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Axiomatic characterizations of the core without consistency

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Abstract

A TU game is totally positive if it is a linear combination of unanimity games with nonnegative coefficients. We show that the core on each cone of convex games that contains the set of totally positive games is characterized by the traditional properties Pareto efficiency, additivity (ADD), individual rationality, and the null-player property together with one new property, called unanimity requiring that the solution, when applied to a unanimity game on an arbitrary coalition, allows to distribute the entire available amount of money to each player of this coalition. We also show that the foregoing characterization can be generalized to the domain of balanced games by replacing ADD by “ADD on the set of totally positive games plus super-additivity (SUPA) in general”. Adding converse SUPA allows to characterize the core on arbitrary domains of TU games that contain the set of all totally positive games. Converse SUPA requires a vector to be a member of the solution to a game whenever, when adding a totally positive game such that the sum becomes totally additive, the sum of the vector and each solution element of the totally positive game belongs to the solution of the aggregate game. Unlike in traditional characterizations of the core, our results do not use consistency properties.

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Notes

  1. An exception is the axiomatic characterization of the restricted core for the specific set of totally positive games (i.e., games that are nonnegative linear combinations of unanimity games) with ordered players by van den Brink et al. (2009).

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Correspondence to Sylvain Béal.

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The authors are grateful to two anonymous referees of this journal for their remarks that helped to improve the writing of this paper. We also thank conference participants at Euro OR Espoo 2022 for valuable comments. This research was supported by “Mathématiques de la décision pour l’ingénierie physique et sociale” (MODMAD).

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Béal, S., Gonzalez, S., Solal, P. et al. Axiomatic characterizations of the core without consistency. Int J Game Theory 52, 687–701 (2023). https://doi.org/10.1007/s00182-022-00835-y

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