In most of this literature, the term ‘ambiguity’ has been treated as a synonym for what Knight called ‘uncertainty’ namely the fact that relative likelihoods are not characterized by well-defined numerical probabilities. The standard method of dealing with ambiguity in decision theory is to endow the decisionmaker with multiple priors as in Gilboa and Schmeidler (1989). This approach may be combined with a variety of preference models, notably including the maxmin model of Gilboa and Schmeidler (1989) and the smooth model of Klibanoff et al. (2005).
For a non-specialist this is puzzling; there is no obvious link to the ordinary meaning (or meanings)1 of ambiguity as a characteristic of propositions with more than one interpretation. In its normal usage, ambiguity is a linguistic concept, but in decision theory it is typically treated as a property of preferences.
The now-standard usage is quite different from that in Ellsberg’s (1961) original article. Ellsberg treated ambiguity, not as a property of preferences or relative likelihoods, but as a property of the information on which judgements of relative likelihoods might be based.“Responses from confessed violators [of the expected utility (EU) axioms] indicate that the difference is not to be found in terms of the two factors commonly used to determine a choice situation, the relative desirability of the possible payoffs and the relative likelihood of the events affecting them, but in a third dimension of the problem of choice: the nature of one’s information concerning the relative likelihood of events. What is at issue might be called the ambiguity of this information, a quality depending on the amount, type, reliability and ‘unanimity’ of information, and giving rise to one’s degree of ‘confidence’ in an estimate of relative likelihoods.”
---Simon Grant, Ani Guerdjikova, John Quiggin "Ambiguity and awareness: a coherent multiple priors model." (17 April, 2020) [quoting p. 657 of Ellsberg's (1961) "Risk, Ambiguity, and the Savage Axiom]
In Yesterday's post I noted that while writing in defense of Knight, Ellsberg (1961) innovatively identified Knightian uncertainty with a class of decisions that can be specified precisely as empirical violations of the (Ramsey-)Savage axioms.* In particular, in such instances (when the formalization is violated) there is "no way to infer meaningful [subjective] probabilities for those events from their choices, and theories which purported to describe their uncertainty in terms of probabilities would be quite inapplicable in that area." (646; [parentheses added].) This identification was an innovation made possible and intelligible by formalization.
As I remarked, Ellsberg noted that it is possible to treat the very same formalization as a [A] normative standard for a certain kind of ideal/idealized agent or [B] as a device to make predictions about agents. Ellsberg recognizes that Savage was (primarily) tempted by the normative interpretation whereas Ellsberg was primarily focused on the empirical interpretation. In particular, he wanted to identify classes of choice circumstances in which the possible rationality (669) of one's decision could not be articulated in or explained by reference to the Savage axioms.
The Savage formalization allows the attribution to agents of precise subjective probabilities about distributions in the world. That is, in certain choice circumstances, the formalism allows for an interpretation of behavior that reveals if not inner states than at least structures that are compatible with such inner states. (It may be useful to compare this with the Samuelsonian strategy of revealed preference.)
With that in place, let's turn to ambiguity. As Grant/Guerdjikova/Quiggin note, in much of decision theory 'knightian uncertainty' is treated as synonymous with 'ambiguity' and has come to mean the absence of well-defined quantitative probabilities.+ And as they correctly note this is not quite what Ellsberg had in mind (and about that more in a second). Before I get to Ellsberg and ambiguity, it is worth noting that the absence of well-defined quantitative probabilities is indeed what Knight meant by uncertainty in the contrast between risk and uncertainty.
However, what the literature means by the absence of well-defined quantitative probability is (and now I quote Gilboa and Schmeidler (1989)) circumstances in which one "considers a set of priors as possible." (emphasis in original). It is important that this set is limited. Now, this is not quite what Knight, generally, intended to mean by the contrast between risk and uncertainty. In paradigmatic cases of uncertainty for Knight (and Keynes), we really have no right to make claims about the priors let alone treating them as well defined possibilities.
As an aside, this is not to deny that, as Richard Pettigrew noted to me, that "there are situations in which your evidence and/or current stage of scientific theorising is not going to deliver a unique probability assignment over the states you consider" but in which one has reasons to limit one's subjective probabilities to a set of possible distributions. And if forced to choose one may well come to think of your own (again quoting Pettigrew) as "your best subjective probabilities and use those while knowing that it would be equally compatible with your evidence to adopt an alternative set of probabilities."** If you and the other involved are experts in more or less the same situation, we can call this situation 'reasonable expert disagreement.' In fact, Savage himself noted (without having to contemplate violations of his axioms) that that “we must be prepared to find reasoning inadequate to bring about complete agreement.” (1954 [1972]:7; see also 3, 67ff.) This is not trivial in the historical context, as I have noted elsewhere, in which the very idea of science involves the arrival of expert consensus.
Now, as Grant/Guerdjikova/Quiggin note, Ellsberg thought ambiguity something else: namely it is a property of the "nature of one’s information." And throughout the paper, Ellsberg goes on to deny that this feature "cannot be conveniently described in terms of a sample distribution." (659) Rather the point of using ''ambiguity" is to try to describe circumstances "when there are questions of reliability and relevance of information, and particularly where there is conflicting opinion and evidence." (659) So, one measure , but it is only one such measure, that one's information is ambiguous in Ellsberg's sense is the kind of reasonable expert disagreement [that is, situations one knows an expert's subjective probability about a distribution is compatible with another expert adopting an alternative set of probabilities.] But it is only one such measure because ambiguity itself cannot (to repeat) be reduced to and articulated in one distribution or another. As Ellsberg noted (660), Savage (57-60), too, had tried to characterize the nature of such ambiguity; Savage had explicitly acknowledged that formal theories known to him lacked resources to do so in satisfying ways.
