But I am not convinced by his proposal, for two reasons. Even if they are, they can make different choices, with different results. But if we draw this distinction clearly it will cast doubt on Roemer's proposal to equalize the prospects of people at the median of different groups.
People can be equally responsible for very different outcomes.
Within each of Roemer's types, people are by hypothesis equally responsible for their position. But because of this, it is arbitrary to use the median within each type to establish comparable degrees of responsibility across types. They simply make different choices. To the extent different results flow from those choices, they are responsible for different results. But across types, differences of persons' positions are due partly to choice and partly to circumstances beyond control, or brute luck.