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Expected utility

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 tex2html_wrap_inline1625 for distinct tex2html_wrap_inline1627 , and if x is chosen over other lotteries, then the unique outcome of the decision is tex2html_wrap_inline1505 , with probability tex2html_wrap_inline1633 , for i = 1,2,3. The expected utility representation for weak order involves a utility function tex2html_wrap_inline1637 for which


It is commonly assumed that X is closed under convex combinations so tex2html_wrap_inline1643 , defined by tex2html_wrap_inline1645 for all tex2html_wrap_inline1647 , is in X whenever tex2html_wrap_inline1049 and tex2html_wrap_inline1653 . This is tantamount to assuming that X is closed under every binary operation tex2html_wrap_inline1657 in a set of operations for tex2html_wrap_inline1659 , with tex2html_wrap_inline1661 .

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 tex2html_wrap_inline1665 for tex2html_wrap_inline1049 and tex2html_wrap_inline1653 into another member of X that is usually denoted by tex2html_wrap_inline1665 or tex2html_wrap_inline1675 or tex2html_wrap_inline1643 . 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 tex2html_wrap_inline1689 under the assumption that every degenerate distribution tex2html_wrap_inline1691 , for which tex2html_wrap_inline1693 , 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 tex2html_wrap_inline1051 on X. They are weak order and the following for all tex2html_wrap_inline1701 and all tex2html_wrap_inline1613 :

independence: tex2html_wrap_inline1705 ,

continuity: tex2html_wrap_inline1707  .
Independence says that tex2html_wrap_inline1051 is preserved under similar mixtures. It is usually defended by an interpretation for tex2html_wrap_inline1711 and tex2html_wrap_inline1713 , whereby x or y obtains with probability tex2html_wrap_inline1719 and z obtains with probability tex2html_wrap_inline1723 . 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 tex2html_wrap_inline1725 for all tex2html_wrap_inline1653 and all tex2html_wrap_inline1095 . Then, whenever tex2html_wrap_inline1151 and tex2html_wrap_inline1733 , we prove that tex2html_wrap_inline1735 for a unique tex2html_wrap_inline1659 . We then fix tex2html_wrap_inline1739 , set tex2html_wrap_inline1741 and tex2html_wrap_inline1743 , define tex2html_wrap_inline1745 when tex2html_wrap_inline1747 and tex2html_wrap_inline1749 , and show that (1) and the linearity property (7) hold for all members of X in the closed preference interval from tex2html_wrap_inline1753 and tex2html_wrap_inline1755 . 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 tex2html_wrap_inline1761 and tex2html_wrap_inline1763 other than 1 and 0.

Modifications of (1) with maintenance of linearity (7) when tex2html_wrap_inline1051 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 tex2html_wrap_inline1767 and tex2html_wrap_inline1769 :


A majority of 94 respondents in their study violated independence with tex2html_wrap_inline1151 and tex2html_wrap_inline1775 . 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 tex2html_wrap_inline1783 . We take X as the function set tex2html_wrap_inline1787 of all maps from S to C and refer to each tex2html_wrap_inline1793 as an act. The constant act that assigns consequence c to every state is denoted by tex2html_wrap_inline1797 . If tex2html_wrap_inline1793 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 tex2html_wrap_inline1807 and tex2html_wrap_inline1809 is a finitely additive probability measure on tex2html_wrap_inline1783 . The measure tex2html_wrap_inline1809 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 tex2html_wrap_inline1783 at the end of the preceding subsection follow from the axioms of preference. This gives tex2html_wrap_inline1809 , which is then used to construct lotteries that correspond to acts with finite consequence sets tex2html_wrap_inline1821 . The natural definition of tex2html_wrap_inline1051 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 tex2html_wrap_inline1051 or tex2html_wrap_inline1057 , 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 tex2html_wrap_inline1837 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 tex2html_wrap_inline1837 approach, let X denote the set of all maps from finite S into tex2html_wrap_inline1837 . State s is said to be nonnull if tex2html_wrap_inline1151 for some tex2html_wrap_inline1049 that differ only in state s. We apply Jensen's axioms of weak order, independence and continuity to tex2html_wrap_inline1055 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 tex2html_wrap_inline1837 , then, for all tex2html_wrap_inline1793 , (x with x(s) replaced by tex2html_wrap_inline1875 with x(s) replaced by tex2html_wrap_inline1879 with x(t) replaced by tex2html_wrap_inline1875 with x(t) replaced by q). The axioms imply that there is a unique probability distribution tex2html_wrap_inline1809 on S and a linear tex2html_wrap_inline1893 unique up to a positive affine transformation such that


with state s null if and only if tex2html_wrap_inline1897 .

Subjective probabilities in (9) arise from two observations. First, Jensen's axioms for tex2html_wrap_inline1055 imply additivity over states:


where each tex2html_wrap_inline1903 on tex2html_wrap_inline1837 is linear, and the tex2html_wrap_inline1903 collectively are unique up to similar positive affine transformations. Second, when tex2html_wrap_inline1909 is defined by


the new independence axiom says that tex2html_wrap_inline1909 is the same for every nonnull state. Since tex2html_wrap_inline1903 preserves tex2html_wrap_inline1909 , it follows for nonnull s and t that tex2html_wrap_inline1903 is a positive affine transformation of tex2html_wrap_inline1925 , say tex2html_wrap_inline1927 with a>0, so tex2html_wrap_inline1931 for (9). That is, tex2html_wrap_inline1933 and tex2html_wrap_inline1935 , and normalization of the tex2html_wrap_inline1937 then gives (9) with tex2html_wrap_inline1939 .

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Next: Cancellation Conditions Up: Preference Representations Previous: Additive measurement