On the possibility of nonaggregative priority for the worst off

We address the question of whether there is a prioritarian moral theory that both (1) avoids giving absolute priority to the worst off (as leximin does) but (2) does not allow arbitrarily small benefits to enough extremely affluent people to take priority over a significant benefit to a worst off person in abject poverty. We argue that this is possible only if one rejects some seemingly plausible conditions.