Decision theory, which traces its philosophical foundations to antiquity, has developed into a mathematically mature subject in recent times. Early evidence of mathematical analysis in decision theory appears in the eighteenth century writings of Daniel Bernoulli (1738) on rational analysis of risky decisions and of Jean-Charles de Borda (1781) and the Marquis de Condorcet (1785) on aggregation of individuals' preferences through voting or algebraic combination for collective action.
The first third of the present century witnessed a new level of mathematical sophistication in Norbert Wiener's (1914) axiomatic analysis of what we now refer to as interval orders, Frank P. Ramsey's (1931) axiomatization for decision under uncertainty, and Bruno de Finetti's (1931, 1937) contributions to subject probability and logical decision making. As mid-century approached, these were joined by the monumental treatise on rational choice and the theory of games by John von Neumann and Oskar Morgenstern (1944, 1947, 1953). Then, in the 1950's, other books that have profoundly influenced mathematical research in decision theory through the rest of the century appeared. These include Kenneth Arrow's (1950, 1963) work on social choice theory, Leonard J. Savage's (1954, 1972) axiomatic foundations for subjective expected utility theory in decision under uncertainty, and Gerard Debreu's (1959) axiomatization of preferences for utility-based economic equilibrium analysis.
The central principle for human judgment and choice in the vast majority of these works and their successors is the notion of order, formalized by transitivity, and the related notion of decision-by-maximization. Even when decision paradigms do not transparently involve maximization, as with Nash equilibria in non-cooperative games (Nash 1951, Hart 1992) and some ballot aggregation procedures, individuals are often assumed to have ordered preferences.
The emphasis on order and maximization has led to a huge body of work
on quantification of preferences, likelihood judgments, and
other qualitative aspects of judgment and choice.
The obvious reason is that quantification facilitates the search for
optimal or near-optimal decisions.
A less obvious reason is that many contributors to decision theory have
been instrumental in developing the representational theory of measurement,
which subsumes but is certainly not limited to representations of
preferences and other aspects of decision theory.
The representational theory of measurement was formalized in
Scott and Suppes (1958) and has received its most complete expression in the
three-volume set by Krantz et al. (1971), Suppes et al. (1989)
and Luce et al. (1990).
Its defining characteristic is the quantitative representation by
analogous numerical structures of qualitative structures that
consist of a ground set X and one or more relations or operations on X.
The set X may have a variety of structural properties, e.g. as a Cartesian
product set or a set of probability distributions, and one of its
relations is often assumed to be a binary ordering relation.
A familiar operation is the binary concatenation operation
where 
denotes the joining together of objects
by placing them
end-to-end for length measurement or putting them in the same
balance pan for weight measurement.
We often use 
to denote an asymmetric and transitive binary
relation on X, in which case 
is a partially ordered set,
and we always define 
as its symmetric complement by
The relation could denote is preferred to, or is more probable than,
or is longer than,
and so forth.
Corresponding interpretations of are is indifferent to, is equally probable as, and is the same length as.
However, if is assumed only to be a partial order without also
being transitive, in which case is not necessarily an equivalence
relation, then 
for 
could signify
incomparability rather than comparable equality.
Positive, closed extensive measurement provides a nice example of a qualitative
structure represented by an analogous quantitative structure.
The qualitative structure is 
with order
relation on X.
We assume also that is transitive, is positive
[for all ,
, is closed under
[for all there is
a 
such that
], and the structure satisfies an
Archimedean condition that is needed for a real valued as opposed to nonstandard
or lexicographic representation.
The analogous quantitative structure is 
, where
denotes the positive reals.
The representation is:
there exists 
such that, for all ,
and
The mapping 
thus preserves on X by > on , and
takes into +.
The representational theory also pays close attention to the uniqueness
status of representing functions.
In the example, is unique up to multiplication by a positive constant:
if satisfies the representation then so does 
if and only
if 
for some 
.
I have described the representational theory of measurement because it provides a general framework for most decision-theoretic representations. Other works that emphasize its approach include Pfanzagl (1968), Roberts (1979) and Narens (1985). Books not cited earlier that adopt the representational tack for decision theory include Fishburn (1970, 1982, 1988) and Wakker (1989), and extensive surveys are available in Fishburn (1968, 1981, 1989, 1994). The sections to follow discuss the representational theory for a variety of preference structures. They are not exhaustive but rather offer a selective survey that illustrates facets of preference theory and includes recent results not found in earlier surveys.
The next section opens with a few definitions of central importance to our subject and then describes basic representations for ordinal utility theory, additive utility theory, and expected utility theory. Section 3 begins our consideration of specific topics with a discussion of cancellation conditions for finite additive measurement. We emphasize recent work on the extent to which such conditions are needed to ensure additivity. Section 4 continues the additivity theme by showing how a general theorem for additive measurement applies to a utility threshold representation for sets of arbitrary cardinality. Section 5 illustrates recent contributions to decision under risk and decision under uncertainty in two areas. The first is a generalization of Savage's subjective expected utility theory in which utilities are real vectors ordered lexicographically and subjective probabilities take the form of real matrices. The second focuses on the role of a binary operation of joint receipt for situations in which holistic alternatives consist of similar but clearly discernible pieces. Section 6 concludes the paper with examples of preference cycles and representations that accommodate cyclic preferences. The representations described in earlier sections assume that preferences are transitive, or at least acyclic.