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Ambiguity Increases and Insurance Deductibles

values are 0.25, 0.5, 0.75, and 1, respectively. The dark dashed line draws G(x;π ), under
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which the cumulative loss probabilities are 0.2, 0.4, 0.6, and 1, respectively. Furthermore,
the dark solid line draws G(x;¯π ), under which the cumulative loss probabilities are 0.3,
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0.6, 0.9, and 1, respectively. Now, suppose that there is a loss distribution, G(x;π ) ∈ Λ (the
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light solid line), under which the cumulative loss probabilities are 0.25, 0.45, 0.8, and 1,
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respectively. There is another distribution, G(x;π ) = G(x;π ) - [G(x;π ) - G(x;π )] (the light
1
1
dashed line), which is constructed by G(x;π ) when α = 0 and, for x > 0, symmetrically
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expanding the center G(x;π ) in the opposite direction with the difference between G(x;π )
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and G(x;π ) (see Panel A). The set Δ is central symmetric because both G(x;π ) and
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G(x;π ) belong to ∈ Δ for all x ∈ [0,3] with the center G(x;π ), which satisfies Definition 1.
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A counterexample is provided in Panel B of Figure 1. Here, we consider a loss
distribution G(x;π ) ∈ Δ (the light solid line), under which the cumulative loss probabilities
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are 0.4, 0.7, 0.9, and 1, respectively, and the distribution G(x;π ) (the light dashed line)
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2
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constructed in a way similar to G(x;π ). Except for the loss x = 3, both G(x;π ) and G(x;π )
2
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2
are outside of Δ for all x. In this case, Definition 1 is not satisfied, and thus, Δ is not
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central symmetric.
We adopt the α-maxmin model to characterize the individual’s decision problem.
The α-maxmin model takes the form of the α-weighted average of the maxmin expected
utility and the maxmax expected utility, where α ∈ [0,1] describes the decision-maker’s
ambiguity preference. Let z denote the net wealth after loss and ν(I(x);u,z,α,G,π) denote
utility under the insurance contract with the indemnity function I(x). To proceed with our
analysis, we make the following assumptions for Δ and v, consistent with the study of the
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optimal coinsurance resulting from an ambiguity increase (Huang, 2025).
Assumption 2: The set Δ ⊆ Λ is compact, convex, and centrally symmetric (Definition 1)
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with a center G(x;π ) ∈ Δ , where π ∈ Π .
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Assumption 3: ν(I(x);u,z,α,G,π) is nondecreasing in π.
We justify these assumptions as follows. As noted by Jewitt and Mukerji (2017),
Rogers and Ryan (2012) prove that, for the α-maxmin preferences under Assumption
1
1
*
2, ambiguity neutrality is described by α = . Since π ∈ Π , the center G(x;π ) ∈ Δ is
2
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2
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G(x;π ) + G(x;¯π ). We make this assumption to ensure that the decision-making
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