On the implementation of the median

In the single-peaked domain, the median rule is strategy-proof but not implementable in (Bayes–)Nash equilibrium by its associated direct mechanism. We define the value-based median mechanism that implements the median rule in (Bayes–)Nash equilibrium in the single-peaked domain under complete and incomplete information. Such a mechanism selects the median of the profile of different values announced by the agents (i.e., ignoring redundant announcements). The value-based median does not depend on agents’ beliefs (in line with robust mechanism design). In the case of incomplete information, it induces truthful revelation of preferences (in line with strategy-proofness) for almost all peaks. We present extensions of our results to generalized median rules and finite policy spaces and their limitations.