Optimal Policies in an Insurance Stackelberg Game: Demand Response and Premium Setting

This paper examines a stochastic Stackelberg differential game between an insurer and a pool of homogeneous policyholders. Policyholders dynamically optimize insurance coverage and risky asset allocations to minimize the probability of wealth shortfall, while the insurer, acting as the leader, sets the premium loading to maximize the expected exponential utility of terminal surplus. Employing dynamic programming techniques, we derive closed-form equilibrium strategies for both parties. The analysis reveals that a strong positive correlation between insurance claims and financial market returns incentivizes full coverage with modest premiums, whereas a strong negative correlation may induce market collapse as insurers exit underwriting to exploit natural hedging opportunities. Furthermore, larger policyholder pools generate diversification benefits that reduce equilibrium premiums and stimulate insurance demand.