  Von Neumann and Morgenstern (1944) claim that preferences are defined over a domain of lotteries. Some of these lotteries are constant, and yield a single prize with certainty. (Prizes might include a banana, a million dollars, a million dollars' worth of debt, death, or a new car.)  Lotteries can also have other lotteries as prizes, so that one can have a lottery with a 40% chance of yielding a banana, and a 60% chance of yielding a 50-50 gamble between a million dollars and death.) The domain of lotteries is closed under a mixing operation, so that if \(L\) and \(L'\) are lotteries and \(x\) is a real number in the \([0, 1]\) interval, then there is a lottery \(x L + (1-x) L'\) that yields \(L\) with probability \(x\) and \(L'\) with probability \(1-x\). They show that every preference relation obeying certain axioms can be represented by the probabilities used to define the lotteries, together with a utility function which is unique up to positive linear transformation.