Endogenous foundations of asymmetric solutions for cooperative games

具有可转移效用的有限合作博弈由有限参与者集和一个联盟函数组成，该联盟函数为任何联盟（参与者集的子集）分配一个价值，这反映了其合作情况下的生产潜力。一个（一分）解决方案分配给任何TU游戏和任何玩家一个收益。（对称的）Shapley值可能是TU游戏最杰出的解决方案。为了解释游戏之外玩家之间的不对称性，Shapley（1953，博士论文）已经提出了他的对称解决方案的加权版本，其中这些不对称性是通过玩家的正权重来建模的——正加权Shapley值。随后，Kalai和Samet（1987）通过将玩家的零权重纳入（一般）加权Shapley值的权重系统，扩展了正加权Shapley值的类别。与加权解的外生表征（表征给定权重的单个解）相反，内生表征为整个解类提供了基础。Kalai和Samet （1987， IJGT）、Hart和Mas-Colell （1989, Econometrica）、Chun （1991， IJGT）、Nowak和Radzik （1995， GEB）、Casajus （2018， JET; 2019， ECOLET, 2021, jmaceco）和Besner （2020， SCWE）提出了正加权Shapley值的内生特征。Casajus （2018， JET; 2019， ECOLET）将Shapley值与（正）加权Shapley值之间的差异精确定位到一个公理，其中公理变得更弱。Shapley值是正加权Shapley值类的众多解之一，该正加权Shapley值本身就是加权Shapley值类的固有子类。人物特征应该反映这一事实。本项目将有助于这一观点。出发点是以下两个问题：(i)将Casajus （2018， JET; 2021， JMATHECO）对正加权Shapley值的表征修改为加权Shapley值的表征，使两类之间的差异仅用一个公理来反映，其中大类的公理弱于小类的公理。（ii） Casajus （2019， ECOLET）对（正）加权Shapley值的描述应在Young （1985， IJGT）对Shapley值描述的精神基础上进一步发展。

A finite cooperative game with transferable utility consists of a finite player set and a coalition function that assigns to any coalition (subset of the player set) a worth, which reflects its productive potential in case of cooperation. A (one-point) solution assigns to any TU game and any player a payoff. The (symmetric) Shapley value (Shapley, 1953) probably is the most eminent solution for TU games. In order to account for asymmetries among the players beyond the game, Shapley (1953, PhD thesis) already suggests weighted versions of his symmetric solution, where theses asymmetries are modelled by positive weights of the players---the positively weighted Shapley values. Later on, Kalai and Samet (1987) extend the class of positively weighted Shapley values by weight systems that allow for zero weights of the players into the class of (general) weighted Shapley values. In contrast to exogenous characterizations of weighted solutions, which characterize a single solution for given weights, endogenous characterizations provide foundations for whole classes of solutions. Endogenous characterizations of the positively weighted Shapley values have been suggested by Kalai and Samet (1987, IJGT), Hart and Mas-Colell (1989, Econometrica), Chun (1991, IJGT), Nowak and Radzik (1995, GEB), Casajus (2018, JET; 2019, ECOLET, 2021, JMATHECO), and Besner (2020, SCWE). Casajus (2018, JET; 2019, ECOLET) pinpoints the difference between the Shapley value and the (positively) weighted Shapley values to one axiom, where the axioms become weaker. The Shapley value is one of many solutions within the class of positively weighted Shapley values that itself is a proper subclass of the class of weighted Shapley values. Characterizations should reflect this fact. This project shall contribute to this view. Starting point are the following two problems: (i) The characterizations of the positively weighted Shapley values due to Casajus (2018, JET; 2021, JMATHECO) shall be modified into characterizations of the weighted Shapley values in a way such that the difference between the two classes is reflected by just one axiom, where the one for the larger class is weaker than the one for the smaller one. (ii) The characterizations of the (positively) weighted Shapley values by Casajus (2019, ECOLET) shall be further developed in the spirit of Young's (1985, IJGT) characterization of the Shapley value.