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: 
,
continuity: 
.
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 
.
