In my article on utility functions (Bernard, 1974) and in my note on 'Two Applications of the CEVR Utility Function' (Bernard, 1978), I analyzed by means of the CEVR, several historical and other examples of 'bets' or 'rational choices' found in the literature, e.g., the St Petersburg, Bernoulli, Morlat, Ellsberg, Allais, and Tversky paradoxes, as well as a simple case of insurance. Further Kavka (1980) presents two other examples, viz. a case of illness and the case of nuclear war or peace between the U.S.A. and the U.S.S.R. This is, of course, of the greatest importance to all of us. We shall begin with Kavka's case of illness, completing and formalizing Kavka's approach. There are three 'states of nature' or 'events' in this case, i.e.: death, total recovery, and permanent paralysis. Further, there are two possible 'actions', i.e.: the new drug is not used or the new drug is used. The probabilities of obtaining the three events in consequence of action 1 (the new drug is not used) are respectively Pl, P~, and pl such that 2~ pl = 1, i = 1, 2, 3. The probabilities of obtaining the three events in consequence of action 2 (the new drug is used) are respectively p~, p~, and p] such that Y, p] = 1, ] = 1, 2, 3. The 'values' of the outcomes for the agent given action 1 are al, a~, and a~ respectively for the three events. And for action 2 they are a~, a~, and a] respectively for the three events. This is illustrated in Table I.
Kavka justifies his cursive presentation of this case by the great difficulty in estimating the probabilities of the events as well as the values of the outcomes which is the result of an absence of a 'utilitarian moral theory of choice'. He nevertheless sets p~ = p] = 0, since the patient will not die if he does not use the new drug and he will not be permanently maimed if he does use it.