  This paper investigates a Stackelberg game between an insurer and a reinsurer
under the $\alpha$-maxmin mean-variance criterion. The insurer can purchase
per-loss reinsurance from the reinsurer. With the insurer's feedback
reinsurance strategy, the reinsurer optimizes the reinsurance premium in the
Stackelberg game. The financial market consists of cash and stock with Heston's
stochastic volatility. Both the insurer and reinsurer maximize their respective
$\alpha$-maxmin mean-variance preferences in the market. The criterion is
time-inconsistent and we derive the equilibrium strategies by the extended
Hamilton-Jacobi-Bellman equations. Similar to the non-robust case in Li and
Young (2022), excess-of-loss reinsurance is the optimal form of reinsurance
strategy for the insurer. The equilibrium investment strategy is determined by
a system of Riccati differential equations. Besides, the equations determining
the equilibrium reinsurance strategy and reinsurance premium rate are given
semi-explicitly, which is simplified to an algebraic equation in a specific
example. Numerical examples illustrate that the game between the insurer and
reinsurer makes the insurance more radical when the agents become more
ambiguity aversion or risk aversion. Furthermore, the level of ambiguity,
ambiguity attitude, and risk attitude of the insurer (reinsurer) have similar
effects on the equilibrium reinsurance strategy, reinsurance premium, and
investment strategy.
