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Ambiguity Increases and Insurance Deductibles

the interval of accident probabilities as the degree of ambiguity to model asymmetric
information and the uncertainty of the accident probabilities in the equilibrium of
insurance markets. The longer the interval, the greater the ambiguity perceived. Based on
these approaches, the size of Π reflects the degree of ambiguity. Accordingly, as the set
F
Π becomes larger, the individual experiences an ambiguity increase. 16
F
*
Next, we define another set of π: Π = {π∣π ≤ π ≤ ¯π } ⊇ Π . Accordingly, π ∈ Π
T
F
T
_ T
F
*
implies that π ∈ Π . We say the individual faces an ambiguity increase when his/her belief
T
shifts from Π to Π . Additionally, let Δ denote the set G(x;π) when π ∈ Π . We also let
F
T
T
T
*
Assumptions 1–3 hold for Δ . Thus, Δ shares the same center G(x;π ) as Δ . Formally, we
F
T
T
define an ambiguity increase as follows.
Definition 2: Under Assumptions 2 and 3, an individual experiences an ambiguity
*
increase if his/her belief shifts from Π to Π , where π ∈ Π ⊆ Π , and Δ
T
T
F
F
F
*
and Δ share the same center G(x;π ).
T
Due to ambiguity neutrality, the loss distribution G(x;π ) used by the insurer for premium
*
pricing is not affected by an ambiguity increase as defined above. Thus, a change in the
premium results solely from a change in the deductible level chosen by the individual.
To provide a better understanding of Definition 2, Figure 3 shows a numerical
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*
example similar to the central symmetry example in Figure 1. Here, G(x;π ) (the
dark dotted line), G(x;π ) (the dark dashed line), and G(x;¯π ) (the dark solid line) are
F
_ F
unchanged. Suppose that the individual’s belief changes from Π to Π for an exogenous
T
F
reason. The sets of cumulative loss probabilities for G(x;π ) (the light dashed line) and
_ T
G(x;¯π ) (the light solid line) are {0.1,0.3,0.6,1} and {0.4,0.7,0.9,1}, respectively. In this
T
case, Assumptions 2 and 3 still hold, and figure shows that Δ ⊃ Δ with the same center
T
F
*
G(x;π ). Therefore, the belief change is an ambiguity increase as described in Definition 2.
16 Huang and Tzeng (2018) use a different definition of an ambiguity increase under the α-maxmin
model. They assume that there are only two possible loss distributions, and the probability π of
the better one (in terms of FSD) follows a distribution. The ambiguity increase is defined on the
distribution of π as the Nth-degree risk increase of Ekern (1980) preserving the α-weighted average of
π. Actually, their definition implies a broader set of π than ours.
17 We appreciate this suggestion proposed by an anonymous reviewer.

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