Sufficient conditions making lexicographic rules over the power set satisfy extensibility

This study aims to clarify sufficient conditions for weak orders on the existing and null alternatives to make leximax and leximin rules over the power set satisfy extensibility. Each null alternative indicates ‘choosing not to choose the corresponding existing alternative’. Extensibility requires that a preference order of any two alternatives is equivalent to that of their singleton sets. Then, the leximax (alternatively, leximin) rule ranks any two subsets by comparing the same-ranked (null) alternatives in the two transformed subsets (which include the existing alternatives in each subset and the null alternatives of other existing alternatives) from top to bottom (alternatively, bottom to top). We then introduce the following two new properties: Semi-inversion desirability requires that a preference of any two null alternatives is not identical to that of their existing alternatives. Consistent desirability requires that a preference order of ‘a null alternative and a non-paired existing alternative’ is not identical to that of their paired (null) alternatives. We show that semi-inversion desirability implies extensibility, and the combination of semi-inversion desirability and consistent desirability is weaker than a traditional property, namely self-reflecting. Furthermore, we clarify the sufficient condition to make the leximax and leximin rules equivalent.