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Ambiguity Increases and Insurance Deductibles

show that, under the optimal insurance contract, not all the losses will be reimbursed by
the insurer when they occur. We present the result in the following proposition:
Proposition 2: Suppose that ambiguity preferences can be described by an α-maxmin
model. Under Assumptions 1–3, for a risk- and ambiguity-averse

individual, there exists at least one x ∈ [0,L] such that I(x) = 0 under the

optimal insurance contract after the introduction of ambiguity for all x ∈
[0,L].

Proof: See Appendix A.

The results of Propositions 1 and 2 are also shown by Gollier (2014) under the smooth
ambiguity aversion model.

Next, we consider when a straight deductible is optimal for the risk- and ambiguity-
averse individual after the introduction of ambiguity for all x ∈ [0,L].
Proposition 3: Suppose that ambiguity preferences can be described by an α-maxmin

model. Under Assumptions 1–3, for a risk- and ambiguity-averse
individual, there exists a straight deductible D ∈ (0,L) such that I(x) =
*
F
max(0,x – D ) for all x ∈ [0,L] which is optimal after the introduction of
*
F
ambiguity for all x ∈ [0,L] if the degree of ambiguity is sufficiently small.

Proof: See Appendix B.

By the continuity of the insurance contract, we prove that a small deviation from π
*
is sufficient to achieve the optimal straight deductible for the risk- and ambiguity-
averse individual. Following Huang (2025), the size of the set Π measures the degree
F
14
of ambiguity according to the α-maxmin model. Under Assumptions 1 and 2, when π
deviates more from π , the individual perceives more ambiguity. As long as the degree
*
of ambiguity is sufficiently small, the optimal insurance contract is a straight deductible.

Gollier (2014) also shows the optimality of a straight deductible, albeit under the

14 This ambiguity measure is explained detailedly in Section 3.

14