 When an agent’s preferences in addition satisfy the independence condition and some other technical conditions, then they can be represented by an expected utility function (Harsanyi 1977b, ch. 4, Hernstein and Milnor 1953, Ramsey 1926, and Savage 1972). Such a function has two important properties. First, the expected utility of a lottery is equal to the sum of the (expected) utilities of its prizes weighted by their probabilities. Second, expected utility functions are unique up to a positive affine transformation. What this means is that if U and V are both expected utility functions representing the preferences of an agent, then for all objects of preference, X, V(X) must be equal to aU(X) + b, where a and b are real numbers and a is positive. In addition, the axioms of rationality imply that the agent’s degrees of belief will satisfy the axioms of the probability calculus.