{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,21]],"date-time":"2025-12-21T10:04:48Z","timestamp":1766311488214,"version":"3.37.3"},"reference-count":24,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,11,25]],"date-time":"2023-11-25T00:00:00Z","timestamp":1700870400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,11,25]],"date-time":"2023-11-25T00:00:00Z","timestamp":1700870400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001778","name":"Deakin University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100001778","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soc Choice Welf"],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we propose a simple axiom which, along with the axioms of additivity (transfer) and dummy player, characterizes the Shapley value (the Shapley\u2013Shubik power index) on the domain of TU (simple) games. The new axiom, cross invariance, demands payoff invariance on symmetric players across \u201cquasi-symmetric games,\u201d that is, games where excluding null players, all players are symmetric. Additionally, we demonstrate that the axiom of additivity can be replaced by a new axiom called strong monotonicity, or it can be completely dropped if a stronger version of cross invariance is employed. We also show that the weighted Shapley values can be characterized using a weighted variant of cross invariance. Efficiency is derived rather than assumed in our characterizations. This fresh perspective contributes to a deeper understanding of the Shapley value and its applicability.<\/jats:p>","DOI":"10.1007\/s00355-023-01490-2","type":"journal-article","created":{"date-parts":[[2023,11,25]],"date-time":"2023-11-25T08:02:05Z","timestamp":1700899325000},"page":"397-418","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Cross invariance, the Shapley value, and the Shapley\u2013Shubik power index"],"prefix":"10.1007","volume":"62","author":[{"given":"Chun-Ting","family":"Chen","sequence":"first","affiliation":[]},{"given":"Wei-Torng","family":"Juang","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7363-3533","authenticated-orcid":false,"given":"Ching-Jen","family":"Sun","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,11,25]]},"reference":[{"key":"1490_CR1","first-page":"317","volume":"19","author":"JF Banzhaf III","year":"1965","unstructured":"Banzhaf JF III (1965) Weighted voting doesn\u2019t work: a mathematical analysis. Rutgers Law Rev 19:317\u2013343","journal-title":"Rutgers Law Rev"},{"key":"1490_CR2","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1007\/s00182-014-0458-2","volume":"44","author":"S B\u00e9al","year":"2015","unstructured":"B\u00e9al S, R\u00e9mila E, Solal P (2015) Axioms of invariance for TU-games. Int J Game Theory 44:891\u2013902","journal-title":"Int J Game Theory"},{"key":"1490_CR3","doi-asserted-by":"publisher","first-page":"346","DOI":"10.1016\/0022-0531(90)90007-7","volume":"50","author":"DH Blair","year":"1990","unstructured":"Blair DH, McLean RP (1990) Subjective evaluations of n-person games. J Econ Theory 50:346\u2013361","journal-title":"J Econ Theory"},{"key":"1490_CR4","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1007\/s11238-009-9171-1","volume":"71","author":"A Casajus","year":"2011","unstructured":"Casajus A (2011) Differential marginality, van den Brink fairness, and the Shapley value. Theory Dec 71:163\u2013174","journal-title":"Theory Dec"},{"key":"1490_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.mathsocsci.2013.12.001","volume":"68","author":"A Casajus","year":"2014","unstructured":"Casajus A (2014) The Shapley value without efficiency and additivity. Math Soc Sci 68:1\u20134","journal-title":"Math Soc Sci"},{"key":"1490_CR6","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1016\/0899-8256(89)90014-6","volume":"1","author":"Y Chun","year":"1989","unstructured":"Chun Y (1989) A new axiomatization of the Shapley value. Games Econ Behav 1:119\u2013130","journal-title":"Games Econ Behav"},{"key":"1490_CR7","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1007\/BF01240278","volume":"20","author":"Y Chun","year":"1991","unstructured":"Chun Y (1991) On the symmetric and weighted Shapley values. Int J Game Theory 20:183\u2013190","journal-title":"Int J Game Theory"},{"key":"1490_CR8","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1007\/BF01780630","volume":"4","author":"P Dubey","year":"1975","unstructured":"Dubey P (1975) On the uniqueness of the Shapley value. Int J Game Theory 4:131\u2013139","journal-title":"Int J Game Theory"},{"issue":"1","key":"1490_CR9","doi-asserted-by":"publisher","first-page":"122","DOI":"10.1287\/moor.6.1.122","volume":"6","author":"P Dubey","year":"1981","unstructured":"Dubey P, Neyman A, Weber RJ (1981) Value theory without efficiency. Math Oper Res 6(1):122\u2013128","journal-title":"Math Oper Res"},{"key":"1490_CR10","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1287\/moor.12.2.185","volume":"12","author":"E Einy","year":"1987","unstructured":"Einy E (1987) Semivalues of simple games. Math Oper Res 12:185\u2013192","journal-title":"Math Oper