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First, a new proportionality principle, called <jats:italic>proportional-balanced treatment<\/jats:italic>, is introduced by strengthening Shapley\u2019s symmetry axiom, which states that if two players make the same contribution to any nonempty coalition, then they receive the amounts in proportion to their stand-alone worths. We characterize the family of values satisfying efficiency, weak linearity, and proportional-balanced treatment. We also show that this family is incompatible with the dummy player property. However, we show that the proportional division value is the unique value in this family that satisfies the <jats:italic>dummifying player property<\/jats:italic>. Second, we propose appropriate <jats:italic>monotonicity<\/jats:italic> axioms, and obtain axiomatizations of the proportional division value without both weak linearity and the dummifying player property. Third, from the perspective of a variable player set, we show that the proportional division value is the only one that satisfies <jats:italic>proportional standardness<\/jats:italic> and <jats:italic>projection consistency<\/jats:italic>. Finally, we provide a characterization of proportional standardness.<\/jats:p>","DOI":"10.1007\/s00355-020-01299-3","type":"journal-article","created":{"date-parts":[[2021,1,11]],"date-time":"2021-01-11T16:24:36Z","timestamp":1610382276000},"page":"35-62","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Axiomatizations of the proportional division value"],"prefix":"10.1007","volume":"57","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1208-4948","authenticated-orcid":false,"given":"Zhengxing","family":"Zou","sequence":"first","affiliation":[]},{"given":"Ren\u00e9","family":"van\u00a0den Brink","sequence":"additional","affiliation":[]},{"given":"Youngsub","family":"Chun","sequence":"additional","affiliation":[]},{"given":"Yukihiko","family":"Funaki","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,1,11]]},"reference":[{"key":"1299_CR1","first-page":"110","volume-title":"Joint cost allocations","author":"R Banker","year":"1981","unstructured":"Banker R (1981) Equity considerations in traditional full cost allocation practices: an axiomatic perspective. 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