{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:42:26Z","timestamp":1760233346375,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,3]],"date-time":"2021-01-03T00:00:00Z","timestamp":1609632000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In many real-world situations, performers always adopt different energy levels (strategies) to participate. Different from pre-existing results, this paper is devoted to proposing several generalized power outputs of the marginal index, the Banzhaf\u2013Coleman index, and the Banzhaf\u2013Owen index, respectively, by assigning different energy levels to all performers. Since these extended power outputs may not be efficacious, we further define the efficacious extensions of these power outputs, respectively. For each of these efficacious power outputs, we demonstrate that there exists a corresponding reduced game and related consonance property that can be used to characterize it. By focusing on the properties of symmetry and accordance, several axiomatic results are also introduced.<\/jats:p>","DOI":"10.3390\/sym13010072","type":"journal-article","created":{"date-parts":[[2021,1,3]],"date-time":"2021-01-03T19:54:46Z","timestamp":1609703686000},"page":"72","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Consonance, Symmetry and Extended Outputs"],"prefix":"10.3390","volume":"13","author":[{"given":"Yu-Hsien","family":"Liao","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, National Pingtung University, Pingtung County 900391, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,3]]},"reference":[{"key":"ref_1","first-page":"317","article-title":"Weighted voting does not work: A mathematical analysis","volume":"19","author":"Banzhaf","year":"1965","journal-title":"Rutgers Law Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"741","DOI":"10.1002\/nav.3800220409","article-title":"Multilinear extensions and the Banzhaf value","volume":"22","author":"Owen","year":"1975","journal-title":"Nav. 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