Restoring Indifference Classes via Ordinal Numbers under the Discrete Leximin and Leximax Preference Orderings

The leximin (leximax) preference ordering compares two n-dimensional real vectors as follows: the coordinates of these vectors are first ordered in ascending (descending) order and then the resulting two vectors are compared lexicographically. It is well known that the leximin (leximax) preference ordering on Rn is not representable (by a utility function). In this paper, given two integers n greater than or equal to 1 and m greater than or equal to 2, we consider the set X of all n -dimensional vectors with integer coordinates assuming values between 1 and m. Equipping X with the leximin (leximax) preference ordering induced from Rn, called the threshold (dual threshold) rule, every vector from X (and its indifference class) is canonically assigned a unique ordinal number in such a way that a vector from X is considered more leximin- (leximax-) preferable if it lies in an indifference class with greater ordinal number. We present a rigorous recursive algorithm for the evaluation of multiplicities of the coordinates in a vector from X via the ordinal number of the indifference class with respect to the ordering, to which this vector belongs. The novelty of our algorithm is twofold: first, it exhibits new properties of the classical binomial coefficients in their interplay with the leximin (leximax) preference ordering and, second, it relies on four integer parameters, each one being obtained by a different cyclic procedure. The joint work of these procedures is based on our main theorem concerning some subtle properties of the enumerating preference function, which represents the leximin (leximax) preference ordering on X.