Bowley solution of a mean–variance game in insurance

In this paper, we compute the Bowley solution of a one-period, mean–variance Stackelberg game in insurance, in which a buyer and a seller of insurance are the two players, and they act in a certain order. First, the seller offers the buyer any (reasonable) indemnity policy in exchange for a premium computed according to the mean–variance premium principle. Then, the buyer chooses an indemnity policy, given that premium rule. To optimize the choices of the two players, we work backwards. Specifically, given any pair of parameters for the mean–variance premium principle, we compute the optimal insurance indemnity to maximize a mean–variance functional of the buyer's terminal wealth. Then, we compute the parameters of the mean–variance premium principle to maximize the seller's expected terminal wealth, given the foreknowledge of what the buyer will choose when offered that premium principle. This pair of optimal choices, namely, the optimal indemnity and the optimal parameters of the premium principle, constitute a Bowley solution of this Stackelberg game. We illustrate our results via numerical examples.