On Fairness and Calibration
The machine learning community has become increasingly concerned with the potential for bias and discrimination in predictive models. This has motivated a growing line of work on what it means for a classification procedure to be "fair." In this paper, we investigate the tension between minimizing error disparity across different population groups while maintaining calibrated probability estimates. We show that calibration is compatible only with a single error constraint (i.e. equal false-negatives rates across groups), and show that any algorithm that satisfies this relaxation is no better than randomizing a percentage of predictions for an existing classifier. These unsettling findings, which extend and generalize existing results, are empirically confirmed on several datasets.
Note: Equal opportunity as used in this paper was originally defined as “Equalized odds” in Hardt et al. 2017. This needs to be fixed in multiple places. 
The paper analyzes the interaction between relaxations of equalized odds criteria and classifier calibration. The work extends and builds on the results of a previous work [23]. 
1. Incompatibility of calibration and non-discrimination: The previous work [23] already establish incompatibility of matching both FPR and FNR (equalized odds) and calibration in exact as well as approximate sense (except in trivial classifiers). This result has been extended to satisfying non-discrimination with respect to two generalized cost functions expressed as a linear combinations of FPR and TPR of respective groups. This generalization while new is not entirely surprising . 
2. The authors show that if the equalized odds is relaxed to non-discrimination with respect to a single cost function expressed as a linear combination of FPR and FNR, then the relaxation and calibration can be jointly achieved under certain feasibility conditions.  
3. Additional experiments are provided to compare the effects of enforcing non-discrimination and/or calibration. 
While there are some new results and additional experiments, the conclusions are fairly similar to [23], essentially establishing incompatibility of satisfying equalized-odds style non-discrimination and classification calibration. 
Additional comments:
1. Why was EO trained not implemented and compared for the heart disease and income datasets?
2. Line 165-166: there seems to be a missing explanation or incorrect assumption here as if only one of a_t or b_t is required to be nonzero then g_t can be zero with just FPR=0 OR FNR=0, respectively. 
[Post rebuttal] Thanks for the clarifications on the contributions and the condition on a_t,b_t. 
I see the significance of the quantification of degradation performance of classifiers even when training is mindful of fairness and calibration conditions -- a similar quantification of accuracy of "best" fair predictor for exclusively EO condition (without calibration) would form a good complementary result (I dont expect the authors to attempt it for this paper, but may be as future work). That said, given the existing literature, I am still not very surprised by the other conclusions on incompatibility of two fairness conditions with calibration.