The Formal, Normative Theories of Savage, De Finetti, and Savage and De Finetti Require Additivity. They Never Introduced Sub or Non Additivity (Interval Valued Probability, Imprecise Probability, Indeterminate Probability) into Their Formal, Normative Theories of Subjective Probability or of Maximizing Subjective Expected Utility

Abstract

George Boole was the first scholar to present a rigorous, mathematically derived, concept of upper and lower probability with a large number of applications in 1854 in The Laws of Thought. Boole made it clear that this was due to the existence of missing or insufficient information, evidence or knowledge. Boole mentioned a number of times that the problems under consideration would always have incomplete or insufficient data. He termed these kinds of problems as being logically indeterminate. If, however, there was no missing or unavailable information, then the upper and the lower probabilities will become equal to each other. There will no longer be any interval valued, or upper and lower, probabilities. There will be a determinate, exact or precise probability. Additivity will always hold. Keynes introduced his weight of the argument,V,a logical relation, in chapter 6 of the A Treatise on Probability. Its purpose was to grade single pieces of data or evidence. Thus V1(h/x1 x2 x3)>V2(h/ x1 x2) means that the weight of the argument, V1, is greater than V2. The comparable mathematical representation of V was accomplished by Keynes with his weight of the evidence variable, w. w was defined on p.315 of the A Treatise on Probability in chapter 26. W was defined on the unit interval [0,1], just like probability is defined on the unit interval between 0 and 1. A w=1 meant that there was no missing or unavailable evidence or knowledge. The evidence is complete. If w=1, then there can’t be any upper and lower probability estimates. Only if w<1 can interval valued probabilities be defined. If w=0, then no probability can be defined. This is G L S Shackle’s argument. Keynes integrated w into his decision weight formula, c. Keynes called c a “conventional coefficient of weight and risk”. C is defined as being equal to p [1/(1 q)][2w/(1 w)]. One obtains a decision rule when c is multiplied by the outcome A or the utility of the outcome, U(A ), and the adjective ‘Maximize’ is appended. [1/(1 q)] models probability preferences, which are assumed to be linear by Subjectivists. [2w/(1 w)] models weight, w, non linearly. W introduces sub and non additivity. Ellsberg, in a manner similar to Keynes, in 1961 modeled what he defined as the ambiguity of the evidence or data by a variable he defined as rho, ρ, where rho measures the degree of ambiguity. ρ, like Keynes’s w, is defined on the interval [0,1]. A rho equal to 0 means that there is no ambiguity. A rho equal to 1 means that complete ambiguity exists. Interval valued probabilities, based on Ellsberg’s analysis of the work of Smith and Good, is recommended by Ellsberg (Ellsberg,pp.111-125), in cases where 0<ρ<1. Feduzi, Runde and Zappia (2012, 2014, 2017) have claimed repeatedly that de Finetti and Savage formally allowed imprecise and indeterminate probabilities to be used by decision makers in their normative theory of decision making. The only way in which this can be formally incorporated into a decision theory is by the use of a variable similar to Keynes’s w or Ellsberg’s ρ. Nowhere in anything written by de Finetti and/or Savage in their lifetimes does any such variable appear in their formal theoretical analysis or in any of their supporting axioms. Savage stated that he did not know how to integrate such a variable, to take account of vagueness, into the additive probability measure that represented his formal, normative theory. Feduzi, Runde and Zappia (2012, 2014, 2017) have misinterpreted the de Finetti and Savage concerns and sympathy for vacillating, indecisive decision makers, confronted with vague and ambiguous evidence, with their recognition that defining one’s personal probabilities can be difficult, with the erroneous and mistaken belief that de Finetti and Savage incorporated a role for imprecise and indeterminate probabilities within their formal, subjectivist, normative theory of probability and their SEU decision theory that rests completely on additive probability. Feduzi, Runde and Zappia need to cite and show exactly where, in any written article, book, or co-authored paper, that Savage, de Finetti, or Savage and de Finetti introduced sub or non additivity into their formal, normative model. If they are unable to do so, then they need to publicly withdraw their claims in the journals in which they published their claims.