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We extend the main results of Moulin (Public Choice 35:437\u2013455, 1980) to our setting and show that the results of Ching (Soc Choice Welf 26:473\u2013490, 1997) cannot always be similarly extended. First, <jats:italic>strategy-proofness<\/jats:italic> and <jats:italic>peaks-onliness<\/jats:italic> characterize the class of generalized median correspondences (Theorem\u00a01). Second, this result neither holds on the domain of symmetric and single-peaked preferences, nor can in this result <jats:italic>min\/max continuity<\/jats:italic> substitute <jats:italic>peaks-onliness<\/jats:italic> (see counter-Example\u00a03). Third, <jats:italic>strategy-proofness<\/jats:italic> and <jats:italic>voter-sovereignty<\/jats:italic> characterize the class of efficient generalized median correspondences (Theorem\u00a02).<\/jats:p>","DOI":"10.1007\/s00182-020-00728-y","type":"journal-article","created":{"date-parts":[[2020,11,17]],"date-time":"2020-11-17T10:03:00Z","timestamp":1605607380000},"page":"1059-1080","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On strategy-proofness and single-peakedness: median-voting over intervals"],"prefix":"10.1007","volume":"49","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9140-4729","authenticated-orcid":false,"given":"Bettina","family":"Klaus","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Panos","family":"Protopapas","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,11,17]]},"reference":[{"key":"728_CR1","doi-asserted-by":"publisher","first-page":"753","DOI":"10.1007\/s00355-018-1137-1","volume":"51","author":"G Achuthankutty","year":"2018","unstructured":"Achuthankutty G, Roy S (2018) On single-peaked domains and min-max rules. 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