Smoothed and Average-case Approximation Ratios of Mechanisms: Beyond the Worst-case Analysis

The approximation ratio has become one of the dominant measures in mechanism design problems. In light of analysis of algorithms, we define the \emph{smoothed approximation ratio} to compare the performance of the optimal mechanism and a truthful mechanism when the inputs are subject to random perturbations of the worst-case inputs, and define the \emph{average-case approximation ratio} to compare the performance of these two mechanisms when the inputs follow a distribution. For the one-sided matching problem, \citet{FFZ:14} show that, amongst all truthful mechanisms, \emph{random priority} achieves the tight approximation ratio bound of $\Theta(\sqrt{n})$. We prove that, despite of this worst-case bound, random priority has a \emph{constant smoothed approximation ratio}. This is, to our limited knowledge, the first work that asymptotically differentiates the smoothed approximation ratio from the worst-case approximation ratio for mechanism design problems. For the average-case, we show that our approximation ratio can be improved to $1+e$. These results partially explain why random priority has been successfully used in practice, although in the worst case the optimal social welfare is $\Theta(\sqrt{n})$ times of what random priority achieves. These results also pave the way for further studies of smoothed and average-case analysis for approximate mechanism design problems, beyond the worst-case analysis.