria to a stochastic control problem, substantially simplifying both analysis and computation. Moreover, the parameter α captures key structural features of the game, including the population size, interaction strength, and degree of heterogeneity across players.

We illustrate the framework through two examples, highlighting recent theoretical and algorithmic developments. For crowd-motion network games, we show that α=0 for all symmetric interaction networks. For asymmetric networks, we quantify the precise polynomial and logarithmic decay rates of α in terms of the number of players, network degree, and the decay rate of interaction asymmetry. We also apply the α-potential game framework to an N-player portfolio selection game under a mean–variance criterion, proving that the game is a potential game and explicitly constructing its Nash equilibrium. Our analysis allows for heterogeneous preference parameters, extending beyond the mean-field interaction structures commonly studied in the literature.

Finally, for general stochastic differential games,  the associated optimization problem is embedded into a conditional McKean–Vlasov control problem to analyze the α-NE. A verification theorem is established to construct α-NE based on solutions to an infinite-dimensional Hamilton-Jacobi-Bellman equation, which is reduced to a system of ordinary differential equations for linear-quadratic games.