There has always been a good deal of skepticism about the behavioral significance of Frank Knight's distinction between "measurable uncertainty" or "risk," which may be represented by numerical probabilities, and "unmeasurable uncertainty" which cannot. Knight maintained that the latter "uncertainty" prevailed - and hence that numerical probabilities were inapplicable - in situations when the decision-maker was ignorant of the statistical frequencies of events relevant to his decision; or when a priori calculations were impossible; or when the relevant events were in some sense unique; or when an important, once-and-for-all decision was concerned.' Daniel Ellsberg "Risk, Ambiguity, and the Savage Axioms," The Quarterly Journal of Economics, Vol. 75, No. 4 (Nov., 1961), pp. 643-669*
Before Daniel Ellsberg become a household name for leaking the Pentagon Papers, he was already known among economists and subsequently philosophers for a paper, published after his service as marine and during his graduate study, that has become associated with a 'paradox' because it is seen as a challenge to Subjective Expected Utility theory, despite the fact that 'paradox' is not to be found in the paper. And, so, while paradoxes naturally attract attention, I'd like to focus the attention of this digression on Ellsberg's way of motivating and situating his own paper.
The quoted paragraph is the first in the essay. After citing Knight's Risk, Uncertainty and Profit, the identity of the main skeptic toward Knight's distinction is identified as Arrow in Ellsberg's first footnote: "see Arrow's comment: "In brief, Knight's uncertainties seem to have surprisingly many of the properties of ordinary probabilities, and it is not clear how much is gained by the distinction. . . Actually, his uncertainties produce about the same reactions in individuals as other writers ascribe to risks." Ellsberg is quoting an important and lengthy (1951) review article by Arrow that both helped discredit Knight among mathematically inclined economists and also was key in creating the (recall Alchian inspired; and here) identification of uncertainty with randomness that became central in theories used in risk and price models in finance. So, the question I wish to pose is in what sense is Ellsberg vindicating Knight?
He is responding, in particular, to those who think that, and Ellsberg is now quoting Ramsey's great (1926) "Truth and Probability," "for a "rational" man all uncertainties can be reduced to risks.3" That is to say, Ellsberg's paper is meant to meet those who prefer to displace uncertainty with probable risk (and so eliminate true uncertainty) on their own ground. In particular, it is meant to engage those who think that (our gambling) "choices are influenced by, or "reflect," differing degrees of belief...[and] to infer those beliefs from the actual choices." It's at this point that Savage's postulates are introduced because these help distinguish systematically in the analysis of observed outcomes whether they track "relative expectations...[or] relative preferences for outcomes." As Ellsberg puts it, "if one picks the right choices to observe, and if the Savage postulates or some equivalent set are found to be satisfied, this distinction can be made unambiguously, and either qualitative or, ideally, numerical probabilities can be determined." (645)
At this point Ellsberg notes that Savage's postulates/axioms can be interpreted normative and empirically. They can be treated as a rational standard for how one ought to behave and as an empirical prediction of how people with "plenty of time to reflect" (646) will generally behave or (as Ellsberg puts it "the rules will be commonly satisfied." (645)) As others have noted, in the 1950s both approaches were not unusual, although in general, as Wade Hands has noted, when Savage was writing alone he tended to emphasize the normative interpretation in terms of an ideal agent. By contrast Savage's sometime co-author, Milton Friedman, tended to treat these as a useful (as-if) device for making predictions.** (More recently, as Hands notes, the normative interpretation has predominated.)
At this stage one may well wonder what the previous paragraph has to do with Knightian uncertainty for we seem to have drifted into an interpretation of rational choice theory. In fact, it is crucial: Ellsberg's strategy is to suggest that "the meaning" of true or Knightian uncertainty "would be that with respect to certain events they did not obey, nor did they wish to obey - even on reflection - Savage's postulates or equivalent rules." In such circumstances "the postulates failed to be acceptable in those circumstances as normative rules, or that they failed to predict reflective choices." (646) Ellsberg adds that he is primarily interested in failures of prediction (and notes that Savage himself is more interested, "no doubt" in their status as "normative rules").
