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.

我们介绍了TU游戏的比例除法值的公理化特征，该分布将大联盟的价值与玩家的独立价值成比例地分配。首先，通过加强Shapley的对称公理引入了一种新的比例原则，即比例均衡处理，该原则指出，如果两个参与者对任何非空联盟做出相同的贡献，那么他们将按其独立资产的比例获得收益。我们表征了满足效率，弱线性和比例均衡处理的一系列值。我们还显示了该家族与虚拟玩家属性不兼容。但是，我们证明比例除法值是该族中满足以下条件的唯一值：模仿玩家财产。其次，我们提出适当的单调性公理，并在不具有弱线性和虚拟化玩家属性的情况下获得比例除法值的公理化。第三，从可变玩家集合的角度来看，我们表明比例除法值是唯一满足比例标准和投影一致性的值。最后，我们提供了比例标准的特征。