  In a many-to-one matching model, with or without contracts, where doctors'
preferences are private information and hospitals' preferences are
substitutable and public information, any stable matching rule could be
manipulated for doctors. Since manipulations can not be completely avoided, we
consider the concept of \textit{obvious manipulations} and look for stable
matching rules that prevent at least such manipulations (for doctors). For the
model with contracts, we prove that: \textit{(i)} the doctor-optimal matching
rule is non-obviously manipulable and \textit{(ii)} the hospital-optimal
matching rule is obviously manipulable, even in the one-to-one model. In
contrast to \textit{(ii)}, for a many-to-one model without contracts, we prove
that the hospital-optimal matching rule is not obviously manipulable.% when
hospitals' preferences are substitutable. Furthermore, if we focus on quantile
stable rules, then we prove that the doctor-optimal matching rule is the only
non-obviously manipulable quantile stable rule.
