de Finetti’s Standpoint
de Finetti was an Italian ‘Financial Mathematician’.
Probability and Induction
On the subjective meaning of probability
de Finetti argues for subjective probability, based on the psychological ‘degree of confidence’.
“If an individual judges two events as equally probable … he would be indifferent to exchanging … advantages or disadvantages deriving from fulfilling one, with the assumed identical consequences of the other.
… the degree of confidence … is expressed via the conditions at which one would bet. … The same criterion … will lead us to the definition on which we shall base the theory of probability.
The first formulation I suggested is based on the notion of mathematical expectation. “
In order to avoid some technical difficulties, de Finetti imagines a bookmaker who is forced to give odds and take bets on all events. de Finetti argues that the bookmaker’s odds determine his ‘probabilities’. de Finetti seems unaware of how books are actually made – to avoid any possibility of loss.
On the probability concept
Discusses budgeting, emphasising the psychological content, and (confusingly) the law of large numbers.
Bayesianism: its Unifying Role for Both the Foundations and Applications of Statistics
“… Bayesian techniques, if considered as no more than formal devices, are no more trustworthy than any other tool … . … any conclusion is arbitrary if the choice (purposively or unadvertedly) responds to formal criteria (inspired, e.g. to simplicity, or to mathematical convenience) rather than to personal advice.”
de Finetti quotes Cornfield:
“The objectivity of science finds its mathematical expression in the fact that individuals starting with quite different prior probabilities will nevertheless compute essentially the same posterior probabilities when faced with a sufficiently large body of data.”
This is only true when the hypotheses are statistical: otherwise the likelihoods depend on the priors and so if the actual case is untypical of true hypothesis its likelihood may be less than some other hypothesis.
“… In some way one can take decision theory as a starting point in conditions of uncertainty and can extract from it the general nucleus as a probability theory.”
That is, one derives a belief in a comparable probabilities from a belief in comparable decisions.
Bayesian Statistical Inference
Elements and steps of a decision process
de Finetti regards Bayesianism as a form of statistical induction in which one makes snap-shot judgments:
“If, on the basis of the acquired information and of the opinions forms from these, a person must or wishes to decide straight away … .”
He recognizes that:
“[The valuation of an uncertain return] will usually be much smaller than [the expected return], depending on the level of risk aversion of the concerned individual and the amount of money at stake in relation to his wealth.”
The value of information
de Finetti notes that subjective probabilities are always conditional on the state of knowledge, so a P(H) is really P(H|K).
Buying-in food for a restaurant. (Thus de Finetti regards his theory as very general.)
de Finetti regards the equivalence between Wald’s admissible procedures and prior probabilities as decisive. He quotes Lindley:
“… although these results [on admissibility] do not show how such initial probabilities must be assigned, they demonstrate that responsible behaviour is equivalent to their determination and vice-versa.”
de Finetti A Short Confirmation of My Standpoint in Expected Utility Hypotheses and the Allais Paradox: Contemporary Discussions and Rational Decisions under Uncertainty with Allais’ Rejoinder. Ed. Allais/Hagen 1979
Bruno de Finetti says:
“No doubt seems to me possible about the validity of the von-Neumann-Morgenstern rule of preference under uncertainty, consisting in maximizing the expected utility.”
He thus conflates uncertainty and (numeric) probability, and overlooks the von-Neumann-Morgenstern objections to the view that he assigns to them. He does, though, make the point that if a person acquires or loses money then their utilities will be likely to change.
de Finetti arguments assume that there are effective decision procedures. With this proviso, he shows that utilities and hence probabilities must be comparable and hence capable of numeric representation. Some such argument seems correct, as in von Neumann & Morgenstern. But he ignore their argument that competition can result in a lack of effective decision procedures and hence a need for strategy.
The Dutch book argument requires the bookmaker to take either side of a bet. Thus,faced with an Ellsberg urn he must ‘post odds’ of evens, so that a gambler who knew for sure the actual mix of balls in the urn could be sure of a profit. More generally, gamblers whose subjective probabilties differ from the bookmakers will expect to win, and the party with the closest subjective probabilties will tend to win. But real bookmakes do not normally rely on their skill at making such estimates.
Suppose that a ‘Keynesian’ bookmaker declares an interval probabilities, and accepts bets accordingly. (E.g., ‘for’ at one odds and ‘against’ at another, with a margin between.) As long as the objective probability lies within the interval, there is no gamble which – objectively – is expected to win. On the other hand, if the objective probability is outside the interval then the bookmaker – objectively – is vulnerable to an expected loss for some gamble. Thus while the Bayesian bookmaker avoids a sure loss, the (well informed) Keynesian bookmaker avoids an expected loss.
See Also (an addendum)
Alberto Feduzi, Jochen Runde and Carlo Zappia De Finetti on uncertainty Cambridge Journal of Economics 2014, 38, 1–21 doi:10.1093/cje/bet054
… Relying on usually overlooked excerpts of de Finetti’s works …. we argue that de Finetti suggested a relevant theoretical case for uncertainty to hold even when individuals are endowed with subjective probabilities. Indeed, de Finetti admitted that the distinction between risk and uncertainty is relevant when different individuals sensibly disagree about the probability of the occurrence of an event. …
.. de Finetti’s view of the foundations of probability, whilst subjective, was also pluralistic, and one that brings him closer to the broader subjective perspective more commonly associated with the likes of Irvin Good (1952) and Cedric Smith (1961).
… outside a strictly subjective interpretation of probability, ‘it would not be at all irrational to interpret this in agreement with Keynes as an absence of comparability’ (de Finetti, 1938 , p 88).
… a practitioner, he was aware that the adoption of subjectivist probabilities does not guarantee complete markets as postulated by the standard economic model of risk exchange.
‘it is often practically impossible to anyone to state that . . . the probability which he can attribute to a certain event has a precise value’ …. [It] may be possible ‘to know imperfectly an opinion, and thus to be capable of identifying only partially the preferences which the opinion implies (in a complete manner) among alternative possible decisions’ (de Finetti and Savage, 1962 , pp 95, 141–2).
Thus the current widespread belief that de Finetti thought that probabilities were necessarilyt precise or that he had an argument to that effect seems false. The most that can be said is that he thought the use of precise probabilities was pragmatic for those applications which he had in mind.