Decision Theory

5Ramsey and Savage

In the previous two chapters we have studied the von Neumann–Morgenstern expected utility theory (NM theory) where all uncertainties are represented by objective probabilities. Our ultimate goal, however, is to understand problems where decision makers have to deal with both the utility of outcomes, as in the NM theory, and the probabilities of unknown states of the world, as in de Finetti’s coherence theory.

For an example, suppose your roommate is to decide between two bets: bet (i) gives $5 if Duke and not UNC wins this year’s NCAA final, and $0 otherwise; bet (ii) gives $10 if a fair die that I toss comes up 3 or greater. The NM theory is not rich enough to help in deciding which lottery to choose. If we agree that the die is fair, lottery (ii) can be thought of as a NM lottery, but what about lottery (i)? Say your roommate prefers lottery (ii) to lottery (i). Is it because of the rewards or because of the probabilities of the two events? Your roommate could be almost certain that Duke will win and yet prefer (ii) because he or she desperately needs the extra $5. Or your roommate could believe that Duke does not have enough of a chance of winning.

Generally, based on agents’ preferences, it is difficult to understand their probability without also considering their utility, and vice versa. “The difficulty is like that of separating two different co-operating forces” (Ramsey 1926, p. 172). However, several axiom systems exist that achieve this. The key is ...