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

Obvious Manipulations in Matching with and without Contracts