# On the Unfortunate Ignorance of Academics regarding the Technical and Mathematical Tools for Dealing with Uncertainty Developed by J M Keynes in the A Treatise on Probability in 1921 and in His Earlier Fellowship Dissertations in 1907 and 1908: Interval Valued Probability, Decision Weights, and Safety First Analysis

50 Pages Posted: 6 Dec 2019

Date Written: November 19, 2019

### Abstract

Keynes developed a number of technical, mathematical tools for dealing with the problems of uncertainty (ambiguity, vagueness, indeterminate probabilities, imprecise probabilities) in his A Treatise on Probability in 1921 (and in his earlier Fellowship Dissertations of 1907 and 1908) that continue to be overlooked by practically all academics in the nearly 100 years since its publication in 1921. Keynes’s technical and mathematical modeling developments took place in chapters 15-17 of Part II of the A Treatise on Probability, chapters 20 and 22 of Part III of the A Treatise on Probability, Chapter 26 of Part IV of the A Treatise on Probability, and chapters 29 and 30 of Part V of the A Treatise on Probability.

Part II of the A Treatise on Probability included Keynes’s original work dealing with non –additive probabilities that was based on the original work of George Boole in his 1854 The Laws of Thought. Keynes’s Boolean upper and lower bounded probabilities, which Keynes categorized as inexact measurement and approximation, established his complete grasp of the concept of interval valued probability. The American Theodore Hailperin showed in 1986 that both Boole and Keynes were using early versions of linear programming techniques to solve both linear and non linear systems of equations and inequations. Keynes used the name “non –numerical probability“ when discussing his no additive, non linear interval valued probability approach. Keynes’s approach has been erroneously identified as an ordinal probability approach by all Post Keynesians, Institutional and heterodox economists, who were greatly influenced by the 1975 paper on Keynes and uncertainty in HOPE by E R Weintraub, which misinterpreted Keynes in terms of G L S Shackle’s theory of possibility.

Part III of the A Treatise on Probability, building on Part II’s interval valued probability concept, developed the concept of finite probability, applicable to both numerical and non numerical (interval valued) probabilities, in order to deal with applications of probability to Keynes’s concepts of induction, based on degrees of analogy, involving degrees of similarity and dissimilarity using human memory ,intuition and pattern recognition.

Part IV of the A Treatise on Probability provided the first decision weight approach to decision making, Keynes’s conventional coefficient of weight and risk, c, that incorporated both non linear probability preferences and non additivity by the use of decision weights .These decision weights incorporated Keynes’s logical analysis, contained in chapter 6 of the A Treatise on Probability of his Evidential weight of the argument approach, V(a/h),which was a logical relation like Keynes P(a/h)=α, 0≤α≤1,but which Keynes waited to provide the comparable mathematical analysis for in chapter 26.In chapter 26,Keynes set V(a/h) equal to w,where w was normalized on the unit interval, so that V(a/h) =w, 0≤w≤1, holds. Keynes then defined uncertainty in the General Theory as an inverse function of V in chapter 12 on page 148 in footnote 1.

Keynes then defined his conventional coefficient of weight and risk, c, to equal the standard EMV and SEU rules multiplied by the decision weights [(2w/1+w)] for non additivity and [(1/(1+q)] for non linearity. F Y Edgeworth recognized Keynes’s approach as breaking new ground. However, his plea for reader assistance from subscribers to Mind was ignored.

Part V of the A Treatise on Probability provided the first “safety first” approach to decision making. As acknowledged by Edgeworth, Keynes used Chebyshev’s Inequality to establish lower bounds for decision problems in order to minimize risk (Keynes, 1921, pp. 353-358). Samuelson, while analyzing Keynes‘s simplified Least Risk analysis on p. 315 of the A Treatise on Probability, unfortunately overlooked Keynes’s generalized approach using Chebyshev’s Inequality in chapter 29. Samuelson also overlooked on this same page Keynes’s development of the logical and mathematical tools needed to specify the investment multiplier in the General Theory, as well as Kahn’s employment multiplier on p.183 of his June, 1931 article in the Economic Journal.

**Keywords:** Keynes, probability, interval valued probability, conventional coefficient, decision weights, safety first

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

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