addressed situations in which the outcomes of choices are not known with certainty, but have probabilities associated with them. A notation for a lottery is as follows: if options A and B have probability p and 1 − p in the lottery, we write it as a linear combination: More generally, for a lottery with many possible options: where . By making some reasonable assumptions about the way choices behave, von Neumann and Morgenstern showed that if an agent can choose between the lotteries, then this agent has a utility function such that the desirability of an arbitrary lottery can be computed as a linear combination of the utilities of its parts, with the weights being their probabilities of occurring. This is termed the expected utility theorem. The required assumptions are four axioms about the properties of the agent's preference relation over 'simple lotteries', which are lotteries with just two options. Writing to mean 'A is weakly preferred to B' ('A is preferred at least as much as B'), the axioms are: completeness: For any two simple lotteries and , either or (or both, in which case they are viewed as equally desirable). transitivity: for any three lotteries , if and , then . convexity/continuity (Archimedean property): If , then there is a between 0 and 1 such that the lottery is equally desirable as . independence: for any three lotteries and any probability p, if and only if . Intuitively, if the lottery formed by the probabilistic combination of and is no more preferable than the lottery formed by the same probabilistic combination of and then and only then . Axioms 3 and 4 enable us to decide about the relative utilities of two assets or lotteries. In more formal language: A von Neumann–Morgenstern utility function is a function from choices to the real numbers: which assigns a real number to every outcome in a way that represents the agent's preferences over simple lotteries. Using the four assumptions mentioned above, the agent will prefer a lottery to a lottery if and only if, for the utility function characterizing that agent, the expected utility of is greater than the expected utility of : . Of all the axioms, independence is the most often discarded. A variety of generalized expected utility theories have arisen, most of which omit or relax the independence axiom. As probability of success Castagnoli and LiCalzi (1996)