THE THRESHOLD DECISION MAKING EFFECTUATED BY THE ENUMERATING PREFERENCE FUNCTION

Based on the leximin and leximax preferences, we consider two threshold preference relations on the set X of alternatives, each of which is characterized by an n-dimensional vector (n ≥ 2) with integer components varying between 1 and m(m ≥ 2). We determine explicitly in terms of binomial coefficients the unique utility function for each of the two relations, which in addition maps X onto the natural 'interval' $\{1,2, \ldots, \vert \tilde{X}\vert\}$, where $\tilde{X} = X/I$ is the quotient set of X with respect to the indifference relation I on X induced by the threshold preference. This permits us to evaluate all equivalence classes and indifference classes of the threshold order on X, present an algorithm of ordering the monotone representatives of indifference classes, and restore the indifference class of an alternative via its ordinal number with respect to the threshold preference order.