A counterexample to Fishburn's conjecture on finite linear qualitative probability
This work deals with set of axioms which give necessary and sufficient conditions for representability of an ordering relation on subsets by an order-preserving probability measure. \textit{C. H. Kraft, J. W. Pratt} and \textit{A. Seidenberg} [Ann. Math. Stat. 30, 408--419 (1959; Zbl 0173.19606)] provided an infinite set of axioms which together with de Finetti's axiom give a necessary and sufficient set of cancellation conditions for the above representability to hold on finite sets. \textit{P. C. Fishburn} [J. Math. Psychol. 40, 64--77 (1996; Zbl 0851.60002); J. Comb. Des. 5, 353--365 (1997; Zbl 0911.92034)] defines \(f(n)\) to be the smallest positive integer \(k\) so that every comparative probability ordering on an \(n\)-element set which satisfies the cancellation conditions \(C_4, \ldots, C_k\) is representable. From the works of the two previously mentioned authors it is known that \(n-1 \leq f(n) \leq n+1\) for all \(n \geq 5.\) Moreover, Fishburn proved that \(f(5)=4\) and conjectured that \(f(n)=n-1\) for all \(n \geq 5.\) This paper confirms that \(f(6)=5\) but gives a counterexample to the Fishburn conjecture by showing that \(f(7) \geq 7,\) using notions of discrete cones and almost representability. It also offers an amended version to Fishburn's conjecture: \(f(n)=n\) for all \(n \geq 7.\)
scientific article; zbMATH DE number 2165014