We present axiomatic characterizations of the proportional division value for TU-games, which distributes the worth of the grand coalition in proportion to the stand-alone worths of the players. First, a new proportionality principle, called proportional-balanced treatment, is introduced by strengthening Shapley’s 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 dummifying player property. Second, we propose appropriate monotonicity 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 proportional standardness and projection consistency. Finally, we provide a characterization of proportional standardness.
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We want to sincerely thank two anonymous referees for their useful comments and suggestions. Youngsub Chun is supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2016S1A3A2924944) and the Housing and Commercial Bank Economic Research Fund for the Institute of Economic Research of Seoul National University. Yukihiko Funaki is supported by JSPS KAKENHI Grant Numbers JP17H02503 and JP18KK0046. Zhengxing Zou is supported by the National Natural Science Foundation of China (Grant Nos. 71771025, 71801016) and the China Scholarship Council (Grant No. 201806030046).
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