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Ambiguity Increases and Insurance Deductibles

ambiguity increase.
Our results augment the literature on ambiguity as follows. First, our results extend
the work of Birghila et al. (2023). They show that under the maxmin model, the optimal
insurance contracts with and without the sabotage condition imposed on the indemnity

function are in the deductible form, although with different shapes of the optimal
indemnity function. We study how ambiguity affects the optimal deductible level under
the more general α-maxmin model, without assuming the sabotage condition. In terms

of studying the optimal deductible level when the risk is ambiguous, our results can be
treated as a dual version of Alary et al. (2013) and Gollier (2014) but under a different
model. Given certain ambiguity structures, these authors study how the optimal deductible
changes under ambiguity aversion compared to ambiguity neutrality for individuals with
smooth-ambiguity-aversion preferences. Alary et al. (2013) show that when ambiguity

occurs only in the no-loss state, ambiguity aversion reduces the optimal deductible relative
to ambiguity neutrality under the smooth ambiguity aversion model. Using the same
model, Gollier (2014) studies a different ambiguity structure. When ambiguity affects only

*
losses below D , he shows that, for possible loss distributions ranked according to FSD,
introducing ambiguity aversion is sufficient to raise the optimal deductible. On the other
hand, given ambiguity aversion, we examine how the optimal deductible changes under an
ambiguity increase for α-maxmin preferences.

3.2 A General Ambiguity Increase
We now discuss a general ambiguity increase. In this case, we do not preserve the
cumulative loss probability at D , which is required in the case of a specific ambiguity
*
F
increase. The general ambiguity increase is described in Definition 2. Let Ġ (x,α) denote
i
αG(x;π ) + (1 - α)G(x;¯π ), where i = F or T. The exogenous variables are suppressed to
i
_ i
*
simplify notations. Accordingly, FOC (8) divided by (1 + τ)(1 - G(D ;π )) becomes
*
F
∗
( ∗ )
(9)
− − 1 ∗ ( ∗ ) ∗ = 0.
( ) ∗ ∗
As shown in Appendix E, finding the determining condition under which the general

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