What are the three fairness criteria that Kleinberg et al. (2016) proved cannot be satisfied simultaneously by a classifier?

Kleinberg et al. (2016) proved that a classifier cannot simultaneously satisfy calibration, equal false positive rates, and equal false negative rates.

Kleinberg et al. (2016) demonstrated the impossibility of a classifier meeting three fairness criteria at the same time: calibration, equal false positive rates, and equal false negative rates. Calibration refers to the alignment of predicted probabilities with actual outcomes. Equal false positive rates mean that the rate at which individuals are incorrectly classified as positive should be the same across groups. Equal false negative rates require that the rate at which individuals are incorrectly classified as negative should also be the same across groups. Achieving all three simultaneously is only possible in trivial cases, necessitating a choice among these criteria when designing fair algorithms.