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Ambiguity Increases and Insurance Deductibles
Note: In Panels A and B of Figure 1, the horizontal axis represents a discrete loss taking a value in the set of {0, 1, 2, 3} and
the vertical axis represents the cumulative loss probability at each value of .
Note: In Panels A and B of Figure 1, the horizontal axis represents a discrete loss taking a value in the set of {0, 1, 2, 3} and
We justify these assumptions as follows. As noted by Jewitt and Mukerji (2017), Rogers and Ryan
the vertical axis represents the cumulative loss probability at each value of .
concerning ambiguity-neutral preferences is not influenced by the presence of ambiguity
(2012) prove that, for the -maxmin preferences under Assumption 2, ambiguity neutrality is described
We justify these assumptions as follows. As noted by Jewitt and Mukerji (2017), Rogers and Ryan
and to characterize ambiguity neutrality with a specific value of α (Jewitt and Mukerji,
by . Since , the center ) Δ is )+ ). We make this assumption
�
�
�
∗
� (2012) prove that, for the -maxmin preferences under Assumption 2, ambiguity neutrality is described
�
�
�
�
2017; Dietz and Walker, 2019; Huang, 2025). Consistent with Assumption 1, when α
�
�
to ensure that the decision-making concerning ambiguity-neutral preferences is not influenced by the
�
by . Since , the center ) Δ is )+ ). We make this assumption
�
�
∗
�
�
�
*
�
= , the loss distribution is G(x;π ) under ambiguity neutrality. Under Assumption 3,
�
�
�
presence of 1 2 ambiguity and to characterize ambiguity neutrality with a specific value of (Jewitt and
to ensure that the decision-making concerning ambiguity-neutral preferences is not influenced by the
G(x;π ) and G(x;¯π ) are the worst and best loss distributions, respectively. We make this
Mukerji, 2017; Dietz and Walker, 2019; Huang, 2025). Consistent with Assumption 1, when , the
�
F
_ F
presence of ambiguity and to characterize ambiguity neutrality with a specific value of (Jewitt and
�
13
loss distribution is ) under ambiguity neutrality. Under Assumption 3, ) and ) are
∗
assumption to identify the minimal and maximal expected utility in decision-making. �
Mukerji, 2017; Dietz and Walker, 2019; Huang, 2025). Consistent with Assumption 1, when , the
�
�
Finally, Assumptions 2 and 3 together imply that, for an increasing utility function,
the worst and best loss distributions, respectively. We make this assumption to identify the minimal and �
loss distribution is ) under ambiguity neutrality. Under Assumption 3, ) and ) are
∗
�
13
maximal expected utility in decision-making. Finally, Assumptions 2 and 3 together imply that, for an �
G(x;π ) ≤ G(x;π ) ≤ G(x;¯π ) for all x ∈ [0,L].
*
the worst and best loss distributions, respectively. We make this assumption to identify the minimal and
F
_ F
increasing utility function, ) ≤ ) ≤ ) for all 0, .
∗
Under Assumptions 1–3, the individual decides the optimal I(x) and P to maximize
13
maximal expected utility in decision-making. Finally, Assumptions 2 and 3 together imply that, for an
�
�
Under Assumptions 1–3, the individual decides the optimal ) and to maximize his/her utility in
increasing utility function, ) ≤ ) ≤ ) for all 0, .
his/her utility in the presence of ambiguity, which is formulated by the α-maxmin model as
∗
�
�
the presence of ambiguity, which is formulated by the -maxmin model as
Under Assumptions 1–3, the individual decides the optimal ) and to maximize his/her utility in
the presence of ambiguity, which is formulated by the -maxmin model as (2)
(2)

