One of the main concerns about fairness in machine learning (ML) is that, in
order to achieve it, one may have to renounce to some accuracy. Having this
trade-off in mind, Hardt et al. have proposed the notion of equal opportunities
(EO), designed so as to be compatible with accuracy. In fact, it can be shown
that if the source of input data is deterministic, the two notions go well
along with each other. In the probabilistic case, however, things change.

As we show, there are probabilistic data sources for which EO can only be
achieved at the total detriment of accuracy, i.e. among the models that achieve
EO, those whose prediction does not depend on the input have the highest
accuracy.

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