A Counterexample to Fishburn’s Conjecture on Finite Linear Qualitative Probability

Kraft, Pratt and Seidenberg (1959) provided an infinite set of axioms which, when taken together with de Finetti’s axiom, gives a necessary and sufficient set of “cancellation” conditions for representability of an ordering relation on subsets of a set by an order-preserving probability measure. Fishburn (1996) defined f(n) to be the smallest positive integer k such that every comparative probability ordering on an n-element set which satisfies the cancellation conditions C4, . . . , Ck is representable. By the work of Kraft et al. (1959) and Fishburn (1996, 1997), it is known that n − 1 ≤ f(n) ≤ n + 1 for all n ≥ 5. Also Fishburn proved that f(5) = 4, and conjectured that f(n) = n − 1 for all n ≥ 5. In this paper we confirm that f(6) = 5, but give counter-examples to Fishburn’s conjecture for n = 7, showing that f(7) ≥ 7. We summarise, correct and extend many of the known results on this topic, including the notion of “almost representability”, and offer an amended version of Fishburn’s conjecture.