Abstract
In this paper we reformulate the necessary and sufficient conditions for the Shapley value to lie in the core of the game. Two new classes of games, which strictly include convex games, are introduced: average convex games and partially average convex games. Partially average convex games, which need not be superadditive, include average convex games. The Shapley value of a game for both classes is in the core. Some Cobb Douglas production games with increasing returns to scale turn out to be average convex games. The paper concludes with a comparison between the new classes of games introduced and some previous extensions of the convexity notion.
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References
Driessen TSH (1986) Solution Concepts of theK-Convexn-Person Games. International Journal of Game Theory 15: 201–229.
Driessen TSH (1988) Cooperative Games, Solutions and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands.
Driessen TSH, Tijs SH (1983) The τ-value, the nucleolus and the core for a subclass of games. Methods Operations Research 46: 395–406.
Driessen TSH, Tijs SH (1985) The τ-value, the Core and Semiconvex Games. International Jounral of Game Theory 14: 229–248.
Grafe F, I∼narra E, Zarzuelo JM (1993) On Externality Games. Documentos de Trabajo 93.01. Universidad del Pais Vasco.
Granot D (1986) A Generalized Linear Production Model: A Unifying Model. Math. Programming 21: 212–222.
Ichiishi T (1981) Super-modularity: Applications to Convex Games and to the Greedy Algorithm for LP. Journal of Economic Theory 25: 283–286.
Legros P (1986) Allocating Joint Costs by means of the Nucleolus. International Journal of Game Theory 15: 109–119.
Owen G (1975) On the Core of Linear Production Games. Mathematical Programming 9: 358–370.
Schmeidler D (1972) Cores of Exact Games. Journal of Mathematical Analysis and Applications 40: 214–225.
Shapley LS (1953) A Value forn-Person Games. Annals of Mathematics Study 28: 307–317.
Shapley LS (1971) Cores and Convex Games. International Journal of Game Theory 1: 1–26.
Sharkey WW (1982) Cooperative Games with Large Cores. International Journal of Game Theory 11: 175–182.
Sprumont Y (1990) Population Monotonic Allocation Schemes for Cooperative Games with Transferable Utility. Games and Economic Behavior 2: 378–394.
Weber RJ (1988) Probabilistic Values for Games. In Roth AE (ed) The Shapley Value: Essays in honor of Lloyd S. Shapley, Cambridge University Press, Cambridge, pp 101–119.
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The authors thank G. Owen, S. Tijs, and J. Ostroy and two anonymous referees of the International Journal of Game Theory for their comments and suggestions. The usual disclamer applies. We are grateful to the Universidad del Pais Vasco-EHU (grant UPV 209.321-H053/90) and the Ministry of Education and Science of Spain (CICYT grant PB900654) for providing reseach support.
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Iñarra, E., Usategui, J.M. The Shapley value and average convex games. Int J Game Theory 22, 13–29 (1993). https://doi.org/10.1007/BF01245567
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DOI: https://doi.org/10.1007/BF01245567