Random dictatorship domains

A domain of preference orderings is a random dictatorship domain if every strategyproof random social choice function satisfying unanimity defined on the domain, is a random dictatorship. Gibbard (1977) showed that the universal domain is a random dictatorship domain. We investigate the relationship between dictatorial and random dictatorship domains. We show that there exist dictatorial domains that are not random dictatorship domains. We provide stronger versions of the linked domain condition (introduced in Aswal et al. (2003)) that guarantee that a domain is a random dictatorship domain. A key step in these arguments that is of independent interest, is a ramification result that shows that under certain assumptions, a domain that is a random dictatorship domain for two voters is also a random dictatorship domain for an arbitrary number of voters.