Abstract
The Shapley value is a well-known solution concept for TU games. The Maschler–Owen value and the NTU Shapley value are two well-known extensions of the Shapley value to NTU games. A hyperplane game is an NTU game in which the feasible set for each coalition is a hyperplane. On the domain of monotonic hyperplane games, the Maschler–Owen value is axiomatized (Hart Essays in game theory. Springer, 1994). Although the domain of hyperplane game is a very interesting class of games to study, unfortunately, on this domain, the NTU Shapley value is not well-defined, namely, it assigns an empty set to some hyperplane games. A prize game (Hart Essays in game theory. Springer, 1994) is an NTU game that can be obtained by “truncating” a hyperplane game. As such, a prize game describes essentially the same situation as the corresponding hyperplane game. It turns out that, on the domain of monotonic prize games, the NTU Shapley value is well-defined. Thus, one can define a value which is well-defined on the domain of monotonic hyperplane games as follows: given a monotonic hyperplane game, first, transform it into a prize game, and then apply the NTU Shapley value to it. We refer to the resulting value as the “generalized Shapley value” and compare the axiomatic properties of it with those of the Maschler–Owen value on the union of the class of monotonic hyperplane games and that of monotonic prize games. We also provide axiomatizations of the Maschler–Owen value and the generalized Shapley value on that domain.
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Notes
Hart (1994) inappropriately uses the following condition to define monotonic NTU games: for each pair \(S,T \in 2^N \setminus \{ \emptyset \}\) with \(S \subseteq T\), \(V(S) \times \{ 0^{T \setminus S} \} \subseteq V(T)\). One can easily see that this definition is inappropriate since it implies that a hyperplane is monotonic only if all associated hyperplanes are parallel. In any case, it can be checked that all statements concerning monotonic NTU games in Hart (1994) remain valid even if we use our definition.
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Acknowledgements
I sincerely thank Toru Hokari for instruction and guidance. I also appreciate comments and suggestions from the editor and an anonymous referee which greatly improved the quality of this paper. All remaining errors are my own.
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Yu, C. Hyperplane games, prize games and NTU values. Theory Decis 93, 359–370 (2022). https://doi.org/10.1007/s11238-021-09846-9
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DOI: https://doi.org/10.1007/s11238-021-09846-9