In this paper we study the flip relation on the set of comparative probability orders on n atoms introduced by Maclagan (1999). With this relation the set of all comparative probability orders becomes a graph G_n. Firstly, we prove that any comparative probability order with an underlying probability measure is uniquely determined by the set of its neighbours in G_n. This theorem generalises the theorem of Fishburn, Peke\v c and Reeds (2002). We show that the existence of the underlying probability measure is essential for the validity of this result. Secondly, we obtain the numerical characteristics of the flip relation in G_6. Thirdly, we prove that a comparative probability order on n atoms can have in G_n up to f{n+1} neighbours, where f(n) is the nth Fibonacci number. We conjecture that this number is maximal possible. This partly answers a question posed by Maclagan.
Keywords. comparative probability, flippable pair, probability elicitation, subset comparisons, simple game, weighted majority game, desirability relation
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Authors addresses:
Marston Conder
Department of Mathematics
University of Auckland
Private Bag 92019
Auckland
Dominic Searles
138 Kapiro Road,
R.D.1.
Kerikeri,
Bay of Islands
Arkadii Slinko
Department of Mathematics
The University of Auckland
Private Bag 92019
Auckland NZ
E-mail addresses:
| Marston Conder | m.conder@auckland.ac.nz |
| Dominic Searles | dnsearles@gmail.com |
| Arkadii Slinko | a.slinko@auckland.ac.nz |