Abstract
A transferable utility (TU) game with n players specifies a vector of \(2^n-1\) real numbers, i.e. a number for each non-empty coalition, and this can be difficult to handle for large n. Therefore, several models from the literature focus on interaction situations which are characterized by a compact representation of a TU-game, and such that the worth of each coalition can be easily computed. Sometimes, the worth of each coalition is computed from the values of single players by means of a mechanism describing how the individual abilities interact within groups of players. In this paper we introduce the class of Generalized additive games (GAGs), where the worth of a coalition \(S { \subseteq } N\) is evaluated by means of an interaction filter, that is a map \(\mathcal {M}\) which returns the valuable players involved in the cooperation among players in S. Moreover, we investigate the subclass of basic GAGs, where the filter \(\mathcal {M}\) selects, for each coalition S, those players that have friends but not enemies in S. We show that well-known classes of TU-games can be represented in terms of such basic GAGs, and we investigate the problem of computing the core and the semivalues for specific families of GAGs.
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Notes
Note that all the previous definitions hold for TU-games where v represents a gain, while the inequalities should be replaced with \(\le \) when v is a cost function.
Since the graph is undirected, we assume by convention that, if an edge between i and j is present, \(w_{i,j} \ne 0\) and \(w_{j,i}=0\) for \(i < j\).
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Acknowledgements
We thank two anonymous referees and the Associate Editor for their valuable suggestions and comments on a former version of this paper. The research of S. Moretti benefited from the support of the Projects CoCoRICo-CoDec ANR-14-CE24-0007 and NETLEARN ANR-13-INFR-004 of the French National Research Agency (ANR).
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Cesari, G., Lucchetti, R. & Moretti, S. Generalized additive games. Int J Game Theory 46, 919–939 (2017). https://doi.org/10.1007/s00182-016-0561-7
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DOI: https://doi.org/10.1007/s00182-016-0561-7