2 Introduction to Imprecise Probabilities
67
But 4 is not necessarily the maximum buying price for g. This leads to the idea of
natural extension, wherein we try to assess a new gamble outside the domain based
on our assessment of the gambles inside.
Definition 2.19 (Natural Extension) Let P be a lower prevision and g i ∈
domP for i = 1, 2, . . . , n; then, for any gamble g, we can define the natural
extension E of P as follows:
E(g) = sup
a ∈ R : g − a ≥
n
i=1
λ i [g i − P (g i )], n ∈ N, λ i ∈ R ≥0
(2.17)
The natural extension in Eq. (2.17) can be derived directly from the desirability
axioms. The λ i [g i −P (g i )]’s are desirable because of the positive homogeneity. The
n
i=1 λ i [g i − P (g i )] is desirable because of the combination of desirable gambles.
The term g − a is desirable because of the monotonicity of desirable gambles.
Definition 2.20 (Coherence) A lower prevision, P , is called coherent if
P (g) = E(g)
(2.18)
for all g ∈ domP .
Coherence means that our supremum buying price of a gamble should not be
raised on the combination of other gambles. For example, in Table 2.2, we can
see that g 7 ≥ g 1 + g 2 . Therefore, we can dispose to buy g 7 for 4; it also avoids
sure loss. However, clearly, we can buy this gamble for 5 without any loss. This is
against the assumption that 4 is the supremum buying price for g 7 ; therefore, it leads
to inconsistency.
2.5.4 Duality
In the previous section, we derived the natural extension E of P . Now, for a finite
number of gambles g 1 , g 2 , . . . , g n and finite set of outcomes Ω ≡ {x 1 , x 2 , . . . , x k },
we can write Eq. (2.17) in the following manner:
67
But 4 is not necessarily the maximum buying price for g. This leads to the idea of
natural extension, wherein we try to assess a new gamble outside the domain based
on our assessment of the gambles inside.
Definition 2.19 (Natural Extension) Let P be a lower prevision and g i ∈
domP for i = 1, 2, . . . , n; then, for any gamble g, we can define the natural
extension E of P as follows:
E(g) = sup
a ∈ R : g − a ≥
n
i=1
λ i [g i − P (g i )], n ∈ N, λ i ∈ R ≥0
(2.17)
The natural extension in Eq. (2.17) can be derived directly from the desirability
axioms. The λ i [g i −P (g i )]’s are desirable because of the positive homogeneity. The
n
i=1 λ i [g i − P (g i )] is desirable because of the combination of desirable gambles.
The term g − a is desirable because of the monotonicity of desirable gambles.
Definition 2.20 (Coherence) A lower prevision, P , is called coherent if
P (g) = E(g)
(2.18)
for all g ∈ domP .
Coherence means that our supremum buying price of a gamble should not be
raised on the combination of other gambles. For example, in Table 2.2, we can
see that g 7 ≥ g 1 + g 2 . Therefore, we can dispose to buy g 7 for 4; it also avoids
sure loss. However, clearly, we can buy this gamble for 5 without any loss. This is
against the assumption that 4 is the supremum buying price for g 7 ; therefore, it leads
to inconsistency.
2.5.4 Duality
In the previous section, we derived the natural extension E of P . Now, for a finite
number of gambles g 1 , g 2 , . . . , g n and finite set of outcomes Ω ≡ {x 1 , x 2 , . . . , x k },
we can write Eq. (2.17) in the following manner: