Gillies’ Uncertainty in Keynes
Donald Gillies Probability and Uncertainty in Keynes’s The General Theory In Jochen Runde and Sohei Mizuhara (eds.) The Philosophy of Keynes’s Economics. Probability, Uncertainty, and Convention,Routledge, pp. 111-129. 2003.
1. The Post-Keynesians and the Problem
In Britain the election of Margaret Thatcher in 1979 signalled the end of the government’s use of Keynesian policies, and the adoption instead of free market policies … . However, some remained convinced of the value of Keynesian ideas in economics.
[The] Post-Keynesians argue that probability and uncertainty were central to the real Keynes who wrote a Treatise on Probability in 1921, and in his General Theory of 1936 made implicit use of probability in his theory of long-term expectation.
Perhaps the first significant Post-Keynesian book was the first volume of Skidelsky’s masterly life of Keynes which appeared in 1983. This covers Keynes’s life up to 1920, and discusses Keynes’s early philosophical work on probability and induction – a topic which had been ignored for many years. Other Post-Keynesian books to appear in the 1980’s include Carabelli, Fitzgibbons, and O’Donnell. In 1985 a collection of papers edited by Lawson and Pesaran appeared. This contains articles by Victoria Chick, Alexander and Sheila Dow, Tony Lawson, and John Pheby. Somewhat younger Post-Keynesians include Bateman, Davis, and Runde.
2. The Logical, Subjective, and Intersubjective Interpretations of Probability
For Keynes probability is degree of rational belief not simply degree of belief.
[The] Principle of Indifference leads to a number of paradoxes. Keynes gives a full account of these in chapter IV of his Treatise, and makes an attempt to solve them. Yet it has to be said that his solution is far from satisfactory.
What is known as the Dutch Book argument then shows that for betting quotients to be coherent, they must satisfy the axioms of probability and so can be regarded as probabilities.
[Intersubjective] beliefs can be treated as probabilities through an extension of the Dutch Book argument.
Theorem. Suppose Ms A is betting against B = (B1, B2, … , Bn) on event E. Suppose Bi chooses betting quotient qi. Ms A will be able to choose stakes so that she gains money from B whatever happens unless q1 = q2 = … = qn.
Informally what this theorem shows is the following. Let B be some social group. Then it is in the interest of B as a whole if its members agree, perhaps as a result of rational discussion, on a common betting quotient rather than each member of the group choosing his or her own betting quotient.
According to the early Keynes, there exists a single rational degree of belief in some conclusion c given evidence e. … Yet in very many cases different individuals come to quite different conclusions even though they have the same background knowledge and expertise in the relevant area, and even though they are all quite rational. A single rational degree of belief on which all rational human beings should agree seems to be a myth.
There appears to be an assumption here that people are actually ‘quite rational’ in the sense that Keynes thought logical.
We very often find an individual human being belonging to a group which shares a common outlook, has some degree of common interest, and is able to reach a consensus as regards its beliefs.
My own reading of Keynes’ Treatise is that he saw his Principle of Indifference as proving a much-needed underpinning of many of the then routine applications of probability theory that were actually valid, but that he recognized that his was not universally applicable, and even gave examples.
3. Probability in Keynes’s Theory of Long-Term Expectation
One point of view is the continuity thesis that Keynes held much the same view of probability throughout his life. This thesis is advocated by (among others) Lawson (1985), Carabelli (1988), and O’Donnell (1989). Opposed to this is the discontinuity thesis that Keynes changed his views on the interpretation of probability significantly between 1921 and 1936. This thesis is advocated by Bateman (1987 & 1996), and Davis (1994).
Ramsey subjected Keynes’s logical interpretation of probability to an extensive criticism.
“[The] calculus of probabilities belongs to formal logic. But the basis of our degrees of belief – or the a priori probabilities, as they used to be called – is part of our human outfit, perhaps given us merely by natural selection, analogous to our perceptions and our memories rather than to formal logic. So far I yield to Ramsey – I think he is right. But in attempting to distinguish ‘rational’ degrees of belief from belief in general he was not yet, I think, quite successful.’”
Gillies notes that Keynes was critical of the Principle of Indifference. He goes on …
Although intersubjective probability is largely an explication of what Keynes says, I think that it does improve on Keynes’s position at one point. Both Keynes and Knight seem to assume that uncertainty is a qualitative concept which cannot be quantified, but, if we use the method of betting quotients and the Dutch book argument, we can quantify uncertainty and treat it using the standard mathematical theory of probability.
