Optimal insurance under maxmin expected utility

Abstract

We examine a problem of demand for insurance indemnification, when the insured is sensitive to ambiguity and behaves according to the maxmin expected utility model of Gilboa and Schmeidler (J. Math. Econ. 18:141–153, 1989), whereas the insurer is a (risk-averse or risk-neutral) expected-utility maximiser. We characterise optimal indemnity functions both with and without the customary ex ante no-sabotage requirement on feasible indemnities, and for both concave and linear utility functions for the two agents. This allows us to provide a unifying framework in which we examine the effects of the no-sabotage condition, of marginal utility of wealth, of belief heterogeneity, as well as of ambiguity (multiplicity of priors) on the structure of optimal indemnity functions. In particular, we show how a singularity in beliefs leads to an optimal indemnity function that involves full insurance on an event to which the insurer assigns zero probability, while the decision maker assigns a positive probability. We examine several illustrative examples, and we provide numerical studies for the case of a Wasserstein and a Rényi ambiguity set.

Appendix: Convexity and compactness of the ambiguity set in Example 4.4

Lemma A.1

For a fixed $Q \in ca^{+}_{1}(\Sigma )$, let $\mathcal{C}_{\mathcal{W}}$ be the set defined as

$$ \mathcal{C}_{\mathcal{W}}:=\bigg\lbrace P\in ca^{+}_{1}(\Sigma ): \frac{dP}{dQ}=\frac{w(X)}{\int w(X)dQ},\, w\in \mathcal{W} \bigg\rbrace , $$

where $\mathcal{W} \subseteq L^{1} (\mathbb{R}, \mathcal{B}(\mathbb{R}), Q \circ X^{-1} )$ is a collection of nonnegative increasing weight functions such that $\int w(X)dQ > 0$ for all $w \in \mathcal{W}$. Then the following hold:

(i) If $\mathcal{W}$ is a convex cone, then $\mathcal{C}_{\mathcal{W}}$ is convex.

(ii) If $\mathcal{C}_{\mathcal{W}}$ is uniformly absolutely continuous with respect to some $\mu \in ca^{+}(\Sigma )$, then $\mathcal{C}_{\mathcal{W}}$ is weak^∗-compact.

Proof

(i) is easy to verify. To show (ii), first note that $\mathcal{C}_{\mathcal{W}}$ is norm-bounded. Since $\mathcal{C}_{\mathcal{W}}$ is also uniformly absolutely continuous with respect to $\mu \in ca^{+}(\Sigma )$, it follows from Dunford [33, Theorem IV.9.2] that $\mathcal{C}_{\mathcal{W}}$ is weakly sequentially compact and hence weak^∗-compact by Maccheroni and Marinacci [54, Theorem 1]. □

Remark A.2

In Lemma A.1, if $\mathcal{C}_{\mathcal{W}}$ is countable, that is, of the form

$$ \bigg\lbrace P_{n}\in ca^{+}_{1}(\Sigma ): n \in \mathbb{N}, \, \frac{dP_{n}}{dQ}=\frac{w_{n}(X)}{\int w_{n}(X)dQ},\, w_{n}\in \mathcal{W}\bigg\rbrace , $$

and if $\lim _{n \rightarrow \infty} P_{n}[A]$ exists for each $A \in \Sigma$, then the requirement of uniform absolute continuity of $\mathcal{C}_{\mathcal{W}}$ is superfluous by the Vitali–Hahn–Saks theorem (Dunford [33, Theorem III.7.2]).

Proposition A.3

If $\mathcal{W}$ is order bounded in the Banach lattice $L^{1} (\mathbb{R}, \mathcal{B}(\mathbb{R}), Q \circ X^{-1} )$ with a constant upper bound and a nonnegative lower bound having nonzero $L^{1}$-norm, then $\mathcal{C}_{\mathcal{W}}$ is uniformly absolutely continuous with respect to $Q$.

Proof

If $\mathcal{W}$ is order bounded in $L^{1} (\mathbb{R}, \mathcal{B}(\mathbb{R}), Q \circ X^{-1} )$ with a constant upper bound and a nonnegative lower bound having nonzero $L^{1}$-norm, then there exist $M \in \mathbb{R}_{+}$ and $f \in L^{1}_{+} (\mathbb{R}, \mathcal{B}(\mathbb{R}), Q \circ X^{-1} )$ such that $\Vert f\Vert _{1} = \int f\,d(Q \circ X^{-1}) > 0$ and $f \leq w \leq M$ for each $w \in \mathcal{W}$. Consequently, for each $P \in \mathcal{C}_{\mathcal{W}}$,

$$ \frac{dP}{dQ} \leq \frac{M}{\Vert f\Vert _{1}} < \infty . $$

Hence for each $P \in \mathcal{C}_{\mathcal{W}}$ and each $A \in \Sigma$,

$$ P[A] \leq \frac{M}{\|f\|_{1}} \,Q[A]. $$

Consequently, for each $\varepsilon > 0$, letting $\delta := \frac{\Vert f\Vert _{1}}{M} \, \varepsilon > 0$, it follows that for each $A \in \Sigma$ and each $P \in \mathcal{C}_{\mathcal{W}}$,

$$ Q[A] < \delta \text{ implies that } P[A] < \frac{M}{\Vert f\Vert _{1}} \, \delta = \varepsilon . $$

Hence $\mathcal{C}_{\mathcal{W}}$ is uniformly absolutely continuous with respect to $Q$. □