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This paper explores many-to-one matching models, both with and withoutcontracts, where doctors' preferences are private and hospitals' preferencesare public and substitutable. It is known that any stable-dominating mechanism--which is either stable or individually rational and Pareto-dominates (fromthe doctors' perspective) a stable mechanism--, is susceptible to manipulationby doctors. Our study focuses on \textit{obvious manipulations} and identifiesstable-dominating mechanisms that prevent them. Without contracts, we show thatmore efficient mechanisms are less likely to be obviously manipulable and thatany stable-dominating mechanism is not obviously manipulable. However, withcontracts, none of these results hold. While we demonstrate that theDoctor-Proposing Deferred Acceptance (DA) Mechanism remains not obviouslymanipulable, we show that the Hospital-Proposing DA Mechanism and any efficientmechanism that Pareto-dominates the Doctor-Proposing DA Mechanism become (very)obviously manipulable, in the model with contracts.

Obvious Manipulations in Matching without and with Contracts