# Refinement Orders for Quantitative Information Flow and Differential Privacy +
## 6 Application: Comparing DP Mechanisms
### Refinement Order between Families of Mechanisms
(We recall however that our result for the exponential mechanism is only a conjecture.) This tells us that, once we have a refinement between mechanisms, it continues to hold for reduced \(\varepsilon\) in the refining mechanism. **Corollary 1**. : _Let \(G,TG,R,E\) be the geometric, truncated geometric, randomized response and exponential mechanisms respectively. Then, for all \(\varepsilon^{\prime}\leq\varepsilon\), we have that \(TG^{\varepsilon}\sqsubseteq^{\mathrm{prv}}R^{\varepsilon^{\prime}}\), \(TG^{\varepsilon}\sqsubseteq^{\mathrm{prv}}E^{\varepsilon^{\prime}}\), \(G^{\varepsilon}\sqsubseteq^{\mathrm{prv}}R^{\varepsilon^{\prime}}\) and \(G^{\varepsilon}\sqsubseteq^{\mathrm{prv}}E^{\varepsilon^{\prime}}\)._  
Thus, it is safe to 'compare epsilons' wrt \(\sqsubseteq^{\mathrm{prv}}\) if we want to replace a geometric mechanism with either a randomized response or exponential mechanism. (As with the previous theorem, note that the results for the exponential mechanism are stated as conjecture only, and this conjecture is assumed in the statement of this corollary.) What this means is that if, for example, we have a geometric mechanism \(TG\) that operates on databases with distance measured using the Hamming metric \(d_{\mathrm{H}}\) and satisfying \(\varepsilon\cdot d_{\mathrm{H}}\)-privacy, then any randomized response mechanism \(R\) parametrized by \(\varepsilon^{\prime}\leq\varepsilon\) will also satisfy \(\varepsilon\cdot d_{\mathrm{H}}\)-privacy. Moreover, if we decide we'd rather use the Manhattan metric \(d_{\mathrm{M}}\) to measure distance between the databases, then we only need to check that \(TG\) also satisfies \(\varepsilon\cdot d_{\mathrm{M}}\)-privacy, as this implies that \(R\) will too. The following Tables 4-6 summarize the refinement relations with respect to the various families of mechanisms. **Theorem 16**. : _Every (truncated geometric, randomized response, exponential) mechanism is 'the safest possible mechanism' when parametrized by \(\varepsilon=0\). That is, \(L^{\varepsilon}\sqsubseteq^{\mathrm{avg}}M^{0}\) for all mechanisms \(L\), \(M\) (possibly from different families) and \(\varepsilon>0\)._  
While this result may be unsurprising, it means that we know that refinement must _eventually_ occur when we reduce \(\varepsilon\). It is interesting then to ask just _when_ this refinement occurs. We examine this question experimentally by considering different mechanisms and investigating for which values of \(\varepsilon\) average-case refinement holds. For simplicity of presentation, we show results for \(5\times 5\) matrices, noting that we observed similar results for experiments across different matrix dimensions, at least wrt the coarse-grained comparison of plots that we do here. The results are plotted in Figure 3.  
The plots show the relationship between \(\varepsilon_{1}\) (\(x\)-axis) and \(\varepsilon_{2}\) (\(y\)-axis) where \(\varepsilon_{1}\) parametrizes the mechanism being refined and \(\varepsilon_{2}\) parametrizes the refining mechanism. For example, the blue line on the top graph represents \(TG^{\varepsilon_{1}}\sqsubseteq^{\mathrm{avg}}E^{\varepsilon_{2}}\). We fix \(\varepsilon_{1}\) and ask for what value of \(\varepsilon_{2}\) do we get a \(\sqsubseteq^{\mathrm{avg}}\) refinement. Notice that the line \(\varepsilon_{1}=\varepsilon_{2}\) corresponds to the same mechanism in both axes (since every mechanism refines itself). Figure 3: \(Cont\). We can see that refining the randomized response mechanism requires much smaller values of epsilon in the other mechanisms. For example, from the middle graph, we can see that \(R^{4}\sqsubseteq^{\mathrm{avg}}TG^{1}\) (approximately) whereas, from the top graph, we have \(TG^{4}\sqsubseteq^{\mathrm{avg}}R^{4}\). This means that the randomized response mechanism is very'safe' against average-case adversaries compared with the other mechanisms, as it is much more 'difficult' to refine than the other mechanisms.