Kadane’s Principles of Uncertainty

Kadane Principle of Uncertainty Chapman Hall 2011

This book is mostly about quantifiable uncertainty, or what is commonly called probability, rather than any broader conception of uncertainty.

Motivating example

The motivating example (1.1) involves anticipating the weather tomorrow, which may be wet or dry, below or above 68°F. One can buy insurance against specified weather conditions, such as being wet. Kadane notes:

We do not consider the issue of default, that either of us will be unable or unwilling to redeem our promises when the time comes to settle. … There is [also] an assumption here that the price at which you offer to buy such a ticket is the same as the price at which you are willing to sell such a ticket.

Thus he sets up a Dutch-book argument.

… we now study what properties your prices must have so that you are assured of not being a sure loser. …

avoiding sure loss is a weak requirement on what it takes to behave reasonably in the face of uncertainty.

 

Kadane shows how these weak assumptions lead to the axioms of probability theory.

Other views

A distinction is drawn in some economics writing between “risk” and “uncertainty” ,the rough idea being that “risk” concerns matters about which there are agreed probabilities, while “uncertainty” deals with the remainder. This distinction is attributed by some to Knight (1921) … . Others attribute it to Keynes ([General Theory of Employment]1937, pp. 213, 214). The view taken in this book is that from the viewpoint of the individual decision-maker, this distinction is not useful, a point conceded by Keynes (ibid, p. 214).

“The sense in which I am using the term uncertain is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence, or the obsolescence of a new invention, or the position of private wealth-owners in the social system in 1970. About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know. Nevertheless, the necessity for action and for decision compels us as practical men to do our best to overlook this awkward fact and to behave exactly as we should if we had behind us a good Benthamite calculation of a series of prospective advantages and disadvantages, each multiplied by its appropriate probability, waiting to the summed.”

Hence the view is that if Knightian uncertainty exists, it does not affect the decision maker.

Comments

The principles of uncertainty are intended as ‘the principles of uncertainty as it affects rational decision-taking’, rather than being more general.

Insurance example

Kadane’s axioms are formally dependent on his assumptions, and so might not apply when:

• we are not sure about the pay-offs we will receive in particular circumstances, for example we may not trust others, or:
• we want to set a margin between buying and selling, if only to avoid nugatory effort. 

Avoiding a sure loss seems like an essential part of being rational, but since the other assumptions are already quite strong, the overall effect is of assumptions that – taken together – are not weak. In most situations I can avoid a sure loss simply by not offering. Is it reasonable to forbid a margin?

My ability to forecast the weather is not the finest. Suppose that I play Kadane’s insurance game every day. If I play against someone whose forecasts are better than mine, they will tend to win money off me in the long-run. In this sense I have a sure (long-run) loss against better forecasters.  To avoid this I would normally wish to set a margin between my buying and selling price, dependent on how confident I was in my forecasts. Kadane forbids this, but otherwise I could follow his exposition to obtain an axiomatization of imprecise probabilities, with a theory much like Binmore’s.

Keynes

Keynes, like Knight, was strong in putting forward the view that there was more to uncertainty than quantified probability, and that the difference mattered. He is sometimes held to have recanted in later life. Keynes admits “the necessity for action and for decision compels us as practical men to … “.  Keynes is not talking about, for example, insurance, but about ordinary business. If we wish to rely on Keynes we must restrict the scope of the theory still further: ‘the principles of uncertainty as it affects rational decision-taking by ordinary businesses acting under competition’. In business there is competitive pressure on margins.  Those who are bad at forecasting go broke. In recent decades there has also been strengthened standards and strong downward pressure on transaction costs. Hence Kadane’s original assumptions might be reasonable of typical contemporary economic activity. That is, the business environment more and more encourages or constrains people to think probabilistically. But that doesn’t mean that shocks can’t happen and that there might not be benefits in thinking more broadly.

Knight

Kaldane makes no mention of the common argument, as in Knight, that as in the above example, people will be less inclined to gamble when they are less confident in their probability estimates. Thus in business people will generally seek to work in areas where they can estimate, or will fail. But in uncertain times they may reasonably become more cautious.

So?

I guess I’m looking for another book, ‘the principles of broad uncertainty’.

Dave Marsay