In a class of three-sided matching problems in which the core is non-empty, we show that no stable mechanism is strategy-proof for those who internalize the trilateral structure of the market in their preferences. This impossibility is related to the incompatibility between stability, one-sided strategy-proofness, and one-sided non-bossiness in two-sided markets. Furthermore, unlike what happens in marriage markets, strong restrictions on preferences are needed to ensure that stability and one-sided strategy-proofness are compatible for each market side.

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