ABSTRACT
In common applications of von Neumann-Morgenstern expected utility theory, the prize space is some subset Z of the real line, with the interpretation that the prizes are money amounts, say reckoned in dollars. To make things as concrete as possible, suppose for the time being that Totrep is entering into gambles involving his net worth (bank balance) - these gambles are based on randomizing devices with objective probabilities, and the outcomes are independent of anything else affecting Totrep, such as other income he may earn. (We’ll see why this assumption is necessary in Chapter 12 .) Moreover, all gambles under consideration are described by simple probability distributions, and Totrep happily ascribes to the three mixture space axioms in this context, so that we know his preferences can be represented by expectation of a utility function u : Z —> R. Interpret the outcome z as Totrep’s bank balance after the gamble is conducted, and not as his net winnings from the gamble. The question is: What can be said about this function u?
