  With the growing awareness to fairness in machine learning and the
realization of the central role that data representation has in data processing
tasks, there is an obvious interest in notions of fair data representations.
The goal of such representations is that a model trained on data under the
representation (e.g., a classifier) will be guaranteed to respect some fairness
constraints.
  Such representations are useful when they can be fixed for training models on
various different tasks and also when they serve as data filtering between the
raw data (known to the representation designer) and potentially malicious
agents that use the data under the representation to learn predictive models
and make decisions.
  A long list of recent research papers strive to provide tools for achieving
these goals.
  However, we prove that this is basically a futile effort. Roughly stated, we
prove that no representation can guarantee the fairness of classifiers for
different tasks trained using it; even the basic goal of achieving
label-independent Demographic Parity fairness fails once the marginal data
distribution shifts. More refined notions of fairness, like Odds Equality,
cannot be guaranteed by a representation that does not take into account the
task specific labeling rule with respect to which such fairness will be
evaluated (even if the marginal data distribution is known a priory).
Furthermore, except for trivial cases, no representation can guarantee Odds
Equality fairness for any two different tasks, while allowing accurate label
predictions for both.
  While some of our conclusions are intuitive, we formulate (and prove) crisp
statements of such impossibilities, often contrasting impressions conveyed by
many recent works on fair representations.
