Symmetric mechanism design: Comment

In an earlier paper published in this journal, Azrieli and Jain (2018) claim that a social choice function (SCF) is symmetrically implementable in a dominant strategy equilibrium only if the SCF is symmetric. This result crucially relies on their notion of dominant strategy, which is used in the game theory literature and is stronger than the one traditionally used in the mechanism design literature. We discuss the limitation of using their notion in mechanism design and provide a set of characterizations of symmetric mechanism design using the (weaker) standard notion that differ from their findings.