Aziz and Stursberg propose an “Egalitarian Simultaneous Reservation” rule (ESR), a generalization of Serial rule, one of the most discussed mechanisms in the random assignment problem, to the more general random social choice domain. This article provides an alternative definition, or characterization, of ESR as the unique most ordinally egalitarian one. Specifically, given a lottery p over alternatives, for each agent i the author considers the total probability share in p of objects from her first k indifference classes. ESR is shown to be the unique one which leximin maximizes the vector of all such shares (calculated for all i, k). Serial rule is known to be characterized by the same property. Thus, the author provides an alternative way to show that ESR, indeed, coincides with Serial rule on the assignment domain. Moreover, since both rules are defined as the unique most ordinally egalitarian ones, the result shows that ESR is “the right way” to think about generalizing Serial rule.