AI 中文总结
本文提出基于α-势博弈的N人随机LQ微分博弈框架，推导了成本函数导数的等价表示，得到α-Nash均衡，并验证其与现有网络LQ博弈方法的一致性。
AI 中文摘要
本文从α-势博弈的视角研究N人随机线性二次（LQ）微分博弈。首先考虑带乘性噪声的闭环LQ博弈，其中漂移项和扩散项均线性依赖于状态和完整控制向量。针对该模型，推导了参与者成本函数一阶和二阶线性导数的概率表示与偏微分方程（PDE）表示，并证明二者等价。随后开发了开环随机LQ α-势博弈框架，利用线性导数构造方法构建α-势函数，并基于模型系数和容许控制半径推导近似参数α的显式上界。此外，通过将状态与变分过程增广，α-势函数的最小化问题被转化为有限维随机控制问题，从而得到开环α-Nash均衡。作为应用，重新研究了文献[GuoLiZhang2025]中考虑的网络LQ博弈，结果表明本文方法得到的反馈表示与现有条件McKean-Vlasov方法中的反馈一致，且本文的刻画直接源自标准有限维LQ控制问题。
英文摘要
This paper studies $N$-player stochastic linear-quadratic (LQ) differential games from the perspective of $α$-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 \(α\)-potential game framework. Using the linear derivative construction, we build an \(α\)-potential function and derive an explicit upper bound for the approximation parameter \(α\) in terms of the model coefficients and the admissible control radius. Moreover, the minimization of the \(α\)-potential function is reduced to a finite-dimensional stochastic control problem by augmenting the state with the variational process, which yields an open-loop \(α\)-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.
Comments38 pages
