Ellsberg’s Risk …
Conventionally, when there is no unique credible probability distribution over outcomes, one should form an average or ‘expected’ probability distribution and use it in maximizing expected utility, applying the Principle of Insufficient Reason if necessary. Ellsberg shows that under such conditions of ‘ambiguity’ even experts like Savage may take some account of the uncertainty in expected utility. Thus ‘conservative’ people will tend to avoid actions whose outcomes are highly ambiguous, even if they have the best ‘expected’ outcome. This violates Savage’s ‘Sure-thing’ axiom, and means that their subjective probabilities cannot be represented just by numbers.
Ellsberg gives some ‘paradoxical’ thought-experiments based on urns, but these have failed to convince probabilists, and so Ellsberg has not had the influence on economics, for example, that we might have wished.
While Ellsberg correctly identifies that while Sure-Thing and the Principle of Sufficient Reason seem commonsense, not even mathematicians are obliged to accept them, but he gives no mathematical argument against them, or for taking account of ambiguity.
This subject was discussed by Keynes in his Treatise at length. For example, if one takes Boole’s notion of probability as a variable, not just a number, then utilities become expressions involving variable probabilities. Thus there is an extra degree of freedom, and the paradox is resolved: one may prefer a definite utility to an ambiguous one.
Interestingly, Ellsberg suggests a link to Whitehead: if one acts ‘rationality’ one is taking no account of the potential for underming the conditions on which the current regularities (including the probability distributions) are dependent. By acting ‘conservatively’ one avoids undermining the basis for one’s own actions. On the other hand, if one is innovative in seeking to make a difference then one can’t expect to rely on one’s routine expectations, but should reason about what might happen some other way.