The expected number of pairwise comparisons needed to learn a partial order on n elements is shown to be at least /4 − (), and an algorithm is given that needs only /4 + () comparisons on average. In addition, the optimal strategy for learning a poset with four elements is presented

It is shown that each continuous transformation from Euclidean -space ( > 1) into Euclidean -space that preserves the equality of distances (that is, fulfils the implication is a similarity map. The case of equal dimensions already follows from the Beckman-Quarles Theorem

Actual individual preferences are neither complete (=total) nor antisymmetric in general, so that at least every must be an admissible input to a satisfactory choice rule. It is argued that the traditional notion of “indifference” in individual preferences is misleading and should be...