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FOURTH
INTERNATIONAL SYMPOSIUM ON

IMPRECISE PROBABILITIES AND THEIR APPLICATIONS

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Carnegie Mellon University

Pittsburgh, PA, USA

July 20-23 2005

#
ISIPTA'05 ELECTRONIC PROCEEDINGS

## Simon Marshall

# On the Existence of Extremal Cones and Comparative Probability Orderings

### Abstract

We study the recently discovered phenomenon of existence
of comparative probability orderings on finite sets that violate Fishburn
hypothesis - we call such orderings and the discrete cones
associated with them extremal. Conder and Slinko constructed an extremal discrete cone on the set of n=7 elements and showed that no extremal cones exist on the set of n< 7 elements. In this paper we construct an extremal cone on a finite set of prime cardinality p if p
satisfies a certain number theoretical condition. This condition has been
computationally checked to hold for 1,725 of the 1,842 primes between 132 and 16,000, hence for all these primes extremal cones exist.

** Keywords. ** Comparative probability ordering, Discrete cone, Quadratic residues

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smc@decsai.ugr.es