Cancellation Conditions for Finite Two-Dimensional Additive Measurement

On Fishburn’s questions about finite two-dimensional additive measurement

“…The lower bound f (3, 5) ≥ 4 is previously reported by Fishburn (2001) (based on his Theorem 3.1). Hence we arrive at f (3, 5) = 4.…”

“…In Fishburn (2001), it is shown that f (3, 3) = 3, f (3, 4) = f (4, 4) = 4 and 4 ≤ f (3, 5) ≤ 7. A call for the exact values of f (3, 5), f (4, 5) and f (5, 5) is made.…”

Preference as fulfilment of desires

“…In order to achieve an additive representation, we introduce the cancellation conditions C p by Tversky (1964) and Fishburn (1970) (See also Fishburn (2001) or Wakker (1989)). They roughly state that if two vectors, as those above, only differ by the distribution of the properties, one vector from W p cannot dominate the other in a strict sense analogous to the strong Pareto principle.…”

Additive Representability of Finite Measurement Structures

“…427-428), who made the initial contribution to this topic, proved that f (2, m 2 ) = 2 and that f (3, 3) ≥ 3. Little else was known about these functions until Fishburn's papers [18,19]. One of the most significant results of [18] was the general upper bound for f (m 1 , m 2 , .…”

“…Apart from obvious questions that these results prompt, Fishburn [18,19] formulated the following interesting ones. An important paper by Fishburn and Roberts [20] studied the uniqueness question, which we do not survey here due to lack of space.…”