Several seemingly different types of representations are grouped together under
this heading because they have an additive character and can be analyzed by similar
mathematical methods.
A general theory of additive measurement is presented in Fishburn (1992b),
where it is applied to a variety of contexts, including positive extensive
measurement, additive utility measurement for multiattribute alternatives,
difference measurement for strength of preference comparisons,
threshold measurement, expected utility,
and comparative probability.
The paper includes a condition for X of arbitrary cardinality that is
necessary and sufficient for the existence of an additive representation.
I will describe its approach for threshold measurement in Section 4.
The present subsection considers only comparative
probability and multiattribute utility to illustrate the additive theme.
I include comparative probability under the preference rubric
because its relation 
is often defined from preference comparisons.
Suppose x and y are uncertain events.
Let 
be the gamble that pays $100 if x obtains and
$0 otherwise, and similarly for 
.
Then the approach promoted by de Finetti (1937) and Savage (1954)
defines 
if is preferred to .
We formulate X for the present discussion as a family of subsets of a universal
set 
.
For comparative probability, is a set of states, members of X
are events, and X usually includes the empty event 
and
universal event .
The event set X may or may not be closed under operations like union,
intersection, and complementation, and its members can have very different cardinalities.
The set of items to be compared by preference in multiattribute utility
theory is a subset A of a Cartesian product set
with 
.
Each 
is a nonempty set, and we assume without loss of generality that
the are mutually disjoint and every 
appears in at
least one n-tuple in A.
The universal set is defined as 
, and
so every member of X is an n-element subset of .
Suppose is finite and on X is a weak order.
The basic additive representation uses 
for
It is common in the multiattribute case to denote the restriction
of u on by 
so, when 
,
.
Then, when (4) holds, it remains valid when the origin of each
is translated by adding a constant 
to all values.
For comparative probability, we assume 
and that the
union 
of and 
is monotonic, so 
.
Then we can take 
and 
when (4) holds, so u becomes a probability distribution on .
A necessary and sufficient condition for (4) referred to as cancellation,
independence, or additivity, was identified first by Kraft, Pratt and
Seidenberg (1959):
CANCELLATION:
For every pair 
and
of finite sequences of members of X such that
it is false that 
for 
and 
for some j.
Necessity of Cancellation for (4) follows from the fact that (5) implies
Hence if (4) holds and if Cancellation is violated by for all j and for some j, summation over j on the
right side of (4) followed by cancellation of identical terms leaves the
contradiction that 0 > 0.
Sufficiency of Cancellation for (4) follows from solution theory for
finite systems of linear inequalities by way of a solution-existence theorem known by
various names, including the separating hyperplane lemma,
the theorem of the alternative, Farkas's lemma, and Motzkin's lemma:
see, for example, Scott (1964), Fishburn (1970, 1996a), or
Krantz et al. (1971).
Essentially the same separation lemma applies when
is only assumed acyclic or a partial order, with slight modifications in
Cancellation.
For example, if (4) is to hold when 
is replaced
by 
, we replace the last line of
Cancellation by ``it is false that for all j.''
When (4) holds for finite under weak order, u
is not generally unique in any simple sense.
Special conditions that are not necessary for (4) but which yield
simple uniqueness forms, such as absolute uniqueness for
subjective probabilities, are described in
Fishburn and Roberts (1989) and Fishburn (1989, 1994).
Additional discussions of Cancellation for finite
appears in the next section.
Theories of additive measurement for infinite usually assume
nicely structured domains,
such as 
for additive utility or 
for comparative probability.
Most also use existence axioms that simplify cancellation conditions,
promote representational uniqueness, and
facilitate the derivation or assessment of u.
Examples for (4) with
in the multiattribute case of both the algebraic and
topological varieties are detailed in
Fishburn (1970), Krantz et al. (1971) and Wakker (1989).
Their cancellation conditions use only m = 2 and m = 3
in Cancellation, and their functions as defined after (4)
are unique up to similar positive affine transformations.
When is infinite for the comparative probability case, the weak
order representation (4) is usually replaced by
(1)
in conjunction with 
,
and
so that u is a finitely additive probability measure on X.
Savage's (1954) elegant axiomatization for this representation assumes
, weak order, ,
for all x, the m = 2 part of Cancellation
which says that
and an Archimedean axiom involving finite partitions of .
The representing measure is unique and satisfies the following
divisibility property:
if 
then for every 
there is an 
such that
.
Proof are given in Fishburn (1970, 1988) as well as Savage (1954).
Survey material on related axiomatizations of comparative probability appears
in Section 6 in Fishburn (1994).
