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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