On Arrow, Hurwicz, and Maskin...Another strategy to displace Knightian Uncertainy

The papers by Arrow and Hurwicz (1972)...Maskin (1979).... all deal with this approach to the problem. The complete ignorance approach retains the states - consequences framework of Savage (1954) but starts with the assumption that a decision-maker has no beliefs about the relative likelihood of the states. An important distinction between these papers and the Savage framework is that they assume that the state space is finite. This means that subjective probabilities cannot be derived by repeatedly sub-dividing events, as in Savage (1954). The decision-maker is assumed to have a utility function over the consequences which may be ordinal or cardinal.--David Kelsey & John Quiggin (1992) Journal of Economic Surveys, "Theories of Choice under Ignorance and Uncertainty" p. 139

For a few years now, I have been blogging about and giving papers on the displacement of Knightian uncertainty in professional economics (and decision-sciences more generally). My argument always boils down to the claim that underneath the increasingly fancy math some structure is imposed on the world when economists and decision theorists model decision under conditions uncertainty. I claim that structure (or probability distribution) is illicit or unearned because that structure rules out unnoticed, albeit genuine physical possibilities. This is especially so when economists treat a decision as a random walk (recall; and here), but it also shows up in other techniques.

But in these approaches, I have followed (recall) Knight and Malinvaud and operationalized uncertainty as circumstances when no grounds for assigning probabilities exist or when, speaking to hard-nosed economists, no long-term profitable insurance is possible. In response, sometimes sophisticated audiences challenge me to consider the treatment of so-called decision under 'complete ignorance.' This was first formulated by Arrow and Hurwicz (both Nobel laureates eventually in a paper they wrote in the mid 1950s, but only published in the 1972s. Arrow's role is significant, because he developed many of the other strategies economist's rely on to model (or as I claim displace) Knightian uncertainty, and he was (recall) polemically opposed to Knight. This work on complete ignorance has made me realize that I have been thinking about true uncertainty in too limited fashion.

Before I get to that just a note. After reading Arrow/Hurwicz and Maskin  (who also won a Nobel), and while writing this post, I encountered the survey article by Kelsey & Quiggin quoted above. This survey article is especially useful to those who try to understand what the mathematical results amount to in their intended applications to decisions. The quoted passage above mentions the key two issues for present purposes: (i) complete ignorance really does not impute probabilities to modeled world; (ii) it assumes "an exhaustive list of possible states of nature." (Maskin 1979: 320)*

On (i), there seem, thus, technical ways around Knight's version of Knightian uncertainty. These are going to be circumstances where the outcomes have clear utilities (or some other reward) attached to them, but the probabilities are unknown. This are going to be contexts with decision procedures that "rely primarily on the maximum and the minimum of an action" (Kelsey & Quiggin, 140.) I think what's key about such circumstances that possible choices are really constrained and known in advance (that's what (ii) also says). Kelsey and Quigging comment that ""it must be admitted that many practical problems do not satisfy the assumptions of the complete ignorance models. Most of the researchers in this area would agree that complete ignorance is a special case and a rather extreme one." (140) I am unsure if it is extreme (in all relevant senses); I rather suspect it is quite common in closed environments with limited options (like games), or in environments where the options are deliberately perhaps artificially limited, but one has no grounds for prior probabilities about outcomes, say because it is an one-off decision.**  

Yet, and this gets me to (ii); the knowledge that one can list all (genuinely) possible outcomes seems to violate the spirit of Knightian uncertainty. I had noted this before (recall), in fact, because Arrow uses the same trick in his effort to displace Knightian uncertainty via so-called contingent commodities (or Arrow commodities/securities), which are used in the context of general equilibrium modeling. In the context of GE modeling this approach really begs the question (against the Knightian critic) because one really does not know either what the possible states of the economy may be nor that these are (ahh) finite.+ 

The main problem is primarily epistemological; economies are not like that (Hayek is, for the time being, right about that.)++ But there is also a sense in which this is a problem about possibility (or in what I like to call modal economics). Why think it is (ordinarily) even possible to list all the possible consequences of a decision (as the Good Place is noting this week, the combinatoric explosion of even ordinary decisions is huge)? That is, this reinforces my thought what most of the mathematically sophisticated ways of displacing Knightian uncertainty have in common is a tendency to assume that possibility is easy and so to impose a modal structure on the world (without this structure being grounded in any pre-existing or robust knowledge of the world's modal properties). 

Anyway, I have said enough. Except that I need to find a definition or operationalization of Knightian uncertainty that also rules out the tendency to assume what the states of the world may be. 

 

 

*In context Maskin recognizes that there are many "conceivable" mappings that can apply to "the same world."  And this raises interesting further questions I ignore here (although Kelsey & Quiggin are worth reading on this, too.)

**Think of the present situation in Britain's parliament regarding Brexit. One may well be able to rank the outcomes, and even -- at some point, let's stipulate -- have an exhaustive list of what the choice options may be --, but the assigning of even subjective probabilities for each outcome is rather arbitrary.  And, indeed, it seems that May is, indeed, playing maximin. (I thank Keith Hankins for discussion.)

+Perhaps, one can rule out that the list will be infinite, but not that it will be indeterminately large (or open ended).

++Why, for the time being? Well, in a planned economy, or one with simulated planning through clever algorithms, the epistemological problem can be narrowed down.

Comments

[John Quiggin]

I've been arguing for twenty years or so that the real issue is unawareness, that is, rejecting the idea that it is possible to list all possible consequences of a decision or all states of nature relevant to the outcome. There is a literature developing on this, already large in absolute terms, but tiny relative to that on Knightian uncertainty (aka ambiguity).

Here's a bibliography last updated in 2014 I think http://faculty.econ.ucdavis.edu/faculty/schipper/unaw.htm

[Eric Schliesser]

Thank you, John. I have been circulating a paper that looks at Arrow's tendency to displace Knight's original account of uncertainty with a finite list of possible outcomes/states. So, I am pleased to learn there is a much larger literature. What is the canonical paper on unawareness in your view (that is, on rejecting the idea that it is possible to list all possible consequences of a decision or all states of nature relevant to the outcome)? Does any of them connect it to Knight's original ideas?

[John Quiggin]

(I thought I posted this earlier, but apparently not) It's a bit hard to tell what Knight really thought, because the language of subjective probability hadn't been developed. Arguably, all Knight meant was to say that some events don't have well-defined frequentist/objective probabilites. That's why people talk more about Ellsberg in this context.