Decision making: Keynes’ Treatise
Keynes (Ch. XXVI) asks:
“… I have argued that only in a strictly limited class of cases are degrees of probability numerically measurable. … If, therefore, the question of right action is under all circumstances a determinate problem, it must be in virtue of an intuitive judgment directed to the situation as a whole, and not in virtue of an arithmetical deduction derived from a series of separate judgments directed to the individual alternatives each treated in isolation.
We must accept the conclusion that, if one good is greater than another, but the probability of attaining the first less than that of attaining the second, the question of which it is our duty to pursue may be indeterminate… . It may be remarked, further, that the difficulty exists, whether the numerical indeterminateness of the probability is intrinsic or whether its numerical value is … simply unknown.
The second difficulty, to which attention is called above, is the neglect of the ‘weights’ of arguments in the conception of ‘mathematical expectation.’ … if two probabilities are equal in degree, ought we, in choosing our course of action, to prefer that one which is based on a greater body of knowledge?
The question appears to me to be highly perplexing, and it is difficult to say much that is useful about it. But the degree of completeness of the information upon which a probability is based does seem to be relevant, as well as the actual magnitude of the probability, in making practical decisions. …”
“Is it certain that a larger good, which is extremely improbable, is precisely equivalent ethically to a smaller good which is proportionately more probable? We may doubt whether the moral value of speculative and cautious action respectively can be weighed against one another in a simple arithmetical way … .”
This anticipates some concerns , e.g. of Taleb, about our financial systems.
“… There seems, at any rate, a good deal to be said for the conclusion that, other things being equal, that course of action is preferable which involves least risk, and about the results of which we have the most complete knowledge.”
In reviewing the classical argument that numeric probability is just what is required to support conventional ‘rational’ decision making, based on utility maximization (Ch. XXVI) Keynes notes:
“… the doctrine that the ‘mathematical expectations’ of alternative courses of action are the proper measures of our degrees of preference is open to doubt on two grounds—first, because it ignores … the amount of evidence upon which each probability is founded; and second, because it ignores the element of ‘risk’ and assumes that an even chance of heaven or hell is precisely as much to be desired as the certain attainment of a state of mediocrity. … “
This anticipates the Ellsberg paradox.
“The old assumptions, that all quantity is numerical and that all quantitative characteristics are additive, can be no longer sustained. Mathematical reasoning now appears as an aid in its symbolic rather than in its numerical character. I, at any rate, have not the same lively hope as Condorcet, or even as Edgeworth, “éclairer les Sciences morales et politiques par le flambeau de l’Algèbre.” In the present case, even if we are able to range goods in order of magnitude, and also their probabilities in order of magnitude, yet it does not follow that we can range the products composed of each good and its corresponding probability in this order.”