The paper explores the inherent trade-offs in achieving fairness in risk score determination. It formalizes three fairness conditions and proves that no method can satisfy them simultaneously except in highly constrained cases. The results show that even approximate satisfaction of these conditions requires data to lie in specific constrained scenarios. The three fairness conditions are: calibration within groups, balance for the positive class, and balance for the negative class. These conditions are distinct from statistical parity and reflect natural notions of fairness. The paper also presents a characterization theorem showing that the only cases where all three conditions can be satisfied are when there is perfect prediction or equal base rates. An approximate version of this theorem is also established, showing that approximate satisfaction of the conditions implies approximate perfect prediction or equal base rates. The paper discusses the implications of these results for fairness in algorithmic decision-making, highlighting the incompatibility of key fairness notions and the need for careful trade-offs in their application.The paper explores the inherent trade-offs in achieving fairness in risk score determination. It formalizes three fairness conditions and proves that no method can satisfy them simultaneously except in highly constrained cases. The results show that even approximate satisfaction of these conditions requires data to lie in specific constrained scenarios. The three fairness conditions are: calibration within groups, balance for the positive class, and balance for the negative class. These conditions are distinct from statistical parity and reflect natural notions of fairness. The paper also presents a characterization theorem showing that the only cases where all three conditions can be satisfied are when there is perfect prediction or equal base rates. An approximate version of this theorem is also established, showing that approximate satisfaction of the conditions implies approximate perfect prediction or equal base rates. The paper discusses the implications of these results for fairness in algorithmic decision-making, highlighting the incompatibility of key fairness notions and the need for careful trade-offs in their application.