Risk, Ambiguity and Decision (Studies in Philosophy)

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Summary: Ellsberg's Ambiguity is the Same as Keynes's Weight Index
Review: Ellsberg does an excellent job of demonstrating the very special nature of the Ramsey-de Finetti-Savage(RFS) subjectivist approach to both probability and decision making.The RFS approach requires the decision maker to be able to specify precise,exact,definite single number answers for all probability estimates.This is supposedly accomplished by an elicitation procedure based on the requirement that every decision is to be modeled as if it were a betting situation.Ellsberg shows that many decision makers will not accept such a betting quotients modeling approach to specifying numerical probabilities because such individuals make imprecise probability assessments.These estimates of probability are intervals.Each interval is made up of a lower probability and an upper probability.Only in the special case where the lower probability equals the upper probability will the RFS approach be sound.Ellsberg makes it very clear that he is building on the work of Good,Koopman,Smith and others in emphasizing the importance of intervals in specifying probability assessments in real world situations.Ellsberg's contribution is in terms of explaining why the vast majority of decision makers rely on intervals and not on precise probability estimates.Ellsberg's answer is that both the quantity and quality of relevant information, data or knowledge is ambiguous ,unclear,conflicting,incomplete or not available.Ambiguity represents a second dimension of decision making.This means that the RFS approach to probability estimation and the Subjective Expected Utility(SEU)theory built upon it is a special limiting case that only obtains when the information base is clear ,complete, available, and non-conflicting.Ellsberg operationalizes his concept of ambiguity by defining a variable called rho,where 0<=rho<=1.Rho is "...a number between 0 and 1 reflecting the decision-maker's degree of confidence in or reliance upon the estimated distribution... in a particular decision problem."(Ellsberg,2001,p.194).Ellsberg then incorporates rho as a linear decision weight in a"resticted Bayes/Hurwicz criterion",i.e.,a decision rule.Unfortunately, little of Ellsberg's work is truly original.Practically everything that Ellsberg does had already been done by J.M. Keynes in his A Treatise on Probability in 1921.Unfortunately, due to the misplaced influence of two error filled reviews of Keynes's approach by Frank Ramsey,95% of the reviews being based on the first 4 chapters of Keynes's book,Keynes's imprecise interval approach,which Keynes called nonnumerical or nonmeasurable probabilities , in order to emphasis the need to use TWO numerals in the estimation of a probability,came to be looked on as some "mysterious nonnumerical probabilities" that did not obey the laws of probability.Finally,in chapter 26 of A Treatise on Probability , on page 315 and page 315,footnote 2,Keynes specifies an index w ,where w equals the weight of the evidence and measures the degree of the completeness of the relevant evidence upon which the probability estimates are based.W is defined as 0<=w<=1.Keynes then incorporates this index into a decision theoretic criterion rule which he called "a conventional coefficient of weight and risk."Letting c designate the "conventional coefficient",the goal of Keynes's decision rule is to maximize cA,where A is equal to some outcome.In this reviewer's opinion,Keynes's decision rule is greatly superior to Ellsberg's rule since Keynes correctly incorporates nonlinearity into his weighting function.It is the nonlinearity effect which is creating all of the anomalies and paradoxes in standard SEU theory.Both Ellsberg's and Keynes's work goes a very long way towards correcting these deficiencies.However,Keynes got to the mountain top first.