{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T04:10:04Z","timestamp":1772770204613,"version":"3.50.1"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,4,7]],"date-time":"2024-04-07T00:00:00Z","timestamp":1712448000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,4,7]],"date-time":"2024-04-07T00:00:00Z","timestamp":1712448000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["522837108"],"award-info":[{"award-number":["522837108"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"name":"HHL Leipzig Graduate School of Management gGmbH"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Int J Game Theory"],"published-print":{"date-parts":[[2024,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>First, we suggest and discuss second-order versions of properties for solutions for TU games used to characterize the Banzhaf value, in particular, of standardness for two-player games, of the dummy player property, and of 2-efficiency. Then, we provide a number of characterizations of the Banzhaf value invoking the following properties: (i) [second-order standardness for two-player games or the second-order dummy player property] and 2-efficiency, (ii) standardness for one-player games, standardness for two-player games, and second-order 2-efficiency, (iii) standardness for one-player games, [second-order standardness for two-player games or the second-order dummy player property], and second-order 2-efficiency. These characterizations also work within the classes of simple games, of superadditive games, and of simple superadditive games.<\/jats:p>","DOI":"10.1007\/s00182-024-00893-4","type":"journal-article","created":{"date-parts":[[2024,4,7]],"date-time":"2024-04-07T16:01:15Z","timestamp":1712505675000},"page":"989-1004","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Second-order productivity, second-order payoffs, and the Banzhaf value"],"prefix":"10.1007","volume":"53","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2650-2650","authenticated-orcid":false,"given":"Andr\u00e9","family":"Casajus","sequence":"first","affiliation":[]},{"given":"Rodrigue Tido","family":"Takeng","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,4,7]]},"reference":[{"issue":"7","key":"893_CR1","doi-asserted-by":"publisher","first-page":"1098","DOI":"10.1016\/j.aml.2012.03.025","volume":"25","author":"JM Alonso-Meijide","year":"2012","unstructured":"Alonso-Meijide JM, \u00c1lvarez-Mozos M, Fiestras-Janeiro MG (2012) A comment on 2-efficiency and the Banzhaf value. 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