Endogenous foundations of asymmetric solutions for cooperative games

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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.