Keynes’sTreatise on Probability
This is the context in which John Maynard Keynes wrote his doctoral dissertation that he discussed in 1908 and published much later in a revised form as A Treatise on Probability (Keynes 1921).
Bradley W. Bateman (1996) argues that Keynes’s Treatise found its origin in reflections on George E. Moore’s Principia Ethica (1903) and, in particular, in his intuition that the weak point of Moore’s argument lay in the use of a naive version of frequentist probability. When Keynes began to study probability theory, he discovered the logical and practical limitations of the frequentist tradition. Keynes’s problem became, therefore, not so much that of criticising Moore’s probability, but rather of finding an alternative theory that made it possible to consider probability as a guide to action (Baccini 2004). Indeed, according to Keynes:the importance of probability can only be derived from the judgement that it is rational to be guided by it in action; and a practical dependence on it can only be justified by a judgement that in action we ought to act to take some account of it. It is for this reason that probability is to us the “guide of life”. (Keynes 1921: 323, original emphasis)
The Treatise contains the development of a new logical theory of probability: it is concerned with the “degree of belief” which is rational to entertain in given conditions of knowledge, and not, as in De Morgan, merely with the actual beliefs of particular individuals “which may or may not be rational” (Keynes 1921: 4). According to Keynes, there is a direct connection between probability, rational belief and action: to have a belief signifies being disposed to act on the basis of it: “the probable is the hypothesis on which it is rational for us to act” (Keynes 1921: 307).
However, the question is not so simple “for the obvious reason that of two hypotheses it may be rational to act on the less probable if it leads to the greater good” (Keynes 1921: 307).
Mathematical expectation and expected utility maximization are not the right tools to solve these kinds of problems. Keynes raised three formidable objections to the theory of mathematical expectation, all from the side of probability. The first was that probability is not fully measurable; the second, that in mathematical expectation the weight of the argument, that is, the amount of evidence upon which probability is based, is not considered; the third, that the element of “risk” is completely ignored, assuming that “an even chance of heaven or hell is precisely as much to be desired as the certain attainment of a state of mediocrity” (Keynes 1921: 312). To overcome these objections, Keynes proposed a “conventional coefficient”, substituting the probability value in the mathematical expectation formula, and considering together the probability of “goodness”, the risk associated with it and the weight of evidence, that is, the degree of unreliability or ambiguity of the information on which the probability value is based (Brady 1993). The Keynesian coefficient may be formulated as c = (1 + q2pW + w) wherep is probability, q = 1 - p is the risk coefficient and 0 ≤ w ≤ 1 the weight of the argument. The introduction of this coefficient generalizes the expected value formula by transforming it, according to Keynes, into a useful tool guiding actions in conditions of uncertainty. In the Keynesian formulation, the standard case - when an agent makes her choice using the expected value of different outcomes - becomes a very particular case. In this case the agent maintains that the information at her disposal is unambiguous and certain, that is, w = 1; and at the same time she is risk neutral, in the particular sense that she prefers not to consider the risk value of her choice, and then she drops q from the Keynesian coefficient.Keynes added to this exposition a paragraph containing a famous disclaimer regarding the usefulness of mathematics in moral sciences that attracted the attention of posterity. As a consequence, Keynes’s rejection of expected utility was categorized, starting at least from Shackle (1952), as a by-product of a general scepticism about the possibility to apply mathematical tools to economic problems. Instead, his contribution may be better understood as an escape from both utilitarianism (the refusal of the use of utility value) and frequentism, and as an anticipation of a modern approach to choice under uncertainty focusing on the problem of weighting probability values with the measures of the reliability of information on which probability values are based.