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Probability, risk and uncertainty

A century later, work on uncertainty by F.H. Knight and J.M. Keynes remain highly relevant in areas ranging from economics and finance to insurance, law and management, says Jochen Runde, Professor of Economics & Organisation at Cambridge Judge Business School.

A view of a down stock market ticker.

By Professor Jochen Runde.

This year marks the centenary of the first publication of two books that have strongly influenced how economists and their colleagues in other disciplines conceptualise uncertainty and think about its consequences, F.H. Knight’s Risk, Uncertainty and Profit, and J.M. Keynes’s A Treatise on Probability. To mark the occasion, The Cambridge Journal of Economics has just published a special issue, edited and introduced by Phil Faulkner, Alberto Feduzi, Chuck McCann Jr., and myself, on the history of the two works, on the ideas in them, and subsequent work they have inspired.

The two books are quite different:

  • Knight’s, written for economists, was an attempt to incorporate profits within the equilibrium theory of the time, and is famous in both economics and management for his account of profit as the reward entrepreneurs receive for bearing non-insurable uncertainties.
  • Keynes’s, in contrast, was aimed at philosophers, is on probability and the foundations of statistics per se, and was largely ignored by economists when it first came out. But the ideas developed there re-emerged in Keynes’s analysis of the impact of uncertainty on investment and the demand for money in his magnum opus The General Theory of Employment Interest and Money, published 15 years later and which laid the foundations of what would become macroeconomics. They also informed his rather sceptical views on the then-emerging discipline of econometrics.

The reason the two books are nevertheless often mentioned in the same breath lies in their shared emphasis on the point that most business and investment decisions are made in the absence of numerically definite probabilities of (the factors that might affect) their consequences. It is for this reason that Knight and Keynes are routinely cited for coining the distinction between risk and uncertainty that has become part of the jargon of economics: between situations in which decisions are informed by objective numerical probabilities of contingent outcomes (decision-making under risk) and situations in which there are no such probabilities to go on (decision-making under uncertainty).

Contrasting approaches of Knight and Keynes

Drawing on standard texts on probability of the time, Knight associated situations of risk with cases in which numerical probabilities can be calculated based on equal probabilities as in classical games of chance (what he called “a priori probability”) or where it is possible to determine relative frequencies empirically (what he called “statistical probability”). Under conditions of uncertainty, he argued, decision-makers form qualitative judgements of probability he called “estimates”, although he allowed that people sometimes express these in numerical form as a proper fraction.

Keynes, in contrast, was proposing a novel and highly distinctive theory of probability of his own. In terms of this theory, probability is interpreted as a measure of the strength of a partial logical relation, what Keynes called the probability relation, between a hypothesis and the available evidence relevant to that hypothesis. A key feature of the theory is that probability relations are not generally capable of numerical measurement.

So, while Keynes also recognised what Knight called a priori probabilities and statistical frequencies, he emphasised that these are special cases. In most other cases, in his view, probabilities are non-numerical, and then sometimes not even capable of being ordered in terms of “more probable than” or “as probable as.” Further, he suggested that when acting on judgements of probability, it is necessary to consider, not only how probable some conclusion may be relative to the evidence, but also the extent, or what he called the “evidential weight”, of that evidence. This idea came to the fore in his later economic writings, where he associated low evidential weight with low confidence and what he called a preference for liquidity. Interestingly, Knight made a similar distinction between a judgment of probability and one’s confidence in that judgement.

A shift in the 1960s

While most people would surely agree that the great majority of business and investment decisions are made under uncertainty rather than risk, the risk-uncertainty distinction slipped from view in much of mainstream economics from the 1960s onwards. The chief reason for this was the rise of the subjectivist or Bayesian interpretation of probability, pioneered by the Cambridge philosopher and younger contemporary of Keynes, Frank Ramsey, and the famous Italian probabilist Bruno de Finetti, and given brilliant expression in the context of subjective expected utility (SEU) theory in Leonard Jimmie Savage’s 1954 The Foundations of Statistics. On the subjectivist approach, probability is interpreted as a measure of an actor’s strength of belief in a contingent outcome, which can be read off the odds at which they would be prepared to bet on that outcome (so long as they are “rational” in the sense of satisfying certain consistency conditions). Since it is possible to bet on just about any contingency, this approach makes it possible to argue that rational decision makers always act “as if” they have sharp numerical probabilities at the back of their minds even in situations in which there are no objective probabilities to go on. And with Savage-type versions of the SEU model—a sophisticated formulation of the old idea that risky options should be valued in terms of the sum of the probability-weighted desirability of their consequences—going on to become widely adopted in mainstream economics, many came to regard the risk/uncertainty distinction as redundant.

