# Refinement Orders for Quantitative Information Flow and Differential Privacy +
## 6 Application: Comparing DP Mechanisms
### Refinement Order between Families of Mechanisms  
Now, we explore whether it is possible to compare mechanisms from different families. We first ask: can we compare mechanisms which have the same \(\varepsilon\)? We assume that the input and output domains are the same, and the intention is to decide whether to replace one mechanism with another. **Theorem 14**. : _Let \(R\) be a randomized response mechanism, \(E\) an exponential mechanism and TG a truncated geometric mechanism. Then, \(TG^{\varepsilon}\sqsubseteq^{\mathrm{prv}}R^{\varepsilon}\) and \(TG^{\varepsilon}\sqsubseteq^{\mathrm{prv}}E^{\varepsilon}\). However, \(\sqsubseteq^{\mathrm{prv}}\) does not hold between \(E^{\varepsilon}\) and \(R^{\varepsilon}\)._  
Proof. : We present a counter-example to show \(E^{\varepsilon}\sqsubseteq^{\mathrm{prv}}R^{\varepsilon}\) and \(R^{\varepsilon}\sqsubseteq^{\mathrm{prv}}E^{\varepsilon}\). The remainder of the proof is in Appendix D.  
Consider the following channels:  
\[ \begin{array}{|c|cccc|}\hline A&x_{1}&x_{2}&x_{3}&x_{4}\\ \hline x_{1}&8/15&4/15&2/15&1/15\\ x_{2}&2/9&4/9&2/9&1/9\\ x_{3}&1/9&2/9&4/9&2/9\\ x_{4}&1/15&2/15&4/15&8/15\\ \hline\end{array}  \tag{34}\]  
Channel \(A\) represents an exponential mechanism and channel \(B\) a randomized response mechanism. Both have (true) \(\varepsilon\) of \(\log(\
icefrac{{12}}{{5}})\). (Channel A was generated using \(\varepsilon=\log(4)\). However, as noted earlier, this corresponds to a lower _true_\(\varepsilon\).) However, \(d_{A}(x_{1},x_{3})>d_{B}(x_{1},x_{3})\) and \(d_{A}(x_{2},x_{3})<d_{B}(x_{2},x_{3})\). Thus, \(A\) does not satisfy \(d_{B}\)-privacy, nor does \(B\) satisfy \(d_{A}\)-privacy. Intuitively, the randomized response mechanism maintains the same (\(\varepsilon\)) distinguishability level between inputs, whereas the exponential mechanism causes some inputs to be _less_ distinguishablethan others. This means that, for the same (true) \(\varepsilon\), an adversary who is interested in certain inputs could learn more from the randomized response than the exponential. In the above counter-example, points \(x_{2}\), \(x_{3}\) in the exponential mechanism of channel \(A\) are _less_ distinguishable than the corresponding points in the randomized response mechanism \(B\). As an example, let's say the mechanisms are to be used in geo-location privacy and the inputs represent adjacent locations (such as addresses along a street). Then, an adversary (your boss) may be interested in how far you are from work, and therefore wants to be able to distinguish between points distant from \(x_{1}\) (your office) and points within the vicinity of your office, without requiring your precise location. Your boss chooses channel \(A\) as the most informative. However, another adversary (your suspicious partner) is more concerned about where exactly you are, and is particularly interested in distinguishing between your expected position (\(x_{2}\), the boulangerie) versus your suspected position (\(x_{3}\), the brothel). Your partner chooses channel \(B\) as the most informative. Regarding the other refinements, we find (experimentally) that in general they do not hold between families of mechanisms. (Recall that we only need to produce a single counter-example to show that a refinement doesn't hold, and this can be done using the methods presented in Section 5.) We next check what happens when we compare mechanisms with _different_ epsilons. We note the following. **Theorem 15**. : _For any (truncated geometric, randomized response, exponential) mechanisms \(M_{1}^{\varepsilon_{1}},M_{2}^{\varepsilon_{2}}\), if \(M_{1}^{\varepsilon_{1}}\sqsubseteq M_{2}^{\varepsilon_{2}}\) for any of our refinements (\(\sqsubseteq^{\mathrm{avg}},\sqsubseteq^{\mathrm{max}},\sqsubseteq^{\mathrm{prv}}\)), then \(M_{1}^{\varepsilon_{1}}\sqsubseteq M_{2}^{\varepsilon^{\prime}_{2}}\) for \(\varepsilon^{\prime}_{2}<\varepsilon_{2}\)._  
Proof. : This follows directly from transitivity of the refinement relations, and our results on refinement with families of mechanisms.