Essays on Uncertainty and Risk: F H Knight on the Role of Strong Versus Weak Evidence in the Uncertainty (Estimates) Versus Risk (Probabilities) Distinction in RUP: It Had Little to Do with the Issue of Whether Uncertainty is Measurable or Not Measurable

Frank Knight was only partly successful in trying to analyze how to discuss the meaning of uncertainty, as well as the impact uncertainty would have on the decision maker. Knight, like Keynes, found the concept of uncertainty to be difficult and problematic. The major reason why Knight was only partially successful in defining and analyzing uncertainty was that he was not able to clearly enunciate a concept of the "weight of the evidence" to support his claim that "estimates" were different from probabilities, even though they took the same form as a fraction a/b between 0 and 1.Knight lacked the full range of logical, mathematical, and statistical skills possessed by J. M. Keynes. Keynes, despite having the aid of the powerful mathematician, William Ernest Johnson, at his side, openly confessed very similar doubts and concerns about the subject matter of weight (uncertainty was an inverse a function of weight), which was analyzed by Keynes using his "weight of the evidence" (and not the weight of the arguments based on the logical relation, V) mathematical variable, w, in the A Treatise on Probability, as Knight had about uncertainty. Knight was unable to completely break away from the form that a probability took, as a ratio (a/b), in order to develop his "estimates" approach as a third category. He needed to be able to have developed a concept of the weight of the evidence. However, he came very close with his concept of confidence, but he was unable to completely achieve his goal. He used the term "weight" one time in Risk, Uncertainty and Profit in exactly the same sense as Keynes had used it in his A Treatise on Probability.Given all of the constraints he faced, Knight, nevertheless, succeeded in bringing in the concept of confidence into play before Keynes did. He also attempted to bring in a concept of interval valued probability into his discussions. Given these developments by Knight, it is impossible for Knight to have been a Ramsey, de Finetti, Savage, or Friedman type exponent of subjective, Bayesian probability, since questions of the confidence one has in an estimate of a probability or in the estimate of the probability of the estimate of a probability, which was the terminology used by Knight in his attempt to bring a concept of weight into play, simply do not and can’t take place because the subjective, precise, point estimate probability is the decision maker’s degree of confidence. Knight’s brief recognition that intervals would be used by decision makers immediately brings non additivity into play. Non additivity is ruled out by subjectivists, since subjectivists define a rational decision maker as one whose evaluations of probability are consistent with and cohere to the purely mathematical laws of the probability calculus. Talking about the degree of confidence a decision maker has in his subjective probability or his doubt about his estimate of probability is a sign that the decision maker is not "probabilistically sophisticated". He needs to be retrained so as to eliminate any concept of vagueness or ambiguity from his probability assessments through Ramsey’s betting quotient approach, combined with proper scoring rules and elicitation methods to assist him in realizing what his true, precise, exact numerical probability is. This paper will define a "sophisticated decision maker" as one who is "not probabilistically sophisticated" and a "not sophisticated decision maker" as one who is "probabilistically sophisticated", a description first used by Mark Machina. The Bayesian subjectivist is forced to be an unsophisticated decision maker because the Bayesian subjectivist requires that all decisions must be in accordance with the purely mathematical laws of the probability calculus. Since interval valued probabilities are not coherent and consistent with these requirements, Ramsey would have to reject and deny the relevance of G. Boole’s interval valued approach to probability that was explicitly used by Keynes in the A Treatise on Probability (TP,1921) in chapters 3, 15, 16, 17, 20, 22, and 29, as well as by Knight in chapter 8 on pp. 235, 237 of Risk, Uncertainty, and Profit. Good examples of "unsophisticated decision makers" are Milton Friedman, L. Savage, B. de Finetti, and F P Ramsey.