Now, so for Ellsberg ambiguity is a "subjective variable" (660) about the nature of one's information. But the circumstances under which it arises can be characterized objectively. The proxy he uses to characterize these circumstances are violations of the Savage axioms can be predicted to occur and, perhaps, thought (somewhat) reasonable.
But that is not all of it (and this connects to Gilboa/Schmeidler).*** In circumstances of ambiguity one has, according to Ellsberg, some subjective grounds to "eliminate" some "possible distributions," but not all such possibilities. (661) For, Gilboa/Schmeidler (and the Bayesian literature) that is treated as the occasion to look for solutions to decision problems in terms of sets of priors as opposed to unique priors. In so doing, they end up echoing what I have called the displacement strategy of Knightian uncertainty.
It is worth noting that there is an interesting difference of emphasis between trying to characterize an optimal decision strategy (in the manner of Gilboa/Schmeidler), say because one must choose or wish to guide policy-makers, and between wishing to promote understanding of how to think about or study one's information about possible distributions. (I have jokingly called the latter modal economics.) Ellsberg is clearly less interested in the former, and more interested in the latter. Ambiguity in Ellsberg's sense does influence choice (664); yet Ellsberg is interested in finding out when and how, and to create "testable propositions" about such states [669], not to tell you what strategies to use in choice.
That is to say, Knight thought of uncertainty in terms of not knowing the (range and nature of) possible distributions. Ellsberg thinks we can develop rigorous thought about what a theorist taught me to call intermediate circumstances in which we have some information about the possibilities. Somewhat surprisingly, according to Ellsberg -- and here he refers also to Chipman's experiments++ -- this thought has to be developed not from normative first principles, but rather experimentally ((655) within what in reviewing Donald Davidson, Suppes, and Siegel, he calls (in 1957) "the new field of experimental economics;" (1009). Of course, Ellsberg soon got preoccupied by different concerns. But that's really for another occasion.
So, Ellsberg has been remembered as generating a "paradox" in subjective expected utility theory. (This seems to be due to Harry V. Roberts.) This way of thinking about him says more about the economists' attitude toward their own past than Ellsberg. We have seen that alone among the responses to Knight that are still acknowledged by contemporary mathematical economists, Ellsberg wrote to vindicate Knight.+++ But he did so by using the canons of rational decision-making to specify when some circumstances of true uncertainty might occur. We could call them Ellsberg circumstances. These Ellsberg circumstances are characterized by constrained decision options in which information about world has the character of what he called ambiguity. In Ellsberg circumstances one grasps, we might say, the possible contours of the underlying distributions. Whether, in Ellsberg circumstances, it might pay to take bets on these contours is a tough challenge (one may well be in context of what he called "treacherous possibilities." (666)) He then opened the door not to better decision rules, but rather to the epistemology of information, which uniquely he thought ought to be explored experimentally.
1. Footnote by Grant/Guerdjikova/Quiggin: "Empson (1930) famously distinguished seven types of ambiguity." [that is, Empson, W. (1930). Seven Types of Ambiguity. Read Books Limited.]
*Such violations of (subjective) expected utility theory were later called a 'paradox,' but to the best of my knowledge Ellsberg was never tempted to do so
+I use 'quantitative' because in Ellsberg the contrast is between qualitative and quantitative probabilities.
**Pettigrew in private correspondence. Quoted with permission. Any mistakes should be attributed to me, alas.
***Grant/Guerdjikova/Quiggin are interested in modeling the lack of awareness of agents of "some possibilities relevant to the outcome of their decisions," in particular, "there are multiple probability distributions that may be applicable, depending on whether or not these possibilities are realized." That is, in one sense indeed much closer to Ellsberg's intentions. But this is not the place to analyze their representation of unawareness.
++Ellsberg explicitly and repeatedly refers to J. S. Chipman, (1960) "Stochastic Choice and Subjective Probability." Chipman was his supervisor at Harvard. In his fine paper on Ellsberg's (1961) paper, Carlo Zappia calls attention to the significance of Ellsberg's 1958 review of Davidson, Donald, Patrick Suppes, and Sidney Siegel. 1957. Decision Making. An Experimental Approach. Based on his reading of Ellsberg's dissertation, Zappia thinks Ellsberg himself was primarily focused on developing a normative theory to replace Savage's. I thank Charles McCann for calling attention to Zappia.
+++I don't mean to suggest he was the only defender of Knight in the period (cf. Shackle).
Minor point on layout. GGQ are cited for the quote within a quote, but should be for the whole thing, Feel free to delete after fixing
Posted by: John Q | 04/21/2020 at 12:42 PM
Thanks for alerting me to this. Do you think the corrected version works?
Posted by: Eric Schliesser | 04/21/2020 at 12:52 PM
Very interisting post, I look forward to reading GGQ. As for the descriptive/normative divide and Ellsberg's clear aim to deal with normative issues, his doctoral thesis has entire sections dedicated to the issue, included one prefiguring the so-called alfaMEU decision rule of Ghilardato-Marinacci-Mascheroni. Ellsberg's thesis was published only as late as 2001: https://www.google.com/search?tbm=bks&q=ellsberg+decision
A paper discussing normative issues in Ellsberg and current decision theory is available here: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3220431
Posted by: Carlo Z | 04/22/2020 at 04:57 PM