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A seminal characterization of strategyproof SDSs by Gibbard (Econometrica 45(3):665\u2013681, 1977) implies that there are no strategyproof Condorcet extensions and that only random dictatorships satisfy <jats:italic>ex post<\/jats:italic> efficiency and strategyproofness. The latter is known as the <jats:italic>random dictatorship theorem<\/jats:italic>. We relax Condorcet-consistency and <jats:italic>ex post<\/jats:italic> efficiency by introducing a lower bound on the probability of Condorcet winners and an upper bound on the probability of Pareto-dominated alternatives, respectively. We then show that the randomized Copeland rule is the only anonymous, neutral, and strategyproof SDS that guarantees the Condorcet winner a probability of at least 2\/<jats:italic>m<\/jats:italic>. Secondly, we prove a continuous strengthening of Gibbard\u2019s random dictatorship theorem: the less probability we put on Pareto-dominated alternatives, the closer to a random dictatorship is the resulting SDS. Finally, we show that the only anonymous, neutral, and strategyproof SDSs that maximize the probability of Condorcet winners while minimizing the probability of Pareto-dominated alternatives are mixtures of the uniform random dictatorship and the randomized Copeland rule.<\/jats:p>","DOI":"10.1007\/s00355-024-01519-0","type":"journal-article","created":{"date-parts":[[2024,4,1]],"date-time":"2024-04-01T19:02:16Z","timestamp":1711998136000},"page":"19-55","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Relaxed notions of Condorcet-consistency and efficiency for strategyproof social decision schemes"],"prefix":"10.1007","volume":"63","author":[{"given":"Felix","family":"Brandt","sequence":"first","affiliation":[]},{"given":"Patrick","family":"Lederer","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6447-7794","authenticated-orcid":false,"given":"Ren\u00e9","family":"Romen","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,4,1]]},"reference":[{"issue":"3","key":"1519_CR1","doi-asserted-by":"publisher","first-page":"689","DOI":"10.2307\/2998580","volume":"66","author":"A Abdulkadiro\u011flu","year":"1998","unstructured":"Abdulkadiro\u011flu A, S\u00f6nmez T (1998) Random serial dictatorship and the core from random endowments in house allocation problems. 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