A Principal-Agent Mean-Field Game Model of Insurance with Risk Interdependence
We study an insurance contract-design problem under moral hazard, endogenous participation, and strategic risk interdependence. Because the resulting $N$-agent game suffers from the curse of dimensionality, we approximate the strategic interactions via a heterogeneous mean-field game. We rigorously establish the existence of a lower-level mean-field Nash equilibrium using measurable selection arguments and the Kakutani fixed-point theorem. By proving the $L^1$-Lipschitz continuity of the aggregate participation threshold, we further establish equilibrium uniqueness via a contraction mapping. We then embed this mean-field response into the insurer's upper-level Stackelberg optimization problem. We formulate the objective through general performance envelopes to accommodate potential equilibrium multiplicity, proving the existence of upper-level $\varepsilon$-optimal contracts, and demonstrating the existence of an exact Stackelberg equilibrium under the uniqueness regime. We conclude by extending the model to finite contract menus, providing numerical evidence that multi-contract screening improves the principal's expected payoff in interdependent risk environments.