That is to say, Ellsberg's intention, or strategy, is to identify Knightian uncertainty with a class of decisions that can be specified precisely as empirical violations of the Savage axioms. Again, on may wonder, why would that count as an instance of Knightian uncertainty? And the answer is: "there would be simply [a] no way to infer meaningful probabilities for those events from their choices, and [b] theories which purported to describe their uncertainty in terms of probabilities would be quite inapplicable in that area." (646; [parentheses added].) This [a-b] is true, as Ellsberg notes, absent "different operations for measuring probability." (646; [emphasis added.]) Here, Knightian uncertainty is associated with well defined choice circumstances in which it is impossible to attribute (with the Savage postulates) subjective probabilities to agents.
There is an instructive irony here. As mentioned above, Ellsberg noted that Arrow was a fierce critic of Rawls. When, later in 1973, (recall) in reviewing Rawls' Theory of Justice in the Journal of Philosophy, Arrow (correctly) discerned Rawls' adherence to Knightian uncertainty, but he misdescribed Knight's position as resting not on the denial of subjective probabilities (in the axiomatic tradition of "Ramsey and by L. J. Savage"), but on the denial of "objective probability." So, from Arrow's later (1973) perspective, Ellsberg can't be seen as defending Knight at all!+ I mention the irony because while the later (1973) Arrow is reductive about Knight, it is true that Knightian uncertainty was not limited to the class of interest to Ellsberg (subjective probabilities).
At this point, I am in the position to explain the relationship between Knightian uncertainty and what Ellsberg called ambiguity. But before I get to that (in a future post, alas), I need to spell out why the irony is instructive. One ground on which Arrow (1951) had criticized Knight is that his "analysis" is "lacking in formal clarity." (416) And in larger context, the implication is that without such formal clarity, Knight's position ought not be taken seriously. In fact, in 1951 Arrow's hopes that then recent "developments" in "the formalization of the theory of choice under uncertainty and its relation to many business phenomena" represent "dramatic breaks in continuity and have given hopes of a much clearer understanding of the problem." (405)++ That is to say, the formalization presents "a revolution in though" (to quote Samuelson, who in turn echoes J.N. Keynes anticipating Kuhn) that is, the "Mathematical-Economics Revolution."
While framed as a defense of Knight, Ellsberg's contribution is in the spirit of the Arrow-Samuelson hopes for economics. For, in fact, Ellsberg makes the very intelligibility of Knightian uncertainty dependent on a prior formalization (e.g. Savage's postulates). For, Knightian uncertainty can in his hands only be constituted by choice instances where this formulazation, when treated as a means to assign subjective probabilities to thoughtful (and willing to bet) agents, is empirically falsified or (if one dislikes the Popperianism) inapplicable. For, Knightian uncertainty now just is or is constituted by the very violation of the (particular) formalization. Whatever one thinks of that it's pretty clear this far removed from Knight's (and Keynes's) own ideas about the nature and epistemology of uncertainty even if it clearly is a case of Knightian uncertainty. Obviously, as Ellsberg notes, the violation is itself also a research opportunity to develop other formalizations that may be predictive of such choice situations, or their systematic violations.***
*This post was prompted by comments on one of my draft paper by Richard Pettigrew, John Quiggen, and M.A. Khan.
**Ellsberg does not mention Friedman, but in the second paragraph of the paper, just before he footnotes Arrow, he alludes to Friedman's as-if approach: "the feeling has persisted that, even in these situations, people tend to behave "as though" they assigned numerical probabilities, or "degrees of belief," to the events impinging on their actions." (643)
+In the 1951 article that Ellsberg is responding to, Arrow had been more careful, and noted “[Knight] states that the estimates are generally based on some classification, either of objective circumstances or, more often, of ability of individuals to judge or of subjective confidence in one’s judgment.”
++Arrow then cites Ramsay as an anticipation of the expected utility theory by Von Neumann and Morgenstern; he discusses statistical inference as developed by Neyman, Pearson, and Wald; and Shackle.
***I don't mean to suggest this is the only research opportunity in his wake. But about that more in my next post focusing on ambiguity.
Notably Ellsberg’s career is characterized by finding or exposing flaws in commonly accepted propositions. All his papers in the AER, EJ, QJE, and finally on the Vietnam debacle in the NYTimes are so characterizable. .
Posted by: Vernon SMITH | 04/20/2020 at 04:44 PM