The simplest version of expected utility formulates X as a set of finite-support probability distributions, also called lotteries, on a set C of consequences. The consequences in C are viewed as mutually exclusive outcomes of decision. If for distinct , and if x is chosen over other lotteries, then the unique outcome of the decision is , with probability , for i = 1,2,3. The expected utility representation for weak order involves a utility function for which
It is commonly assumed that X is closed under convex combinations so , defined by for all , is in X whenever and . This is tantamount to assuming that X is closed under every binary operation in a set of operations for , with .
A straightforward generalization of the simple version under convex closure takes X as a function set closed under a mixture operation which maps each triple for and into another member of X that is usually denoted by or or . This is essentially the approach taken in the axiomatization of von Neumann and Morgenstern (1944), whose utility representation is (1) in conjunction with the linearity property
When the simple version applies and u is extended from X to C by under the assumption that every degenerate distribution , for which , is in X, (6) follows from (1) and (7).
After early confusion about the von Neumann-Morgenstern approach was clarified by Malinvaud (1952) (see also Fishburn and Wakker 1995), several equivalent axiomatizations for (1) and (7) were developed. My favorite is Jensen's (1967) which, in addition to properties for the mixture operation, uses three assumptions for on X. They are weak order and the following for all and all :
Independence says that is preserved under similar mixtures. It is usually defended by an interpretation for and , whereby x or y obtains with probability and z obtains with probability . Continuity ensures the existence of real-valued utilities as opposed to nonstandard or multidimensional lexicographically ordered utilities. We discuss the lexicographic case in Section 5.
The proof in Fishburn (1970, 1982, 1988) that (1) and (7) follow from weak order, independence and continuity shows that these axioms imply independence for for all and all . Then, whenever and , we prove that for a unique . We then fix , set and , define when and , and show that (1) and the linearity property (7) hold for all members of X in the closed preference interval from and . The representation is then extended to the rest of X in the only way that preserves linearity under indifference. The resulting u is unique up to a positive affine transformation obtained, for example, by choosing values for and other than 1 and 0.
Modifications of (1) with maintenance of linearity (7) when is assumed only to be a partial order are presented in Fishburn (1970, 1982), Vincke (1980) and Nakamura (1988), and integral forms for (6) are axiomatizated for certain classes of probability measures in Fishburn (1970, 1982).
Beginning around 1980, several investigators developed nonlinear versions of the theory that weaken its independence axiom, which is often inconsistent with actual preferences (Allais 1953, MacCrimmon and Larsson 1979, Kahneman and Tversky 1979, Tversky and Kahneman 1986). Consider an example from Kahneman and Tversky (1979) with and :
A majority of 94 respondents in their study violated independence with and . Fishburn (1988, Chapter 3) reviews weak order and partial order theories due to Machina (1982), Chew (1983), Fishburn (1983) and Quiggin (1993), among others, that accommodate this and other failures of independence. Section 6 describes another theory that also allows violations of transitivity.
The preceding theories are grouped under the heading of decision under risk because their consequence probabilities are given as part of the formulation and are not derived from axioms. Theories of expected utility that derive subjective probabilities of uncertain events along with utilities from axioms are grouped under the heading of decision under uncertainty. The best known of these is Savage's (1954) theory of subjective expected utility. A full account also appears in Fishburn (1970), and a summary is given in Fishburn (1994, Section 7).
Savage's theory is based on a set C of consequences and a set S of states of the world that describe the decision maker's areas of uncertainty and are outside his or her control. Subsets of S are uncertain events, and the relevant set of events is assumed to be the entire power set . We take X as the function set of all maps from S to C and refer to each as an act. The constant act that assigns consequence c to every state is denoted by . If is the chosen act and state s obtains, then x(s) is the resulting consequence. Savage's representation is (1) and
where u(c) on the right denotes and is a finitely additive probability measure on . The measure is unique and satisfies the divisibility property at the end of the preceding subsection; u is bounded and unique up to a positive affine transformation.
The representation of (1) and (8) is derived from axioms that include weak order, independence assumptions, and an Archimedean partition axiom. The proof shows first that the axioms for comparative probability on at the end of the preceding subsection follow from the axioms of preference. This gives , which is then used to construct lotteries that correspond to acts with finite consequence sets . The natural definition of on the lottery set is shown to satisfy expected utility axioms that yield a linear u on lotteries and establish Savage's representation for all acts with finite consequence sets. The representation is then extended to all acts.
Savage's contribution stimulated a number of alternative axiomatizations of subjective expected utility for decision under uncertainty, including theories in Anscombe and Aumann (1963), Pratt, Raiffa and Schlaifer (1964) and Fishburn (1967). A comprehensive review is given in Fishburn (1981). Theories that weaken additivity for subjective probability or transitivity of or , including those in Schmeidler (1989), Luce and Narens (1985), Gilboa (1987) and Fishburn and LaValle (1987), are described in Chapter 8 in Fishburn (1988).
Savage's theory has the disadvantage of requiring S to be infinite. The most popular alternatives to his theory that retain weak order and additive subjective probability, and which yield unique subjective probabilities for finite S, use lotteries in their formulations. This is done either by replacing C by the set of all lotteries on C, so that acts map states into lotteries, or by constructing mixed acts as lotteries whose outcomes are Savage acts: see, for example, Anscombe and Aumann (1963) and Fishburn (1967, 1970). To illustrate the approach, let X denote the set of all maps from finite S into . State s is said to be nonnull if for some that differ only in state s. We apply Jensen's axioms of weak order, independence and continuity to and add a nontriviality condition and a new independence axiom which says that if s and t are nonnull states, and p and q are lotteries in , then, for all , (x with x(s) replaced by with x(s) replaced by with x(t) replaced by with x(t) replaced by q). The axioms imply that there is a unique probability distribution on S and a linear unique up to a positive affine transformation such that
with state s null if and only if .
Subjective probabilities in (9) arise from two observations. First, Jensen's axioms for imply additivity over states:
where each on is linear, and the collectively are unique up to similar positive affine transformations. Second, when is defined by
the new independence axiom says that is the same for every nonnull state. Since preserves , it follows for nonnull s and t that is a positive affine transformation of , say with a>0, so for (9). That is, and , and normalization of the then gives (9) with .