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This paper studies the stability and strategy-proofness aspect of thetwo-sided one-to-one matching market. Agents have single-peaked preferences ontrees. In this setting, we characterize all rich anonymous tree-single-peakeddomains where a stable and (weakly group) strategy-proof matching rule exists.We also show that whenever there exists a stable and strategy-proof matchingrule on a rich anonymous tree-single-peaked domain, one or both of the deferredacceptance rules (Gale and Shapley, 1962) satisfy stability and weak groupstrategy-proofness on that domain. Finally, we show that for markets with asize of at least five, there is no rich anonymous domain where a stable andnon-bossy matching rule exists. As a corollary, we show incompatibility betweenstability and group strategy-proofness on rich anonymous tree-single-peakeddomains for markets with a size of at least five.

Compatibility between stability and strategy-proofness with single-peaked preferences on trees