race to be coherent, the sum of the probabilities over all the horses must be 1. This property characterises a ‘reasonable individual’. An example is presented in Section 1.7.6.
For a historical and philosophical discussion of subjective probabilities and a commentary on the work of de Finetti and Savage in the middle of the twentieth century, see Lindley (1980), Lad (1996), Taroni et al. (2001), Dawid (2004), Dawid and Galavotti (2009), Galavotti (2016, 2017), and Zynda (2016).
Savage, like de Finetti, viewed a personal probability as a numerical measure of the confidence a person has in the truth of a particular proposition. This opinion is viewed with scepticism today and was viewed with scepticism then (Savage 1967), as illustrated by Savage (1954).
I personally consider it more probable that a Republican president will be elected in 1996 than it will snow in Chicago sometime in the month of May, 1994. But even this late spring snow seems to me more probable than that Adolf Hitler is still alive. Many, after careful consideration, are convinced that such statements about probability to a person mean precisely nothing or, at any rate, that they mean nothing precisely. At the opposite extreme, others hold the meaning to be so self‐evident []. (p. 27)7
1.7.6 The Quantification of Probability Through a Betting Scheme
The introduction of subjective probability through a betting scheme is straightforward. The concept is based on hypothetical bets (Scozzafava 1987):
The force of the argument does not depend on whether or not one actually intends to bet, yet a method of evaluating probabilities making one a sure loser if he had to gamble (whether or not he really will act so) would be suspicious and unreliable for any purposes whatsoever. (p. 685)
Consider a proposition that can only take one of two values, namely, ‘true’ and ‘false’. There is a lack of information on the actual value of and an operational system is needed for the quantification of the uncertainty about imparted by the lack of information. A value is regarded as an amount to be paid to bet on with the conditions that a unit amount will be paid if is true and nothing will be paid if is false. In other words, is the amount to be paid to obtain an amount equal to the value of , that is associating the value 1 with ‘true’ and the value 0 with ‘false’. This idea was expressed by de Finetti (1940) in the following terms.
The probability of eventis, according to Mr NN, equal to 0.37, meaning that if the person was forced to accept bets for and against event, on the basis of the betting ratiowhich he can choose as he pleases, this person would choose. (p. 113)8
Coherence, as briefly described in Section 1.7.2, is defined by the requirement that the choice of does not make the player a certain loser or a certain winner. Denote an event which is certain, sometimes known as a universal set, as and an event which is impossible, sometimes known as the empty set, as so that if and the two possible gains are
When or , there is no uncertainty in the outcome of the corresponding bet and so the coherence (in the absence of uncertainty) requires the respective gains to be zero. The values of the gains are therefore
This happens when for and for . Therefore if the subjective probability of , that represents our degree of belief on , is defined as an amount , which makes a personal bet on the event or proposition coherent, then the probability satisfies two conditions.
1 (1) ;
2 (2) .
Consider the case of possible bets on events that partition ; i.e. are mutually exclusive and exhaustive (Scozzafava 1987, p. 686). Let , be the amount paid for a coherent bet on . These bets can be regarded as a single bet on with amount . Another condition may be specified from the requirement of coherence, namely
1 (3) .
These conditions are the axioms of probability. Further details are given by de Finetti (1931b) and in
