Axiomatizations of the Proportional Division Value

Abstract

We present axiomatic characterizations of the proportional division value for TU-games, a value that distributes the worth of the grand coalition in proportion to the stand-alone worths of its members. First, a new proportionality principle, called 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 linearty, and 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 three appropriate monotonicity axioms by considering two games in which the stand-alone worths of all players are equal or in the same proportion to each other, and obtain three 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

Sharing the surplus and proportional values

“…This family of values is in line with a recent and growing literature that combine different allocation principles by considering convex combinations of two extreme values, such as the egalitarian Shapley values (being convex combinations of the Shapley value and equal division value, see Joosten (1996) and van den Brink et al (2013)), the consensus values (being convex combinations of the Shapley value and equal surplus division value, see Ju et al (2007)) and the family of convex combinations of the equal division value and the equal surplus division value (axiomatized in, e.g., van den Brink and Funaki (2009); van den Brink et al (2016); Xu et al (2015); Ferrières (2017)). Also, our family of values is in line with a recent and growing literature on non-symmetric surplus sharing values, such as the weighted division value (Béal et al, 2015(Béal et al, , 2016a, the weighted surplus division value Llerena, 2017, 2019), the weighted equal allocation of non-separable contributions value (Hou et al, 2019), and the PD value (Zou et al, 2019).…”

Sharing the Surplus and Proportional Values

“…This family of values is in line with a recent and growing literature that combine different allocation principles by considering convex combinations of two extreme values, such as the egalitarian Shapley values (being convex combinations of the Shapley value and equal division value, see Joosten (1996) and van den Brink et al (2013)), the consensus values (being convex combinations of the Shapley value and equal surplus division value, see Ju et al (2007)) and the family of convex combinations of the equal division value and the equal surplus division value (axiomatized in, e.g., van den Brink and Funaki (2009);van den Brink et al (2016); Xu et al (2015); Ferrières (2017)). Also, our family of values is in line with a recent and growing literature on non-symmetric surplus sharing values, such as the weighted division value (Béal et al, 2015(Béal et al, , 2016a, the weighted surplus division value Llerena, 2017, 2019), the weighted equal allocation of non-separable contributions value (Hou et al, 2019), and the PD value (Zou et al, 2019).…”

Compromising between the Proportional and Equal Division Values: Axiomatization, Consistency and Implementation

“…It is shown that the affine combinations of the ED and ESD values can be characterized using projection consistency by van den Brink et al (2016). Zou et al (2019) use projection consistency to characterize the PD value. Several values satisfy projection consistency, see, e.g., Funaki and Yamato (2001); Llerena (2017, 2019).…”

“…Undoubtedly, an extreme egalitarian value is the equal division value (characterized in van den Brink (2007)), which allocates the total worth equally among all players. As an alternative to this equality principle, the proportional division value (Zou et al, 2019) relies on the proportionality principle: it gives payoffs in proportion to players' stand-alone worth.…”