{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:38:47Z","timestamp":1760236727921,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,12,16]],"date-time":"2021-12-16T00:00:00Z","timestamp":1639612800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Symmetry exists in a multitude of phenomena in varying forms. The main aim of this article is to analyze the plausibility of the equal allocation non-separable costs, the efficient Banzhaf\u2013Owen index and the efficient Banzhaf\u2013Coleman index from the perspective of symmetry. First, based on the difference between \u201cparticipation processes\u201d and \u201callocating results\u201d, different forms of symmetry are proposed. Next, building on these forms of symmetry, axiomatic results are put forth for the three power indexes, whereby the plausibility of the three power indexes is analyzed. Finally, on the basis of these different forms of symmetry and related axiomatic results, this article introduces different dynamic processes to analyze how an initial allocation result approaches the results derived from the three power indexes through dynamically modification.<\/jats:p>","DOI":"10.3390\/axioms10040345","type":"journal-article","created":{"date-parts":[[2021,12,16]],"date-time":"2021-12-16T11:27:36Z","timestamp":1639654056000},"page":"345","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Axiomatic and Dynamic Results for Power Indexes under Symmetry"],"prefix":"10.3390","volume":"10","author":[{"given":"Ruey-Rong","family":"Huang","sequence":"first","affiliation":[{"name":"General Education Center, Chung-Jen Junior College of Nursing, Health Sciences and Management, Chia-Yi 62241, Taiwan"}]},{"given":"Yu-Hsien","family":"Liao","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, National Pingtung University, Pingtung 90039, Taiwan"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,16]]},"reference":[{"key":"ref_1","first-page":"317","article-title":"Weighted voting doesn\u2019t 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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Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/s00182-009-0155-8","article-title":"An NTU value under complement reduced game","volume":"38","author":"Hwang","year":"2009","journal-title":"Int. J. Game Theory"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"589","DOI":"10.2307\/1911054","article-title":"Potential, value and consistency","volume":"57","author":"Hart","year":"1989","journal-title":"Econometrica"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1287\/moor.4.2.99","article-title":"Mathematical properties of the Banzhaf power index","volume":"4","author":"Dubey","year":"1979","journal-title":"Math. Oper. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/BF01254541","article-title":"An axiomatization of the Banzhaf value","volume":"17","author":"Lehrer","year":"1988","journal-title":"Int. J. Game Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1007\/BF01247318","article-title":"Collusion properties of values","volume":"23","author":"Haller","year":"1994","journal-title":"Int. J. Game Theory"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1007\/s003550050125","article-title":"Axiomatizations of the normalized Banzhaf value and the Shapley value","volume":"15","year":"1998","journal-title":"Soc. Choice Welf."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1007\/BF01358800","article-title":"The consonant Shapley value for hyperplane games","volume":"18","author":"Maschler","year":"1989","journal-title":"Int. J. Game Theory"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/BF01769258","article-title":"On the reduced game property and its converse","volume":"15","author":"Peleg","year":"1986","journal-title":"Int. J. Game Theory"},{"key":"ref_15","unstructured":"Kuhn, H.W., and Tucker, A.W. (1953). A value for P-person game. Distinctions to the Theory of Games II, Princeton Press."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/BF01781371","article-title":"Conference structures and fair allocation rules","volume":"9","author":"Myerson","year":"1980","journal-title":"Int. J. Game Theory"},{"key":"ref_17","first-page":"449","article-title":"Convergent Transfer Schemes for P-Person Games","volume":"134","author":"Stearns","year":"1968","journal-title":"Trans. Amer. Math. 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