Why are we still fascinated by these two men 100 years on?

So, given all this, why are we still fascinated with the ideas of Knight and Keynes ideas a century later?

One reason, surely, is that non-probabilistic forms of uncertainty so obviously remain an ever-present fact of life. As the disruptive effects of events like Brexit and COVID-19 once again remind us, we are usually unable even to list all the contingencies that might affect the consequences of our actions, never mind assigning, or acting “as if” we are assigning, sharp numerical probabilities to all of them. In short, the strictly “rational” decision-making and consistent Bayesian updating envisaged by proponents of the SEU model are simply beyond most of us in practical situations, and it is therefore misleading to assume the contrary. Indeed, many non-mainstream traditions in economics such as the Post-Keynesian, Austrian, Institutionalist and neo-Schumpeterian schools, often distinguish themselves from the mainstream by explicitly rejecting the SEU model and putting uncertainty in the sense of Knight and Keynes, limits on knowledge, coordination problems, bounded rationality and so on, at the centre of their research agendas. Several of the contributions to the Special Issue take this kind of line.

The role of Daniel Ellsberg – long before the Pentagon Papers

That said, there does remain an interest in the ideas of Knight and Keynes in the more mathematical reaches of economics, psychology, and decision theory. Behavioural economics has been an important driver here for demonstrating experimentally that, however appealing the SEU model may be as a modelling tool or on normative grounds, most people don’t behave in accordance with it. Many of the empirical violations of the model occur even in situations of risk in which probabilities are objectively given. But others occur specifically where probabilities are absent or “ambiguous”. The seminal contribution here is a 1960 paper by Daniel Ellsberg, the same Daniel Ellsberg who became famous for releasing the Pentagon papers in 1971. Beginning with Knight—it was only later that Ellsberg discovered Keynes’s A Treatise on Probability—Ellsberg used versions of an example also mentioned by Knight and Keynes in which the proportion of a given number of differently coloured balls in each of a pair of urns are known in one case and unknown in the other. Ellsberg was able to show that choices people make between gambles involving random drawings from those urns tend to be inconsistent with their assigning precise numerical probabilities (or at least classical additive probabilities that sum up to one) in the way assumed in SEU theory. This paper, cited in several of the contributions to the Special Issue and the focus of one of them, spawned a still-ongoing stream of research devoted to developing mathematical generalisations of the SEU model that drop the assumption of classical probabilities.

Knight and Keynes’s ideas on uncertainty remain alive in other contexts too. One of these is the literature on uncertainty and idiosyncratic risks in finance and insurance (two of the contributions to the Special Issue focus on finance). The legal arena is another, where Knight comes up in discussions of the power over the disposal of property under conditions of uncertainty, and Keynes’s work on probability and evidential weight has informed work on the burden of proof and the limitations on standard probabilities in law courts. And their ideas are never far from the surface in areas of management research such as entrepreneurship, strategic decision-making, and the management of innovation. In the study of entrepreneurship, consistent with its Knightian origins, uncertainty is often seen as a precondition for the existence of profitable business opportunities and thus for entrepreneurial action.

These connections are currently stimulating research on various topics including more finely grained conceptualisations of uncertainty, different facets of entrepreneurial cognition and action, and an ‘entrepreneurial’ theory of the firm integrating Knight’s view of entrepreneurship as judgment-based decision-making under uncertainty with economic and managerial theories of the firm. In strategic decision making, there is interest in the kinds of over-arching ‘logics’ best-suited to coping with uncertainty, the cognitive and perceptual dimensions of such logics, and the steps often employed in forming judgments and taking decisions.

Finally, in the management of innovation, there is considerable interest in the limits of traditional planning techniques under conditions of uncertainty, with many experts now recommending that possible accidents, errors, and surprises be accommodated and even at times embraced when they arise.

One hundred years on, Knight and Keynes’s ideas about non-probabilistic forms of uncertainty and their consequences are as relevant as they ever were. Somehow, it wouldn’t be surprising if this were still the case a century from now.