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Equivalence theorem, consistency and axiomatizations of a multi-choice value. (English) Zbl 1189.91020

Summary: This paper is devoted to the study of solutions for multi-choice games which admit a potential, such as the potential associated with the extended Shapley value proposed by C.-R. Hsiao and T. E. S. Raghavan [Int. J. Game Theory 21, No. 3, 301–312 (1992; Zbl 0777.90090); Games Econ. Behav. 5, No. 2, 240–256 (1993; Zbl 0795.90092)]. Several axiomatizations of the family of all solutions that admit a potential are offered and, as a main result, it is shown that each of these solutions can be obtained by applying the extended Shapley value to an appropriately modified game. In the framework of multi-choice games, we also provide an extension of the reduced game introduced by S. Hart and A. Mas-Colell [Econometrica 57, No. 3, 589–614 (1989; Zbl 0675.90103)]. Different from the works of Hsiao and Raghavan [loc. cit.], we provide two types of axiomatizations, one is the analogue of R.B. Myerson’s [Int. J. Game Theory 9, 169–182 (1980; Zbl 0441.90117)] axiomatization of the Shapley value based on the property of balanced contributions. The other axiomatization is obtained by means of the property of consistency.

MSC:

91A12 Cooperative games
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