Endogenous foundations of asymmetric solutions for cooperative games

效用可转移的有限合作博弈由一个有限博弈者集合和一个联盟函数组成，该联盟函数为任何联盟(博弈者集合的子集)赋值，反映了合作情况下的生产潜力。一个(1分)解决方案给任何TU比赛和任何玩家分配一个回报。(对称的)Shapley值(Shapley，1953)可能是TU游戏最好的解决方案。为了解释游戏之外的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的加权版本，其中这些不对称性由玩家的正权重-正权重Shapley值来建模。后来，Kalai和Samet(1987)通过允许玩家零权重的加权系统将正加权Shapley值类扩展到(一般)加权Shapley值类。与加权解的外源刻画不同的是，对于给定的权重，外生刻画只刻画单个解，而内生刻画为整类解提供基础。Kalai和Samet(1987，IJGT)、Hart和Mas-Colell(1989，Economrica)、Chun(1991，IJGT)、Nowak和Radzik(1995，GEB)、Casajus(2018，JET；2019，ECOLet，2021，JMATHECO)和Besner(2020，SCWE)提出了正加权Shapley值的内生特征。Casajus(2018，JET；2019，ECOLET)将Shapley值和(正)加权Shapley值之间的差异归结为一个公理，其中公理变得更弱。Shapley值是正加权Shapley值类中的许多解之一，而Shapley值本身是加权Shapley值类的真子类。描述应该反映这一事实。该项目将有助于这一观点。(I)Casajus(2018，JET；2021年，JMATHECO)的正加权Shapley值的刻画应修改为加权Shapley值的刻画，使得两类之间的差异仅由一个公理反映，其中较大类的公理弱于较小类的公理。(2)Casajus(2019，ECOLET)对(正)加权Shapley值的刻画应本着Young(1985，IJGT)刻画Shapley值的精神进一步发展。

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.