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Such rules take the form of a two-stage, multiple-round voting mechanism where the output of the first stage is the default alternative for the second stage and the vote-share thresholds used in every round of binary voting (a)<\/jats:italic>\u00a0vary with the alternative on the table for a public-good level and (b)<\/jats:italic>\u00a0require a qualified majority for approving the alternative on the table by stopping the procedure. We show that these mechanisms implement the ex post utilitarian optimal public-good level, provided valuations can only be high or low. This public-good level is chosen after all potential socially optimal alternatives have been picked for a voting round. We explore ways to reduce the number of voting rounds and develop a compound mechanism when there are three or more valuation types.<\/jats:p>","DOI":"10.1007\/s00355-024-01508-3","type":"journal-article","created":{"date-parts":[[2024,4,2]],"date-time":"2024-04-02T12:01:47Z","timestamp":1712059307000},"page":"677-715","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Semi-flexible majority rules for public good provision"],"prefix":"10.1007","volume":"63","author":[{"given":"Hans","family":"Gersbach","sequence":"first","affiliation":[]},{"ORCID":"http:\/\/orcid.org\/0000-0002-4819-0259","authenticated-orcid":false,"given":"Oriol","family":"Tejada","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,4,2]]},"reference":[{"issue":"4","key":"1508_CR1","doi-asserted-by":"publisher","first-page":"328","DOI":"10.1086\/256963","volume":"58","author":"KJ Arrow","year":"1950","unstructured":"Arrow KJ (1950) A difficulty in the concept of social welfare. 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