The article critiques three formal definitions of fairness in machine learning: anti-classification, classification parity, and calibration. It argues that these definitions have significant statistical limitations and may not effectively ensure equitable outcomes. Anti-classification, which prohibits the use of protected attributes in decision-making, can inadvertently harm the groups it aims to protect. Classification parity, which requires equal error rates across groups, may not reflect true fairness because differences in risk distributions can lead to disparities in error rates. Calibration, which requires outcomes to be independent of protected attributes given risk estimates, provides only a weak guarantee of fairness. The authors propose an alternative approach: treating similarly risky individuals similarly based on the most accurate risk estimates. This strategy aligns with policy objectives and often violates anti-classification and classification parity. However, it requires careful construction of risk estimates to avoid biases in the data. The article highlights the challenges of addressing biases in training data, including measurement error and sample bias. It also discusses the limitations of formal fairness definitions in the context of legal and economic principles, noting that they often fail to capture the complexities of real-world decision-making. The article emphasizes the importance of considering the costs and benefits of decisions when designing fair algorithms. It argues that threshold rules, which apply a fixed decision threshold to risk scores, can achieve equitable outcomes by maximizing social welfare. However, these rules may not satisfy classification parity or anti-classification. The authors also discuss the incompatibility of some fairness definitions, such as calibration and classification parity, and the trade-offs involved in achieving fairness. Overall, the article calls for a more nuanced understanding of fairness in machine learning that accounts for the complexities of real-world decision-making.The article critiques three formal definitions of fairness in machine learning: anti-classification, classification parity, and calibration. It argues that these definitions have significant statistical limitations and may not effectively ensure equitable outcomes. Anti-classification, which prohibits the use of protected attributes in decision-making, can inadvertently harm the groups it aims to protect. Classification parity, which requires equal error rates across groups, may not reflect true fairness because differences in risk distributions can lead to disparities in error rates. Calibration, which requires outcomes to be independent of protected attributes given risk estimates, provides only a weak guarantee of fairness. The authors propose an alternative approach: treating similarly risky individuals similarly based on the most accurate risk estimates. This strategy aligns with policy objectives and often violates anti-classification and classification parity. However, it requires careful construction of risk estimates to avoid biases in the data. The article highlights the challenges of addressing biases in training data, including measurement error and sample bias. It also discusses the limitations of formal fairness definitions in the context of legal and economic principles, noting that they often fail to capture the complexities of real-world decision-making. The article emphasizes the importance of considering the costs and benefits of decisions when designing fair algorithms. It argues that threshold rules, which apply a fixed decision threshold to risk scores, can achieve equitable outcomes by maximizing social welfare. However, these rules may not satisfy classification parity or anti-classification. The authors also discuss the incompatibility of some fairness definitions, such as calibration and classification parity, and the trade-offs involved in achieving fairness. Overall, the article calls for a more nuanced understanding of fairness in machine learning that accounts for the complexities of real-world decision-making.