Divorce as a Saddle-Node Bifurcation in a Stochastic Differential Game under Dual Uncertainty: A Theoretical Framework

Abstract We propose a fully theoretical framework in which marital dissolution emerges as a nonlinear phase transition in a stochastic differential game. Each spouse is modeled as a strategic agent whose cooperation level evolves continuously under mutual influence and external uncertainty. We introduce the concept of dual uncertainty— structural and normative—which reshapes the payoff landscape. At the micro level, we formulate the interaction as a two-player stochastic differential game and characterize feedback Nash equilibria via coupled Hamilton-Jacobi-Bellman equations. By deriving a reduced-order dynamical system for an aggregate cooperation variable, we prove the existence of a saddle-node bifurcation separating stable cooperation from inevitable defection. At the macro level, we construct a McKean-Vlasov-type mean-field model describing the population distribution of marital states. The framework establishes divorce as a critical transition, not a gradual accumulation of risk factors, and provides a foundation for threshold-based interventions.