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.