Ambiguity, Risk, and the Savage Axioms Daniel Ellsberg The Quarterly Journal of Economics, Vol. 75, No. 4. (Nov., 1961), pp. 643-669.
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http://www.jstor.org Mon Mar 10 11:54:44 2008
RISK, AMBIGUITY, AND T H E SAVAGE AXIOMS*
I. Are there uncertainties that are not risks? 643. - 11. Uncertainties that are not risks, 647. - 111. Why are some uncertainties not risks? - 656.
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 t,hat 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.' Yet 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. However, it is hard either to confirm or to deny such a proposition in the absence of precisely-defined procedures for measuring these alleged "degrees of belief." What might it mean operationally, in terms of refutable predictions about observable phenomena, to say that someone behaves "as if" he assigned quantitative likelihoods to events: or to say that he does not? An intuitive answer may emerge if we consider an example proposed by Shackle, who takes an extreme form of the Knightian
* Research for this paper was done as a member of the Society of Fellows, Haward University, 1957. I t was delivered in essentially its present form, except for Section 111, a t the December meetings of the Econometric Society, St. Louis, 1960. In the recent revision of Section 111, I have been particularly stimulated by discussions with A. Madansky, T. Schelling, L. Shapley and S. Winter. 1. F. H. Knight, Rzsk, Uncertaznty and Profit (Boston: Houghton hlifflin, 1921). But see Arrow's comment: "In brief, Knight's uncertainties seem to have surprisingly many of the properties of ordinary probabilities, and i t 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." K. J. Arrow, "Alternative Apprbaches to the Theory of Choice in Risk-taking Situations," Ewnometrzca, Vol. 19 (Oct. 1951), pp. 417, 426.
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position that statistical information on frequencies within a large, repetitive class of events is strictly irrelevant to a decision whose outcome depends on a single trial. Shackle not only rejects numerical probabilities for representing the uncertainty in this situation; he maintains that in situations where all the potential outcomes seem "perfectly possible" in the sense that they would not violate accepted laws and thus cause "surprise," it is impossible to distinguish meaningfully (i.e., in terms of a person's behavior, or any other observations) between the relative "likelihoods" of these outcomes. In throwing a die, for instance, it would not surprise us a t all if an ace came up on a single trial, nor if, on the other hand, some other number came up. So Shackle concludes:
Suppose the captains in a Test Match have agreed that instead of tossing a coin for a choice of innings they will decide the matter by this next throw of a die, and that if i t shows an ace Australia shall bat first, if any other number, then England shall bat first. Can we now give any meaningful answer whatever to the question, "Who will bat first?" except "We do not know?"*
Most of us might think we could give better answers than that. We could say, "England will bat first," or more cautiously: "I think England will probably bat first." And if Shackle challenges us as to what we "mean" by that statement, it is quite natural to reply: "We'll bet on England; and we'll give you good odds." I t so happens that in this case statistical information (on the behavior of dice) is available and does seem relevant even to a "single shot" decision, our bet; it will affect the odds we offer. As Damon Runyon once said, "The race is not always to the swift nor the battle to the strong, but that's the way to bet." However, it is our bet itself, and not the reasoning and evidence that lies behind it, that gives operational meaning to our statement that we find one outcome "more likely" than another. And we may be willing to place bets - thus revealing "degrees of belief" in a quantitative form - about events for which there is no statistical information at all, or regarding which statistical information seems in principle unobtainable. If our pattern of bets were suitably orderly - if it
2. G. L. S. Shackle, Uncertainty in Economics (London: Cambridge University Press, 1955), p. 8. If this example were not typical of a number of Shackle's works, it would seem almost unfair to cite it, since i t appears so transparently inconsistent with commonly-observed behavior. Can Shackle really believe that an Australian captain who cared about batting first would be indifferent between staking this outcome on "heads" or on an ace?
R I S K , A M B I G U I T Y , AND T H E S A V A G E A X I O M S
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satisfied certain postulated constraints -it would be possible to infer for ourselves numerical subjective probabilities for events, in terms of which some future decis...
EllsbergQJE61RiskAmbiguitySavage