max � m ) , , , , ) + 1 )max ) , , , , )� ,
� �)��,� � � � � � �
max � m ) , , , , ) + 1 )max ) , , , , )� , (2)
subject to as defined in Equation (1). The term to be maximized in the objective function (2) can be
� �)��,�
subject to P as defined in Equation (1). The term to be maximized in the objective function
� � �
� � �
rearranged as
subject to as defined in Equation (1). The term to be maximized in the objective function (2) can be
(2) can be rearranged as
rearranged as
�
) , , , , , ) � + )) � ) + 1 ) )�, (3)
�
�
�
�
�
(3)
� � ) )�, (3)
where the utility function , is assumed to be continuous and differentiable, with >0 and <0
) , , , , , ) � + )) � ) + 1
��
�
�
�
�
�
�
representing risk aversion. Moreover, , 1 represents ambiguity aversion, and ) + 1
where the utility function , is assumed to be continuous and differentiable, with >0 and <0
� �
��
�
where the utility function u, is assumed to be continuous and differentiable, with u' > 0
) ) can be viewed as the loss distribution distorted by ambiguity aversion under ambiguity. �
�
representing risk aversion. Moreover, , 1 represents ambiguity aversion, and ) + 1
�
�
) ) can be viewed as the loss distribution distorted by ambiguity aversion under ambiguity.
1
and u'' < 0 representing risk aversion. Moreover, α ∈ ( ,1] represents ambiguity aversion,
2.2 The Optimal Insurance Contract 2
�
and αG(x;π ) + (1 - α)G(x;¯π ) can be viewed as the loss distribution distorted by ambiguity
_ F
F
2.2 The Optimal Insurance Contract
aversion under ambiguity.
In the absence of ambiguity, under Assumption 1, the loss distribution is ). Arrow (1971)
∗
proves that, for a risk-averse individual who maximizes his/her expected utility, the optimal insurance
In the absence of ambiguity, under Assumption 1, the loss distribution is ). Arrow (1971)
∗
contract is a straight deductible 0, ) such that ) max 0, ) for all 0, . Under the
∗
∗
2.2 The Optimal Insurance Contract
proves that, for a risk-averse individual who maximizes his/her expected utility, the optimal insurance
deductible design, for a loss ≤ , the individual obtains no indemnity and takes on all the loss. For a
∗
*
contract is a straight deductible 0, ) such that ) max 0, ) for all 0, . Under the
In the absence of ambiguity, under Assumption 1, the loss distribution is G(x;π ).
∗
∗
deductible design, for a loss ≤ , the individual obtains no indemnity and takes on all the loss. For a
∗

13 Under the smooth ambiguity aversion model, Alary et al. (2013) imply a similar relationship between the expected utility

and the decision maker’s beliefs, with a negative sign arising from the assumption that the loss probability increases with
13 Under the smooth ambiguity aversion model, Alary et al. (2013) imply a similar relationship between
the belief. Gollier (2014) implies a similar relationship with a positive sign by assuming the loss distribution decreased with
13 Under the smooth ambiguity aversion model, Alary et al. (2013) imply a similar relationship between the expected utility
the expected utility and the decision maker’s beliefs, with a negative sign arising from the assumption
and the decision maker’s beliefs, with a negative sign arising from the assumption that the loss probability increases with
the belief. If we assume that �� � �) �,�,�,�,�) ≤0 like Alary et al. (2013), then the worst and best loss distributions are
that the loss probability increases with the belief. Gollier (2014) implies a similar relationship
the belief. Gollier (2014) implies a similar relationship with a positive sign by assuming the loss distribution decreased with
��
� ) and � ), respectively. On the other hand, under the maxmin expected utility, Birghila et al. (2023) find a
with a positive sign by assuming the loss distribution decreased with the belief. If we assume that
�� � �) �,�,�,�,�)
≤0 like Alary et al. (2013), then the worst and best loss distributions are
the belief. If we assume that
closed-form solution for the probability in the worst case and obtain the probability through a numerical approach.
∂ν(I(x);u,z,α,G,π)
��
like Alary et al. (2013), then the worst and best loss distributions are G(x;¯π F ) and
≤ 0
� ) and � ), respectively. On the other hand, under the maxmin expected utility, Birghila et al. (2023) find a
∂π
G(x;π F ), respectively. On the other hand, under the maxmin expected utility, Birghila et al. (2023)
closed-form solution for the probability in the worst case and obtain the probability through a numerical approach.
11
_
find a closed-form solution for the probability in the worst case and obtain the probability through a
numerical approach. 11
12