Probabilities: A Historical Perspective - Information Fusion in Signal and Image Processing - page 266

De Finetti developed a betting theory to explain his argument: the probability p
attributed by an individual to an event E is given by the conditions on which this indi-
vidual would be willing to bet on this event, i.e. on which he would wager the sum
pS in order to win S if the event E occurred. Based on this definition, de Finetti first
showed that the sum of the probabilities of incompatible events has to be equal to 1.
Let

{

E 1 ,...E n }

be a complete class of incompatible events, let p i be their probabili-
ties (always assessed by an individual) and let S i be the stakes that correspond to each
of them. If the event E k

occurs, the gain G k is defined as the difference between the

corresponding stake S k and the sum of the wagers, meaning that:

n

G k = S k − p i S i . [A.10]

i =1

We obtain n equations of this type, corresponding to the n possible outcomes. If
we consider these equations as a system of n equations with n unknowns, i.e. the S i ,
the determinant of this system is equal to:

⎡
⎣

⎤
⎦

1 − p 1 − p 2 ··· − p n

− p 1 + p 2 + + p n .

− p 1 1 − p 2 ··· − p n

D = =1 ··· [A.11]

··· ··· ··· ···

− p 1 − p 2 ··· 1 − p n

If the determinant is not equal to zero, the system has a solution for any G k ,even
if every one of them is positive. This would not be consistent with the concept of
betting. It is difficult to conceive of a game that could always be won or in which it is
possible to give the opponent the possibility of certainly winning! Therefore, the only
consistent solution is that obtained when the determinant is equal to zero, i.e. when:

n

p i =1 . [A.12]

i =1

Furthermore, this condition is sufficient, since we then have:

n

p i G i =0 , [A.13]

i =1

and not all of the gains are positive.

De Finetti interprets the result this way: each evaluation (which is subjective) of

the p i such that i =1 p i =1is an acceptable evaluation, in other words one that
corresponds to a consistent opinion. The choice of an evaluation among those that are
acceptable is then no longer the objective at all.