P[ :C]: Value
Possible probabilities are a generalisation of conventional, precise, probabilities in which one supposes that more than one probability function may be possible. It has been argued that, ‘rationally’, probabilities must be precise, for example using a ‘Dutch Book’ argument. These suppose that anything has a price, such that you would but it for less than its price, and sell it for more. A more general assumption is that things have a buying price, u, and selling price, w, with u ≤ w and s= w-u the ‘margin’ or spread.
Law of large numbers, spreads in expectation
We can think of this in terms of the law of large numbers. If you keep gambling a fixed amount on a fixed random mechanism then in the long-run the total pay-off will tend to its probable value, and this is described as the ‘expected’ amount. The usual assertion is that you should value the gamble at this expected amount. But an alternative is that even if you ‘expect’ a certain probability / value, you also ‘anticipate’ a certain ‘probable error’, represented by the spread. Thus if you buy between your buying and selling price you may ‘expect’ to make a profit (in the conventional sense), but you risk making a systematic, sustained, loss. Similarly, if you sold lottery tickets too cheaply you risk having to pay out too much.
Dutch books with spreads
Now, consider a partition of all possibilities into Ai, with lower and upper probabilities of li, ui. If we have options to buy gambles at price pi for all but one Aj, then the final gamble is worth 1-Σi≠jpi, with the usual Dutch-Book assumptions and argument, and one should buy at that price. Similarly, if one has all but one option to sell then one should value selling the last one according to a similar formula. Hence:
Σi≠jli + uj = 1, etc.
This variation on the Dutch-Book argument justifies the constraints on possible probabilities, starting with just the lower and upper probabilities. The usual argument assumes, in addition, that the spread is 0. But this assumes, for example, that there is some definite, knowable, value that the pound/dollar exchange rate is tending to. This hardly seems reasonable.
Commerce, with imprecision
A more positive example is where a baker makes bread each day in a town where the price of flour and bread varies within a stable probability distribution. The ‘buying’ price is then the cost of the bread, with the buying and selling prices set to cover costs, including the baker’s labour. The baker will want maxima for buying prices and minima for selling prices. Other participants in the market will set the complimentary constraints, and the market will determine the actual prices. As long as the prices remain within bounds the market participants’ needs will be satisfied, and this is the ‘context’ within which the Baker’s plans are valid. If they are not satisfied then there may need to be substantive change. The notion of ‘possible probabilities’ differs from the conventional assumptions of the Dutch Book argument in allowing the Baker a ‘spread’, allowing for some variability and uncertainty in market conditions. Bakers can surivive as long as they can make a profit: they do not have to be able to determine the exact minimum price for each item.