Construction of an Optimal Strategy: An Analytic Insight Through Path Integral Control Driven by a McKean–Vlasov Opinion Dynamics

In this paper, we have constructed a closed-form optimal strategy within a social network using stochastic McKean–Vlasov dynamics. Each agent independently minimizes their dynamic cost functional, driven by stochastic differential opinion dynamics. These dynamics reflect agents’ opinion differences from others and their past opinions, with random influences and stubbornness adding to the volatility. To gain an analytic insight into the optimal feedback opinion, we employed a Feynman-type path integral approach with an appropriate integrating factor, marking a novel methodology in this field. Additionally, we utilized a variant of the Friedkin–Johnsen-type opinion dynamics to derive a closed-form optimal strategy for an agent and conducted a comparative analysis.