# An Examination of Bruno De Finetti's Understanding of J M Keynes's a Treatise on Probability: Close but No Cigar

22 Pages Posted: 14 Jun 2017 Last revised: 15 Jun 2017

Date Written: June 13, 2017

### Abstract

Bruno de Finetti came close to understanding, but still disagreeing with, what Keynes was doing in the A Treatise on Probability, which Keynes had started working on in 1904 and continued developing through August, 1914. Keynes then put the work aside until he reviewed it for publication in 1920. It was published in 1921.

Bruno de Finetti thought that, perhaps, Keynes ‘s views on “non numerical” probability might or could be supported if the decision maker had to use or resort to interval valued probabilities in the initial or beginning stages of an analysis of a decision problem. He gave an example where this is the case given conflicting evidence. His example is an interval valued probability. However, it is not an example involving missing, unavailable, relevant evidence, which is what Keynes meant when discussing the making of decisions under uncertainty. Unfortunately, de Finetti, like Emile Borel, F Y Edgeworth, and E B. Wilson before him, skipped the analysis in Part II of the A Treatise on Probability that would have allowed him to conclude that Keynes’s views on “non numerical probabilities” were not mysterious at all, but were interval valued probabilities based on the non (sub) additivity of probability applications in the real world. This means that Keynes was arguing that non-additivity is the general case in the real world of decision making not additivity.

There is not a single reference in the corpus of de Finetti’s lifetime works that connects Keynes’s non numerical probabilities to both Boole’s and Keynes’s interval valued probability, as done explicitly by Keynes in chapters 3, 15, 16, 17, 20, 22, 26, 29, and 30 of the A Treatise on Probability.

On the other hand, de Finetti always makes it clear that he believed that Keynes’s “non numerical probabilities” could never be the general case in decision making. The general case had to end up always leading to a precise, exact probability estimate and never a non numerical probability, either imprecise or logically indeterminate, even if it did turn out to be the case that what Keynes meant by non numerical probability was interval valued probability. For de Finetti, all intermediate and final probability estimates must be point estimate probabilities. Thus, what de Finetti meant by uncertainty is the existence of measurement error resulting from the concept of inner and outer measures in measure theory when dealing with initial conditions that would likely involve a paucity of relevant evidence. This is completely different from what Keynes meant by the word uncertainty. For Keynes, uncertainty meant that there was missing or unavailable relevant evidence or knowledge. However, Keynes clearly restricted the existence of ignorance, where there is no relevant evidence, to the distant future where technological change, innovation and advance threatened to make current technologies obsolete overnight. This threat of obsolescence is limited to long run investment in physical durable capital good and financial products. Keynes completely and totally rejected the Post Keynesian claims of GLS Shackle, Joan and Austin Robinson, and Paul Davidson that there was complete and total uncertainty about the future so that the weight of the evidence, w, had to equal 0.

Note that de Finetti also ignored Frank Ramsey’s two highly critical, but error filled, reviews of Keynes and A Treatise on Probability. Of course, de Finetti supported Ramsey’s additive and linear approach to the use of numerical probability using betting quotients because it established the coherence and consistency of the better’s beliefs with the purely mathematical laws of the probability calculus and prevented a Dutch book from being able to be made theoretically against the better.

Therefore, rationality had only one meaning for de Finetti. It meant consistency with the purely mathematical laws of the probability calculus, while for Keynes rationality meant basing one’s estimate of a probability on all of the available, relevant evidence or knowledge. Note that the usual definition of rationality used by economists is Maximizing Utility or Maximizing Subjective Expected Utility (SEU theory).

Finally, de Finetti ultimately changed his mind as he grew older. Bruno de Finetti came to regard all logical theories of probability as being partly defective.

**Keywords:** De Finetti Keynes, Borel, Part II, Interval Valued Probability, Approximation

**JEL Classification:** B10, B12, B14, B16, B20, B22

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