On the existence of extremal cones and comparative probability orderings

We study the recently discovered phenomenon [Conder, M. D. E., & Slinko, A. M. (2004). A counterexample to Fishburn's conjecture. Journal of Mathematical Psychology, 48(6), 425-431] of existence of comparative probability orderings on finite sets that violate the Fishburn hypothesis [Fishburn, P. C. (1996). Finite linear qualitative probability. Journal of Mathematical Psychology, 40, 64-77; Fishburn, P. C. (1997). Failure of cancellation conditions for additive linear orders. Journal of Combinatorial Designs, 5, 353-365]—we call such orderings and the discrete cones associated with them extremal. Conder and Slinko constructed an extremal discrete cone on a set of n=7 elements and showed that no extremal cones exist on a set of n?6 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 1725 of the 1842 primes between 132 and 16,000, hence for all these primes extremal cones exist.