Although it is obviously very uncertain what the rate of interest will be in twenty years’ time, there is nothing to prevent us getting a particular individual, or a social group, to propose a betting quotient on this price lying in a specified interval in twenty years’ time. Thus we can by the standard Dutch book procedure introduce probability distributions for the rate of interest in twenty years’ time. These probabilities will, however, be subjective (or intersubjective), and not objective.
This analysis in fact accords quite well with what Keynes and Knight themselves say. Keynes says about examples such as the rate of interest in twenty years’ time:
‘About these matters there is no scientific basis on which to form any calculable probability whatever.’ (my italics – D. G.) Certainly there is no scientific basis to form a calculable probability, and so we cannot have an objective probability, but there is nothing to prevent individuals (or groups) betting, and so forming a subjective (or intersubjective) probability.
Knight does actually say:
‘We can also employ the terms “objective” and “subjective” probability to designate the risk and uncertainty respectively, as these expressions are already in general use with a signification akin to that proposed.’
It was thus natural for Knight to think of subjective probability in his sense, i.e. uncertainty, as: ‘indeterminate, unmeasurable’. This is no longer necessary today.
My own reading is not that Keynes changed his view about the strict logic of probability, but that he came to see the non-routine cases as typical of economics and many other complex problems. Thus there was a change of emphasis, but not – I think – mathematics.
In Keynes’ examples, it could be that the Intersubjective probability of a crash is minute, yet the logical probability very high. In principle the Intersubjective probability is measureable, but that is no help in estimating the logical probability. Of course, the notion of a logical probability is problematic, but in cases where it makes sense there seem to be grounds for wanting to estimate it. Similarly for objective probability. In particular, as Gillies quotes from Keynes, there does seem to be a qualitative difference between being confident that one has a sound estimate of objective probability, and knowing that one’s estimate is far from reliable. Thus even where one has a clear Intersubjective group-think, it is not clear how that should inform action. It may be reasonable not to gamble.
4. Some Concluding Remarks in favour of the Post-Keynesians
Keynes was not a Keynesian, though he may have been a Post-Keynesian!
In his Treatise, Keynes attempted to develop a logical justification for the conventional notion of numeric probability. Along the way he identified some more general theories (such as interval-valued probability) and then presented various assumptions, such as the principle of indifference, under which the conventional theory would be true. He seems to have regarded these assumptions as being reasonable, at least for scientific reasoning.
Ramsey had a similar agenda. But both came to acknowledge that their various assumptions were not completely general, and Keynes clearly regarded this is a significant, if not core, issues for economics. It is this mature view that matters to me, not so much the actual historical development. But I do not find Gillies reading compelling.
My own view is that in his Treatise Keynes was looking for a normative rationality, and did not assume that it was always rational to go along with any prevailing group-think. Similarly, when Keynes describes group-think in stock markets I imagine that his aim was to improve it, not go along with it. Thus Gillies’ ‘calculation’ of intersubjective probability, while important to an understanding of what is going on, is not necessarily related to Keynes’ logical concept. For example, one could do a survey to establish a population’s perception of the prevalence of various accidents, but the result would not necessarily be true.
Gillies has an important result, which seems to show that one does best to accept the status quo, even if wrong. He uses a military analogy, suggesting that everyone has to act ‘as a team’, as if they shared beliefs. But this seems to me a recipe for disaster. It is surely more usual to divide the command of forces, so that while tactical Gillies-like teams are fighting ‘coherently’, others may be taking a broader view, not ‘locked in’ to the current conception. Perhaps it is fair to say that teams need to organise themselves around the challenges that they face, and ensure adequate coherence of action. But just because a team has agreed on a common ‘betting quotient’ and hence displays group rationality need not (and perhaps should not) imply that they are individually rational in the sense that their actions are consistent with their actual beliefs.
If we do accept Gillies intersubjective probabilities then we can reconstruct something like Boole and Keynes’ non-numeric probabilities by taking probability distributions from different groups, or by trying to predict how the probabilities of one group – currently diverse – will converge. From a game theory perspective, one might also consider how a group of people might from into cohesive groups, and what their probabilities might be. The results of these various musings would seem to me to be similar to the conceptions of uncertainty that Keynes explores. The Dutch book argument does not apply, since people in different groups do not have to accept the bets.
I comment on the Dutch Book argument quite a few times. Why can’t gamblers decline to bet if they aren’t confident about their odds?