By the late 1960's, weak-order additive representations for infinite X with nice uniqueness properties were well understood (Fishburn 1970, Krantz et al. 1971), but two noticeable gaps existed for finite-X representations. The first concerned conditions that imply nice uniqueness structures comparable to those of some infinite-X representations. This was partly rectified by the late 1980's in a series of papers surveyed in Fishburn and Roberts (1989).
The second gap concerned Cancellation.
To focus this concern, we reformulate Cancellation from subsection 2.2
as a sequence of conditions based on the number J of
distinct pairs 
involved in (5).
The condition for J is denoted by C(J).
C(J): For every sequence 
of distinct members of 
and
corresponding sequence 
of positive
integers such that
it is false that 
for 
and
for some j.
Condition C(1) is vacuous since its hypotheses require 
, and
C(2) is tantamount to the first-order independence condition which says that
if 
and if every 
appears in 
and same number of times it
appears in 
, then 
.
An example for 
that satisfies C(2)
is the linear order
but this violates C(3) because
In this example,
and
, and so forth.
The 
in C(J) are used for repetitions of the same (x,y)
pair in the sequence 
, 
of
Cancellation, which is clearly equivalent to the conjunction of 
.
Our concern for Cancellation is the smallest J such that every weak-ordered
set 
of a given size has an
additive representation if it satisfies C(2) through C(J).
We revert here to the product formulation of multiattribute preference,
which applies also to comparative probability when
for all i and an event is characterized
by the vector 
which has
if state i is in the event and
otherwise.
We define the size of X, or of , as the n-tuple 
for
which 
for each i.
To avoid trivial 
, we assume along with 
that
for all i.
We then define
as the smallest positive
integer 
such that every weak order on X of size
that violates
Cancellation does so for some C(J) with 
.
In other words, if 
, then:
(i) there is a weak order 
on X of size
that violates 
but
satisfies C(J) for all
;
(ii)
every weak order on an X of size
that satisfies C(J) for 
also
satisfies C(K) for all 
for which C(K) is defined for
that size and therefore has an additive representation as in (4).
In the comparative probability setting for weak orders, Kraft, Pratt and
Seidenberg (1959) proved that X has an additive
representation if 
and first-order independence holds,
so f(2,2) = f(2,2,2) = f(2,2,2,2) = 2.
They showed also that 
for all 
.
In the multiattribute setting, Krantz et al. (1971, pp. 427-428) noted
that 
for all
.
Little else was known about f until recently.
We summarize here results in Fishburn (1996a, b, c)
and note topics for further research.
The first two papers focus on 
for all i.
Let 
denote 
with n entries.
The first paper shows that 
and 
for n = 6,7,8.
The latter result is extended to all 
in
Fishburn (1996b) by explicit constructions based on a theorem
in the first paper that is designed to identify structures that
violate C(J) for relatively large J but satisfy all C(J')
for small J'.
Fishburn (1996a) also shows that for every there are weak order cases
of comparative probability that violate C(4) but have
additive representations whenever one state is deleted, and that there are
failures of Cancellation that require 
for some
i and j in any corresponding failure of a C(J).
In other words, (4) can have no solution when every
applicable C(J) holds under the
restriction that 
.
Fishburn (1996c) considers 
as well as
and proves the following upper bound on f:
This is ineffective for the case of , but shows
in conjunction with the lower bound of the preceding paragraph that
for all .
We also prove for n = 2 that 
for all
even 
, and 
for all odd 
.
The upper bound for these cases is 
.
Two areas for further research are my conjecture that
for all , and derivation of good
lower bounds on 
for
general sizes.
It seems plausible that is
very close to the upper bound 
for most sizes, but
this awaits further study.