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Ambiguity Increases and Insurance Deductibles

increase if the ambiguity increase is consistent with Definition 2 and
preserves the cumulative loss probability at the initial optimal deductible
*
D .
F

Under this assumption, at D , the expected utility in the loss state covered by the insurer
*
F
is unaffected by the ambiguity increase. In other words, the net marginal utility benefit of
lowering the deductible is unchanged. This leads to the same result as the assumption of

*
an ambiguity increase affecting the loss below D ; that is, G(x;π ) = G(x;π ) and G(x;¯π ) =
_ F
F
F
_ T
*
G(x;¯π ) for losses above D , which implies a specific ambiguity increase. Similarly, Gollier
T
F
*
(2014) considers the ambiguity affecting losses below D to study the optimal insurance
contract under ambiguity aversion formulated by the smooth ambiguity aversion model.
19
To clarify Definition 3, we present a numerical example in Figure 4. Consider a
discrete loss x that takes a value in the set {0,1,2,3,4} and w = 5. The full specifications
of the loss distributions are provided in Panels A and B in Appendix C. We assume τ =
0.2, the CARA coefficient γ = 0.5, and α = 0.56, as assumed for the numerical example
of Proposition 3. Before an ambiguity increase, under the loss distributions G(x;π ) and
_ F
*
G(x;¯π ) (the dark dashed and solid lines, respectively) with center G(x;π ) (the dark
F
dotted line), we have D = 2. After an ambiguity increase, the loss distributions become
*
F
G(x;π ) and G(x;¯π ) (the light dashed and solid lines, respectively) with the same center. In
_ T
T
this case, Assumptions 2 and 3 hold. The figure indicates that G(2;π ) = G(2;π ) = 0.9285
_ T
_ F
and G(2;¯π ) = G(2;¯π ) = 0.9315, which satisfies Definition 3. Thus, the ambiguity increase
F
T
is the specific one.
Suppose that a specific ambiguity increase occurs. We show how a risk- and
ambiguity-averse individual respond to this specific ambiguity increase in the following
*
proposition, where D denotes the optimal deductible after the ambiguity increase.
T
Proposition 4: Suppose that ambiguity preferences can be described by an α-maxmin
model. Under Assumptions 1–3, a risk- and ambiguity-averse individual
reacts to a specific ambiguity increase as defined in Definition 3 as

follows:

19 We are grateful for an anonymous reviewer’s suggestion to illustrate this definition with a numerical
example.

20