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ISIPTA'07 -
FIFTH INTERNATIONAL SYMPOSIUM ON

IMPRECISE PROBABILITY: THEORIES AND APPLICATIONS

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Charles University, Faculty of Mathematicsand Physics

Prague, Czech Republic

16-19 July 2007

## ELECTRONIC PROCEEDINGS

## Marston Conder, Dominic Searles, Arkadii Slinko

# Comparative Probability Orders and the Flip Relation

### Abstract

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: **

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