Problem 3 In this problem, we explore tensions between two common fairness denitions. Suppose that the pop ulation consists of two groups, one with sensitive attribute given by a1 and the other with sensitive attribute given by a;;. Let A be a discrete random variable that captures this sensitive attribute. We consider two fairness criteria: 0 Sufciency: Y is independent of A given Y 0 Separation: la" is independent of A given Y. Assume that 13' is a binary classier with nonzero false positive rate and nonzero true negative rate. Suppose that the population has unequal base rates, i.e. ]P'[Y = 1 | A = a1] g ]P'[Y = 1 | A = a2]. Suppose also that separation holds. 1. Prove that the true positive rate is equal for the two groups and the false positive rate is also equal for the two groups. (Hint: use the denition of separation.) 2. Let TPR be the true positive rate and let FPR be the false positive rate. Prove that: TPR - pa\". my=1|Y=LA=ad=m' where Pa..- = IP[Y = 1 | A = ai] denotes the base rate for each group. Deduce that TPR = 0 if sufciency holds. 3. Using that TPR = 0, prove that lP'[Y = 0 | f\" = 0,A = a1] 7E lP'[Y = 0 | f\" = 0,A = a2]. (Hint: use a similar equation to that in part 2.) 4. Conclude that sufciency cannot hold. Discuss the implications for a decisionmaker who wishes to satisfy multiple fairness criteria