18 Letter from Robert Aumann to Leonard Savage and Letter fromLeonard Savage to Robert Aumann

Suppose your utility for an umbrella is 1, and for no umbrella is 0; suppose further that your subjective probability for rain is 1/2. Then acts A and B have utilities 1/2 and 1 respectively. On the other hand, I don’t think it would be unreasonable for you to be indi¤erent between the two acts, since an umbrella is useless in fine weather. Obviously, the answer is that your utility for umbrellas depends on the weather, i.e. on the state of the world. But that leads rather quickly to the conclusion that your postulate P3 is unreasonable; for example, all in all, I prefer an umbrella to a nickel, but if it does not rain, I prefer the nickel. The only conclusion seems to be that it is improper to call an umbrella a ‘‘consequence.’’ An umbrella is really an act; the consequence is getting wet or not. But that does not quite get us out of trouble. Most people prefer to get wet in fine weather; the fact is, most swimming pools are poorly attended when it rains. So even the utility of getting wet is statedependent. The appropriate answer then seems to be: ‘‘Getting wet is also an act; the consequence is getting wet in the rain or getting wet in the sunshine.’’ But you will agree that that isn’t very satisfactory; we have now made the description of the consequence state-dependent. By the same token, sunshine and rain themselves could be called ‘‘consequences,’’ and then one could construct nonsensical acts such as ‘‘You get sunshine if it rains, and rain otherwise.’’ It seems that the notions of ‘‘state,’’ ‘‘act,’’ and ‘‘consequence’’ have rather fuzzy interpretations; in particular, it is not always easy conceptually to distinguish between them. But to make sense of the axioms, it is essential to have a fairly sharp idea of what these notions mean. The main question that is puzzling me is more basic, though; it does not concern your derivation of the subjective probability notion, but the very possibility of defining this notion—in any way—via preferences. Suppose Mr X loves his wife very much, he feels that if he should lose