# Privacy-Preserving Monotonicity of Differential Privacy Mechanisms
## 2 Related Works
[PERSON] and [PERSON] [14] gave an equivalent definition of privacy using mutual information that made plain some of the subtleties of differential privacy. Mutual information differential privacy ensured the trade-off between privacy and informational utility to be gained from the output of any query mechanism. Renyi differential privacy could also reach the trade-off between privacy and utility [39]. For specific situations, this method outperformed existing methods, but itself as a relaxation of differential privacy needed to further investigation. Although these generalized definitions of differential privacy based on information theory can achieve the trade-off, these works cannot provide the measurement method of the trade-off, which is solved by using the definition of privacy-preserving monotonicity. The unified privacy definition provides a way of achieving the trade-off between privacy and utility in differential privacy. [PERSON] et al. [15] proposed Blowfish privacy with the goal of seeking a better trade-off between privacy and utility. The key feature of Blowfish was a policy, where users could specify sensitive information that needed to be protected and knowledge about their databases which had been released to potential adversaries. [PERSON] et al. [40] proposed Pufferfish which could be used to address privacy of correlated data. Although Pufferfish offered a good solution to privacy of correlated data, modeling privacy of users connected to social networks and privacy of spatiotemporal information gathered from sensors into rigorous privacy framework of Pufferfish and designing novel mechanism for these models were still a key challenge. On the whole, unified privacy definition does not show how to measure the trade-off of differential privacy. Our definition of privacy-preserving monotonicity can address the problem. Considering multiplayer circumstance, multiparty differential privacy protocol can get the trade-off. [PERSON] et al. [16] obtained general upper bounds on accuracy of two parties differential privacy protocol of computing any Boolean function. The major result was a new general geometric technique for obtaining non-trivial accuracy bounds for any Boolean functionality. [PERSON] et al. [41] proposed multiparty XOR computation framework of addressing the trade-off between privacy and accuracy. This optimality result of the protocol was very general, which it held for all types of functions, heterogeneous privacy conditions on the parties, all types of cost metrics, and both average and worst-case measures of accuracy, but it was no longer true to correlated data and \((\varepsilon,\delta)\)-differential privacy in multiparty context. Moreover, none of these multiparty differential privacy protocols has offered the way to measure the trade-off. We account this problem with our definition of privacy-preserving monotonicity. By using different methods, local differential privacy can obtain the trade-off. [PERSON] et al. [17] proposed local differential privacy that provided strong confidentiality guarantees in the contexts where aggregate information was released about a database containingsensitive information about individuals. This notion could also reach the trade-off between privacy and utility. [PERSON] et al. [42] studied the fundamental trade-off between local differential privacy and information-theoretic utility functions. They introduced a combinatorial family of extremal privatization mechanisms called staircase mechanisms; however, solving this linear program could be computationally expensive for multivariate random response. [PERSON] et al. [43] investigated the constrained minimization problem and showed that randomized response [9] achieved the optimal trade-off between privacy and utility under minimax distribution estimation. Although they presented the hash \(k\)-ary randomized response that empirically met or exceeded the utility of existing mechanisms at all privacy levels, these mechanisms did not consider the correlation between data.