A counterexample to Fishburn's conjecture on finite linear qualitative probability
type: Item
instance of: scholarly article
title: A counterexample to Fishburn's conjecture on finite linear qualitative probability
zbMATH Open document ID: 1066.60003
published in: Journal of Mathematical Psychology
publication date: 2005-05-04T00:00:00Z
review text: 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 \(n \geq 7.\)
reviewed by: Nelson I. Tanaka
Mathematics Subject Classification ID: 60A05
Mathematics Subject Classification ID: 03B48
zbMATH DE Number: 2165014
zbMATH Keywords: comparative probability
zbMATH Keywords: cancellation conditions
zbMATH Keywords: discrete cones
author: Marston D. E. Conder
author: Arkadii Slinko
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: j.jmp.2004.08.004
OpenAlex ID: W2014494178
Wikidata QID: Q123276724
cites work: The Magma algebra system. I: The user language
cites work: Aggregation of dichotomic preferences
cites work: Finite linear qualitative probability
cites work: Q4221885
cites work: Intuitive Probability on Finite Sets
cites work: Q5639713
cites work: Q5732164
cites work: Measurement structures and linear inequalities
cites work: Q4344108
cites work: Q4775205
DOI: 10.1016/J.JMP.2004.08.004
description: scientific article; zbMATH DE number 2165014