Abstract
In this work we explore games with externalities, where our basic approach is rooted in the concept of marginal contributions of players to coalitions. We considered the general case where a player (in a coalition S) may join another coalition after leaving S. We then show that the standard translation of Shapley’s four axioms to games with externalities is not sufficient to obtain a unique value. Finally, we provide an axiomatic characterization for the family of solutions for games with externalities satisfying those axioms that traditionally are used to characterize the Shapley value in the absence of externalities. In particular, we show that every such solution is a linear combination of marginal contributions of players and provide an interpretation as a bargaining process.
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Notes
- 1.
Part of this chapter is based on the paper “A note on a class of solutions for games with externalities generalizing the Shapley value” (2015).
- 2.
The precise definitions will be provided in Sect. 15.3.
- 3.
- 4.
A game is with no externalities if and only if the payoff that the players in a coalition S can jointly obtain if this coalition is formed is independent of the way the other players are organized. This means that in a game with no externalities, the characteristic function satisfies w(S, Q) = w(S, Q ′) for any two partitions Q, Q ′ ∈ PT and any coalition S which belongs both to Q and Q ′ Hence, the worth of a coalition S can be written without reference to the organization of the remaining players, w(S): = w(S, Q) for all Q ∋ S, Q ∈ PT.
- 5.
More precisely, the Shapley value for games with no externalities happens to be calculated as the weighted average of marginal contributions of players to coalitions.
- 6.
However, their strong version of symmetry implies our symmetry axiom. It strengthens the symmetry axiom by requiring that the payoff of a player should not change after permutations in the set of players in N∖S, for any embedded coalition structure (S, Q).
- 7.
Originally provided by Young (1985) for games with no externalities. He formulated the marginality principle as an axiom, that is, that the solution should pay the same to a player in two games if his or her marginal contributions to coalitions are the same in both games. Marginality is an idea with a strong tradition in economic theory.
References
Albizuri, M.J., Arin, J., Rubio, J.: An axiom system for a value for games in partition function form. Int. Game Theory Rev. 7 (1), 63–72 (2005)
Bolger, E.M.: A set of axioms for a value for partition function games. Int. J. Game Theory 18 (1), 37–44 (1989)
De Clippel, G., Serrano, R.: Marginal contributions and externalities in the value. Econometrica 6, 1413–1436 (2008)
Fujinaka, Y.: On the marginality principle in partition function form games, Unpublished Manuscript, Graduate School of Economics, Kobe University (2004)
Grabisch, M., Funaki, Y.: A coalition formation value for games in partition function form. Eur. J. Oper. Res. 221 (1), 175–185 (2012)
Hernández-Lamoneda, L., Sánchez-Pérez, J., Sánchez-Sánchez, F.: The class of efficient linear symmetric values for games in partition function form. Int. Game Theory Rev. 11 (3), 369–382 (2009)
Hu, C.C., Yang, Y.Y.: An axiomatic characterization of a value for games in partition function form. SERIEs 1 (4), 475–487 (2010)
Lucas, W.F., Thrall, R.M.: n-Person games in partition function form. Nav. Res. Logist. Q. 10, 281–298 (1963)
Macho-Stadler, I., Pérez-Castrillo, D., Wettstein, D.: Sharing the surplus: an extension of the Shapley value for environments with externalities. J. Econ. Theory 135, 339–356 (2007)
Myerson, R.B.: Values of games in partition function form. Int. J. Game Theory 6 (1), 23–31 (1977)
PhamDo, K., Norde, H.: The Shapley value for partition function form games. Int. Game Theory Rev. 9 (2), 353–360 (2007)
Sánchez-Pérez, J.: A note on a class of solutions for games with externalities generalizing the Shapley value. Int. Game Theory Rev. 17 (3), 1–12 (2015)
Shapley, L.: A value for n-person games. Contrib. Theory Games 2, 307–317 (1953)
Young, H.P.: Monotonic solutions of cooperative games. Int. J. Game Theory 14, 65–72 (1985)
Acknowledgements
I thank the participants of the XV Latin-American Workshop on Economic Theory (JOLATE) for comments, interesting discussions, and encouragement. J. Sánchez-Pérez acknowledges financial support from CONACYT research grant 130515.
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Sánchez-Pérez, J. (2016). Marginal Contributions in Games with Externalities. In: Pinto, A., Accinelli Gamba, E., Yannacopoulos, A., Hervés-Beloso, C. (eds) Trends in Mathematical Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-32543-9_15
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