Thursday, March 29, 2012

A Puzzler on Statistical Risk and Fundamental Uncertainty

Let’s compare two situations. In situation A there will be a coin toss. The coin is known with complete certainty to be fair. The odds are obviously .5 that it will come up heads and .5 that it will be tails.

In situation B there will also be a coin toss, but in this case we have no information whatever regarding the fairness of the coin. It could be rigged to always come up heads or tails, or it could be biased toward one or the other, or only in windy weather, or who knows. We don’t. What are the odds? With only two possible outcomes and no reason to expect either to be more likely than the other, it is still .5 and .5.

And that’s the puzzler. Does it really make no difference at all whether we know the true odds? From a decision point of view, are situations A and B effectively the same?

Although this is a greatly stripped-down puzzle, it contains the core of the risk vs uncertainty problem. Consider the typical problem faced by a firm of whether to make an investment. There are many possible outcomes of this investment, and its profitability will be affected by events we may not even conceive of in the present—the so-called unknown unknowns. Nevertheless, if the firm has a hurdle rate of return, its decision will boil down to whether it thinks the expected return is above or below the bar. In situation A it is able to calculate an expected rate of return with full confidence, in situation B with partial, and perhaps very little, confidence. Again, from a decision perspective, is there any reason for the confidence of the expectation to matter? Note that, in the absence of relevant information in situation B, there is no reason to expect the variance of the ROI to be greater or less than in situation A.

Did I mention that this is a tremendously important topic not only in macroeconomics, but also in areas of micro like the justification for the precautionary principle?

Is there a literature that frames the problem this way? I haven’t seen it, and I would be very happy if someone could lead me to it and allow me to achieve enlightenment, since this puzzle has kept me awake off an on for years.  (But I have chipped away at it and even published a paper on it a while back...)


marcel said...

you may well be familiar with this, but ...

When I was at the start of my career, I was interested in this from a macro perspective (big mistake on my part!) and read fairly widely. Leonard Savage wrote a book (one of those green and white texts, the name of whose publisher I cannot recall at the moment) that covered decision theory under risk and uncertainty, and argued, IIRC, that in the face of complete ignorance about probabilities, so that you don't even have any subjective distribution, the uniform was the only one that was reasonable.

On a related note, I also read up on fuzzy sets and fuzzy logic, and, 20 years ago, thought that this was a promising approach as a way to incorporate vague information and subjective beliefs into decision theory.

Zlati Petroff said...

Could you discuss the role of (real) options and limited liability in the second case you describe?

I intuitively think that real options are crucial when you don't know the distribution of future outcomes because they allow you to adjust your "bet" on the future as you receive more information about it. In other words, you don't have to "lock in" any particular assumed distribution at the point of making the bet (which makes the firm's investment choice inherently different from the coin toss example).

At the same time, limited liability and other institutional mechanisms set a lower limit on the distribution of outcomes resulting from any given decision.

Also, could you discuss the role of financial liquidity in the context of uncertainty? I mean to imply that liquidity is essential to optionality: when you make financial investments whose payoffs have an unknown distribution, you have the option to withdraw from the bet as long as the investment you've made is liquid, which means you can set known limits on the outcomes' distribution.

I guess path dependence matters here. If each bet is discrete and independent, optionality is irrelevant. But if the ultimate outcome is path dependent, optionality is important?


Peter Dorman said...

Zlati, you have added another element to this puzzle, time. In the static version there is only one coin toss, and so we are stuck with the unsatisfactory equivalence of absence of knowledge and knowledge of absence (knowing there is no rigging of the coin). Once we add time, you are right that we can begin to make progress.

Option value arises if one can delay a commitment on the expectation of new information; we have a theory for that. Unfortunately, that theory isn't helpful if multiple options are equally committing, as they often are. (An investment opportunity may be unprofitable if it is delayed, giving a competitor more time to lock in market share.) It is also true that liquidity constraints can introduce asymmetry -- but the puzzle rules out differences in the variance of returns according to the degree of certainty over the probability function.

But having come this far, I have to add another thought: in the context of repeated iterations and learning, the difference between A and B shows up most clearly in a Bayesian formulation....

Zlati Petroff said...

This is a pretty breath-taking topic and I think I underestimated it a bit until your response.

I started writing a re-response about the role of social institutions as devices for storing Bayesian updates that help people within those institutions make choices, about the role of society as a hedge that limits the downside of decisions made under uncertainty, and so on...

...But my thoughts don't address the ultimate question, I now realize. I think society and institutions (like the firm) ENABLE us to make choices under complete uncertainty by placing limits on the distribution of outcomes. Yet that tells us nothing about WHICH choice people would make if they are confronted with multiple options with unknown outcomes, so I conclude that I have nothing of marginal value to add here.

Could you add more of your thoughts? At time T=0, you are confronted with several projects with unknown outcome distributions. Which do you pick? How does Bayesian learning help here, given that we need to make the choice now, at T=0?

Walt French said...

There Ain't No Such Thing As A Known-Fair Penny.

You can test it until you're blue in the face, but all you've done is (a) establish a tighter confidence interval around your measured result (presumably, 0.500), and (b) ding up the coin in many tiny ways that could alter the subsequent aerodynamics and bouncing.

The known-distribution, I'll guess, does NOT exist in real life. Oh, yes, you could fill a jar with exactly 1000 red and exactly 1000 black balls, then draw with replacements, but that does not describe any economic activity *I* can think of.

So the textbook problem is the oddball one. In the real world, you have to assess the likelihood of being given a penny with a particular distribution, then add up all those distributions. Very likely, you have little or no information on which to base that. You never step into the same river twice and you never have a distribution replicated perfectly in the future.

So you've indeed found an interesting puzzle: why do some statisticians think that textbook stats applies to the real world without an informed guesstimate about the magnitude of the unknowns?

Peter Dorman said...


Sorry to be delayed. My rumination on this problem (although it is not mention explicitly) was published as "Evolving Knowledge and the Precautionary Principle" in Ecological Economics, 2005. At the time I wrote it I didn't know much about Bayesian analysis, so I didn't express my position as clearly as I could have. Also, be aware that EE managed to munge the one and only equation in the article. The published a correction, but still....

dwb said...

good, experienced project finance people when they sit around and discuss a project add in haircuts for all the things they *cant* think of (and did not think of last time, but which ended up burning them).

thats why we are humans not comupter algorithms.

Paul Schure said...

Although I am not deeply into this it appears this has to do with the difference between risk and fundamental uncertainty.

Have a look at:
Banerjee, Snehal, Ron Kaniel and Ilan Kremer. 2009. Price Drift as an Outcome of Differences in Higher Order Beliefs. Review of Financial Studies. 22(9): 3707-3734.

Abstract of this paper:

Motivated by the insight of Keynes (1936) on the importance of higher order beliefs in financial markets, we examine the role of such beliefs in generating drift in asset prices. We show that in a dynamic setting, a higher order difference of opinions is necessary for heterogeneous beliefs to generate price drift. Such drift does not arise in standard difference of opinion models, since investors' beliefs are assumed to be common knowledge. Our results stand in contrast to c and others, as we argue that in rational expectation equilibria, heterogeneous beliefs do not lead to price drift.

Also have a look at AMS 2006, or: