{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:14:12Z","timestamp":1760235252992,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,2]],"date-time":"2021-08-02T00:00:00Z","timestamp":1627862400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100008431","name":"Consejer\u00eda de Educaci\u00f3n, Junta de Castilla y Le\u00f3n","doi-asserted-by":"publisher","award":["CLU-2019-03"],"award-info":[{"award-number":["CLU-2019-03"]}],"id":[{"id":"10.13039\/501100008431","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008530","name":"European Regional Development Fund","doi-asserted-by":"publisher","award":["CLU-2019-03"],"award-info":[{"award-number":["CLU-2019-03"]}],"id":[{"id":"10.13039\/501100008530","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>There is a substantial strand of literature about ranking the subsets of a set of alternatives, usually referred to as opportunity sets. We investigate a model that is dependent on the preference of a grand set of alternatives. In this framework, the indirect-utility criterion ranks the opportunity sets by the following rule: a subset A is weakly preferred to another subset B if and only if A contains at least one preference maximizing element from A\u222aB. This criterion leads to the indifference of each subset of alternatives to a singleton; symmetry appears at this stage, as the property holds true for any one of the maximizers in A. Conversely, suppose that a ranking of opportunity sets satisfies the property that each opportunity set is indifferent to a singleton contained within it. Then, we prove that such a ranking must use a generalized form of the indirect-utility criterion, where maximization is applied to a selection of the alternatives. Altogether, these results produce a characterization of the advised indirect-utility criterion for ranking opportunity sets.<\/jats:p>","DOI":"10.3390\/sym13081404","type":"journal-article","created":{"date-parts":[[2021,8,2]],"date-time":"2021-08-02T08:44:11Z","timestamp":1627893851000},"page":"1404","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Advised Indirect-Utility Ranking of Opportunity Sets"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1430-3603","authenticated-orcid":false,"given":"Mar\u00eda Dolores","family":"Garc\u00eda-Sanz","sequence":"first","affiliation":[{"name":"BORDA Research Unit and Multidisciplinary Institute of Enterprise (IME), University of Salamanca, E37007 Salamanca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4533-9281","authenticated-orcid":false,"given":"Jos\u00e9 Carlos R.","family":"Alcantud","sequence":"additional","affiliation":[{"name":"BORDA Research Unit and Multidisciplinary Institute of Enterprise (IME), University of Salamanca, E37007 Salamanca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Anysz, H., Nical, A., Stevi\u0107, \u017d., Grzegorzewski, M., and Sikora, K. 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