Differential privacy mechanisms can offer a trade-off between privacy and utility by using privacy metrics and utility metrics. The trade-off of differential privacy shows that one thing increases and another decreases in terms of privacy metrics and utility metrics. However, there is no unified trade-off measurement of differential privacy mechanisms. To this end, we proposed the definition of privacy-preserving monotonicity of differential privacy, which measured the trade-off between privacy and utility. First, to formulate the trade-off, we presented the definition of privacy-preserving monotonicity based on computational indistinguishability. Second, building on privacy metrics of the expected estimation error and entropy, we theoretically and numerically showed privacy-preserving monotonicity of Laplace mechanism, Gaussian mechanism, exponential mechanism, and randomized response mechanism. In addition, we also theoretically and numerically analyzed the utility monotonicity of these several differential privacy mechanisms based on utility metrics of modulus of characteristic function and variant of normalized entropy. Third, according to the privacy-preserving monotonicity of differential privacy, we presented a method to seek trade-off under a semi-honest model and analyzed a unilateral trade-off under a rational model. Therefore, privacy-preserving monotonicity can be used as a criterion to evaluate the trade-off between privacy and utility in differential privacy mechanisms under the semi-honest model. However, privacy-preserving monotonicity results in a unilateral trade-off of the rational model, which can lead to severe consequences.
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