Skidelsky does a fine job in this book as long as he is not dealing with Keynes's approach to probability as presented by Keynes in Dec.,1908 in his second fellowship dissertation at Cambridge or in his A Treatise on Probability (1921;TP).Skidelsky is a historian.He does not have the training in mathematics,statistics,probability,logic and philosophy needed to assess Keynes's TP.In fact, to correctly assess Keynes's TP requires the reader of the TP to have, first of all, mastered George Boole's contributions to probability and logic.NO economist,logician,philospher, statistician,mathematician,psychologist or decision theorist in the 20th century was able to do so.Only F Y Edgeworth's 1922 book review in Mind was on the right track.
Skidelsky relies primarily on two very poor book reviews ,written by Frank Ramsey in 1922 and 1926,respectively,on the TP plus Keynes's 4 page eulogy published in the New Statesman and Nation in October ,1931,in his analysis of Keynes's approach.Keynes showed in the TP that Ramsey's theory is a very special case of Keynes's general theories of probability and decision making that occurs if all probabilities are linear and additive.Keynes's logical theory of probability is an extension of Boole's 1854 The Laws of Thought .It established that the general case is one of non linearity and non additivity.I add Keynes's mathematical proof for the interested reader at the end of this review.
The following pages in this book are incorrect-(a)p.20,(b)pp.38-47,especially p.44,and (c)pp.149-152.
In (a) ,Skidelsky claims that Keynes rejected the frequency interpretation,also called limiting frequency or relative frequency, of probability.In fact,Keynes rejected Venn's frequency theory.Keynes accepted that a logically based frequency theory,as presented by Keynes himself in the TP in chapter 8 of the TP,was a special theory that was sound and valid ,but of limited applicability.
In (b), Skidelsky claims that Keynes accepted Ramsey's criticisms and modified his approach to probability.Of course, Keynes did NO SUCH THING.What Keynes stated was that he " yielded " to Ramsey in the case where the calculus of probabilities applied.The calculus of probabilities applies only in the special case where the probabilities are linear and additive.NOWHERE IN THE 30 VOLUMES OF THE COLLECTED WRITINGS OF JOHN MAYNARD KEYNES did Keynes ever state that he rejected the calculus of probabilities.
In(c),Skidelsky claims that uncertainty exists when probabilities can't be known or measured.Skidelsky's error here is to confuse the case of ignorance,or complete and total uncertainty,with uncertainty ,which is a range .Ignorance
exists if Keynes's weight of the evidence index,w,where w is defined on the unit interval [0,1],has a value of 0.Uncertainty exists if w < 1.The probabilities will become intervals that
are nonlinear and nonadditive.
The reader is advised to buy the
book if he has not already purchased
one of Skidelsky's
other books on Keynes.
Keynes's 1921 A Treatise on Probability(TP) analysis of decision making can be found in sections 6-8 of chapter 26 and chapters 15,17,20 and 22.We will concentrate on the conventional coefficient of risk and weight in chapter 26,as opposed to the interval estimate approach of the other chapters,because of the greater explanatory power exhibited by the conventional coefficient.The technical details can be found on p.315 and in footnote 2 on p.315 .Keynes presented a very precise analysis demonstrating that an analysis of uncertainty introduced non additivity and non linearity into the formal representation of decision making. The subjectivist, Bayesian approach regards decision making as another name for the application of the purely mathematical laws of the probability calculus that require additivity and linearity. The Subjectivist approach makes the crucial error of conflating probability theory with decision theory.Keynes realized that ,due to the impact of the weight of the evidence (confidence)on decision makers ,as well as the optimism-pessimism of the decision maker,decision theory would have to be able to take into account the importance of non linearity and non additivity. The concept of expected value or expected utility is crucial to the Ramsey-De Finetti-Savage-Friedman approach.Keynes demonstrated that expected value or expected utility can ,at best, only be a special case of a much more general theory .
The Ramsey-De Finetti-Savage-Friedman approach is the mathematical translation of Jeremy Bentham's Benthamite Utilitarian approach.
Bentham's approach was that the whole can never be anything more than the sum of the individual ,atomic parts. However, this requires the assumptions of additivity and linearity.Bentham assumed also that all decision makers can calculate the odds all the time.Keynes showed that this was not the case because this requires a w=1. Keynes's demonstration ,taken from chapter 26 of his A Treatise on Probability(1921;TP),of the special case nature of any expected value(utility) approach ,based on the purely mathematical laws of the probability calculus,shows this to be a very special case that rarely,if ever,occurs in the real world.This is why public policy based on utilitarianism fails . Bentham claimed that all individuals have the capability to calculate the odds and outcomes and act on the expected utility (the probability times the utility of the outcome) in a rational(optimizing) way.This is where the rationality postulate comes from.
This can be expressed by the following maximization problem ,where p is the probability of success,q is the probability of failure, and A is the outcome:
The modern version of this is to Maximize pU(A),where p is a subjective probability that is additive,linear,precise,and exact and U(A) is a Von Neumann-Morgenstern Utility function. The goal is to
The modern name for Benthamite Utilitarianism in neoclassical economics is SEU theory(Subjective Expected Utility). Therefore,a microeconomic foundation based on Utility Maximization is just Benthamite Utilitarianism updated with modern mathematical probability techniques.Modern macroeconomics is all disguised SEU theory.
Keynes rejected Benthamite Utilitarianism as a very special case that would only hold under the special assumptions of the subjectivist, Bayesian model-that all probabilities were additive,linear,precise,single number answers that obeyed the purely mathematical laws of the probability calculus.
Keynes specifies his conventional coefficient of risk and weight,c, model in chapter 26 of the TP on p.314 and footnote 2 on p.314,as a counter weight to the Benthamite Utilitarian approach of Ramsey.
Essentially, Keynes's generalized model is given by
where w is Keynes's weight of the evidence variable that measures the completeness of the relevant, available evidence upon which the probabilities p and q are calculated.(Benthamite Utilitarians always assume that the value of w is always 1.)w is an index defined on the unit interval between 0 and 1,p is the probability of success,and q is the probability of failure.p+q sum to 1 if they are additive.This requires that w=1.Keynes's c coefficient can be rewritten as
Now multiply the above by A or U(A).One obtains
cA =p[1/(1+q) ][2w/(1+w)] A or
The goal is to Maximize cU(A) as opposed to the special Ramsey-Savage case of Maximize pU(A).
If w = 1 and all probability preferences are linear,then one obtains Ramsey's special result ,which was