Res"},{"issue":"2","key":"1490_CR11","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1016\/j.geb.2011.03.007","volume":"73","author":"E Einy","year":"2011","unstructured":"Einy E, Haimanko O (2011) Characterization of the Shapley\u2013Shubik power index without the efficiency axiom. Games Econ Behav 73(2):615\u2013621","journal-title":"Games Econ Behav"},{"key":"1490_CR12","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1007\/s001820100080","volume":"30","author":"G Hamiache","year":"2001","unstructured":"Hamiache G (2001) Associated consistency and the Shapley value. Int J Game Theory 30:279\u2013289","journal-title":"Int J Game Theory"},{"key":"1490_CR13","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1007\/BF01756292","volume":"16","author":"E Kalai","year":"1987","unstructured":"Kalai E, Samet D (1987) On weighted Shapley values. Int J Game Theory 16:205\u2013222","journal-title":"Int J Game Theory"},{"issue":"1","key":"1490_CR14","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1287\/moor.26.1.89.10589","volume":"26","author":"A Laruelle","year":"2001","unstructured":"Laruelle A, Valenciano F (2001) Shapley\u2013Shubik and Banzhaf indices revisited. Math Oper Res 26(1):89\u2013104","journal-title":"Math Oper Res"},{"key":"1490_CR15","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511794216","volume-title":"Game theory","author":"M Maschler","year":"2013","unstructured":"Maschler M, Solan E, Zamir S (2013) Game theory. Cambridge University Press, Cambridge"},{"issue":"3","key":"1490_CR16","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1007\/BF01781371","volume":"9","author":"RB Myerson","year":"1980","unstructured":"Myerson RB (1980) Conference structures and fair allocation rules. Int J Game Theory 9(3):169\u2013182","journal-title":"Int J Game Theory"},{"key":"1490_CR17","doi-asserted-by":"publisher","first-page":"657","DOI":"10.2307\/1911680","volume":"45","author":"AE Roth","year":"1977","unstructured":"Roth AE (1977) The Shapley value as a von Neumann\u2013Morgenstern utility. Econometrica 45:657\u2013664","journal-title":"Econometrica"},{"issue":"2","key":"1490_CR18","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1016\/S0899-8256(05)80008-9","volume":"8","author":"AS Nowak","year":"1995","unstructured":"Nowak AS, Radzik T (1995) On axiomatizations of the weighted Shapley values. Games Econ Behav 8(2):389\u2013405","journal-title":"Games Econ Behav"},{"key":"1490_CR19","unstructured":"Shapley LS (1951) The value of an n-person game. The Rand Corporation Working Paper RM 670"},{"key":"1490_CR20","first-page":"307","volume-title":"Contributions to the theory of games II","author":"LS Shapley","year":"1953","unstructured":"Shapley LS (1953a) A value for n-person games. In: Tucker AW, Luce RD (eds) Contributions to the theory of games II. Princeton University Press, Princeton, NJ, pp 307\u2013317"},{"key":"1490_CR21","unstructured":"Shapley LS (1953b) Additive and non-additive set functions. Ph.D. Thesis, Department of Mathematics, Princeton University"},{"key":"1490_CR22","doi-asserted-by":"publisher","first-page":"787","DOI":"10.2307\/1951053","volume":"48","author":"LS Shapley","year":"1954","unstructured":"Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Polit Sci Rev 48:787\u2013792","journal-title":"Am Polit Sci Rev"},{"key":"1490_CR23","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/s001820100079","volume":"30","author":"R van den Brink","year":"2001","unstructured":"van den Brink R (2001) An axiomatization of the Shapley value using a fairness property. Int J Game Theory 30:309\u2013319","journal-title":"Int J Game Theory"},{"key":"1490_CR24","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/BF01769885","volume":"14","author":"HP Young","year":"1985","unstructured":"Young HP (1985) Monotone solutions of cooperative games. Int J Game Theory 14:65\u201372","journal-title":"Int J Game Theory"}],"container-title":["Social Choice and Welfare"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00355-023-01490-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00355-023-01490-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00355-023-01490-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,3,5]],"date-time":"2024-03-05T17:03:27Z","timestamp":1709658207000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00355-023-01490-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,25]]},"references-count":24,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,3]]}},"alternative-id":["1490"],"URL":"https:\/\/doi.org\/10.1007\/s00355-023-01490-2","relation":{},"ISSN":["0176-1714","1432-217X"],"issn-type":[{"type":"print","value":"0176-1714"},{"type":"electronic","value":"1432-217X"}],"subject":[],"published":{"date-parts":[[2023,11,25]]},"assertion":[{"value":"16 June 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 September 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 November 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no relevant or material financial interests that relate to the research described in this paper, \u201cCross Invariance, the Shapley Value, and the Shapley\u2013Shubik Power Index\u201d.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}