Bayesian Mechanism Design with Efficiency, Privacy, and Approximate Truthfulness

Recently, there has been a number of papers relating mechanism design and privacy (e.g., see \cite{MT07,Xia11,CCKMV11,NST12,NOS12,HK12}). All of these papers consider a worst-case setting where there is no probabilistic information about the players' types. In this paper, we investigate mechanism design and privacy in the \emph{Bayesian} setting, where the players' types are drawn from some common distribution. We adapt the notion of \emph{differential privacy} to the Bayesian mechanism design setting, obtaining \emph{Bayesian differential privacy}. We also define a robust notion of approximate truthfulness for Bayesian mechanisms, which we call \emph{persistent approximate truthfulness}. We give several classes of mechanisms (e.g., social welfare mechanisms and histogram mechanisms) that achieve both Bayesian differential privacy and persistent approximate truthfulness. These classes of mechanisms can achieve optimal (economic) efficiency, and do not use any payments. We also demonstrate that by considering the above mechanisms in a modified mechanism design model, the above mechanisms can achieve actual truthfulness.