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...)