An $\alpha$-Potential Game Approach to $N$-Player Stochastic Linear-Quadratic Differential Games

Abstract

This paper studies $N$-player stochastic linear-quadratic (LQ) differential games from the perspective of $\alpha$-potential games. We first consider a closed-loop LQ game with multiplicative noise, where both the drift and the diffusion coefficients depend linearly on the state and the full control vector. For this model, we derive probabilistic and partial differential equation (PDE) representations for the first- and second-order linear derivatives of the players'cost function and prove the equivalence between them. We then develop an open-loop stochastic LQ \(\alpha\)-potential game framework. Using the linear derivative construction, we build an \(\alpha\)-potential function and derive an explicit upper bound for the approximation parameter \(\alpha\) in terms of the model coefficients and the admissible control radius. Moreover, the minimization of the \(\alpha\)-potential function is reduced to a finite-dimensional stochastic control problem by augmenting the state with the variational process, which yields an open-loop \(\alpha\)-Nash equilibrium. As an application, we revisit a network LQ game considered in \cite{GuoLiZhang2025} and show that the feedback representation obtained from our approach coincides with the feedback in the existing conditional McKean--Vlasov approach, while our characterization follows directly from a standard finite-dimensional LQ control problem.