2 Introduction to Imprecise Probabilities
65
2.5.2 Lower Previsions
From our example, we see that the dealer is offering five different gambles, which
are listed in Table 2.2. Now, we are willing to buy g 1 from the dealer. We are already
informed that C is injured, so the chance of winning for C is less than that of A.
Therefore, we can pay a higher price than the reward on C, as we expect C not to
win. However, for B, we are not sure, as we have no information about B. Therefore,
we will not spend more than three, as otherwise, if A wins, then we will end up
losing some amount. However, even if B wins, then we will earn some reward. We
show the transactions in the following Table 2.3.
Our supremum buying price 3 for g 1 can be seen as our lower prevision for g 1 .
We can formulate this behavioural interpretation of lower previsions as follows:
Definition 2.17 (Lower Prevision) Lower prevision P (g) of a gamble can
be seen as the supremum buying price of the gamble. Or in other words, P (g)
is the lowest value, such that we are willing to buy the gamble for all t <
P (g).
Similarly, we associate another map P (g) called upper prevision from the set of
gambles to the real numbers. Here, upper prevision stands for infimum selling price.
That is, if we own a gamble g, then we can sell the gamble for all t > P (g).
For any bounded gamble g, the following conjugacy property holds:
P (g) = −P (−g)
(2.14)
This allows us to express one type of functional in terms of the other [39].
Suppose after buying g 1 , we decide to buy another gamble g 2 for 3. Then, we
can see our total rewards in the following table.
We can see from Table 2.4 that we will end up losing some amount irrespective
of the outcome. Therefore, we incur a sure loss.
Definition 2.18 (Avoiding Sure Loss) A lower prevision is said to avoid sure
loss if for every gamble g 1 , g 2 , . . . , g n and for all non-negative real numbers
(continued)
Table 2.3 Betting on g 1
A
B C
Reward on g 1 3
5 −1
Buy g 1 for 3
0
2 −4
Buy g 1 for 5
−2 0 −6
65
2.5.2 Lower Previsions
From our example, we see that the dealer is offering five different gambles, which
are listed in Table 2.2. Now, we are willing to buy g 1 from the dealer. We are already
informed that C is injured, so the chance of winning for C is less than that of A.
Therefore, we can pay a higher price than the reward on C, as we expect C not to
win. However, for B, we are not sure, as we have no information about B. Therefore,
we will not spend more than three, as otherwise, if A wins, then we will end up
losing some amount. However, even if B wins, then we will earn some reward. We
show the transactions in the following Table 2.3.
Our supremum buying price 3 for g 1 can be seen as our lower prevision for g 1 .
We can formulate this behavioural interpretation of lower previsions as follows:
Definition 2.17 (Lower Prevision) Lower prevision P (g) of a gamble can
be seen as the supremum buying price of the gamble. Or in other words, P (g)
is the lowest value, such that we are willing to buy the gamble for all t <
P (g).
Similarly, we associate another map P (g) called upper prevision from the set of
gambles to the real numbers. Here, upper prevision stands for infimum selling price.
That is, if we own a gamble g, then we can sell the gamble for all t > P (g).
For any bounded gamble g, the following conjugacy property holds:
P (g) = −P (−g)
(2.14)
This allows us to express one type of functional in terms of the other [39].
Suppose after buying g 1 , we decide to buy another gamble g 2 for 3. Then, we
can see our total rewards in the following table.
We can see from Table 2.4 that we will end up losing some amount irrespective
of the outcome. Therefore, we incur a sure loss.
Definition 2.18 (Avoiding Sure Loss) A lower prevision is said to avoid sure
loss if for every gamble g 1 , g 2 , . . . , g n and for all non-negative real numbers
(continued)
Table 2.3 Betting on g 1
A
B C
Reward on g 1 3
5 −1
Buy g 1 for 3
0
2 −4
Buy g 1 for 5
−2 0 −6