Proportional Shapley levels values

Besner, Manfred (2018): Proportional Shapley levels values.

Abstract

The proportional Shapley value (Besner 2016; Béal et al. 2017; Gangolly 1981) is an proportional counterpart to the Shapley value (Shapley 1953b) in cooperative games. As shown in Besner (2017a), the proportional Shapley value is a convincing non-linear alternative, especially in cost allocation, if the stand alone worths of the players are plausible weights. To enable similar properties for cooperative games with a level structure, we generalize this value. Therefore we adapt the proceeding applied to the weighted Shapley values in Besner (2017b). We present, analogous to the four classes of weighted Shapley levels values in Besner (2017b), four different values, the proportional Shapley hierarchy levels value, the proportional Shapley support levels value, the proportional Shapley alliance levels value and the proportional Shapley collaboration levels value, respectively.

• Item Type: MPRA Paper
• Language: English
• Keywords: Cooperative game · Level structure · (Proportional) Shapley (levels) value · Proportionality · Component substitution · Dividends
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C70 - General; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C71 - Cooperative Games

References

Banker, R. D. (1981). Equity Considerations in Traditional Full Cost Allocation Practices: An Axiomatic Perspective. Carnegie-Mellon University.

Béal, S., Ferrières, S., Rémila, E., & Solal, P. (2017). The proportional Shapley value and applications. Games and Economic Behavior.

Besner, M. (2016). Lösungskonzepte kooperativer Spiele mit Koalitionsstrukturen, Master’s thesis, FernUniversität Hagen (in German).

Besner, M. (2017a). Axiomatizations of the proportional Shapley value.

Besner, M. (2017b). Weighted Shapley levels values.

Calvo, E., Lasaga, J. J., & Winter, E. (1996). The principle of balanced contributions and hierarchies of cooperation, Mathematical Social Sciences, 31(3), 171–182.

Derks, J. J., & Haller, H. H. (1999). Null players out? Linear values for games with variable supports. International Game Theory Review, 1(3–4), 301–314.

Gangolly, J. S. (1981). On joint cost allocation: Independent cost proportional scheme (ICPS) and its properties. Journal of Accounting Research, 299–312.

Gómez-Rúa, M., & Vidal-Puga, J. (2010). The axiomatic approach to three values in games with coalition structure. European Journal of Operational Research, 207(2), 795–806.

Gómez-Rúa, M., & Vidal-Puga, J. (2011). Balanced per capita contributions and level structure of cooperation. Top, 19(1), 167–176.

Hammer, P. L., Peled, U. N., & Sorensen, S. (1977). Pseudo-boolean functions and game theory. I. Core elements and Shapley value. Cahiers du CERO, 19, 159–176.

Harsanyi, J. C. (1959). A bargaining model for cooperative n-person games. In: A. W. Tucker & R. D. Luce (Eds.), Contributions to the theory of games IV (325–355). Princeton NJ: Princeton University Press.

Huettner, F. (2015). A proportional value for cooperative games with a coalition structure. Theory and Decision, 78(2), 273–287.

Kalai, E., & Samet, D. (1987). On weighted Shapley values. International Journal of Game Theory 16(3), 205–222.

Moriarity, S. (1975). Another approach to allocating joint costs, The Accounting Review, 50(4), 791–795.

Myerson, R. B. (1980). Conference Structures and Fair Allocation Rules, International Journal of Game Theory, Volume 9, Issue 3, 169–182.

Shapley, L. S. (1953a). Additive and non-additive set functions. Princeton University.

Shapley, L. S. (1953b). A value for n-person games. H. W. Kuhn/A. W. Tucker (eds.), Contributions to the Theory of Games, Vol. 2, Princeton University Press, Princeton, 307–317.

Tijs, S. H., & Driessen, T. S. (1986). Extensions of solution concepts by means of multiplicative ε-tax games. Mathematical Social Sciences, 12(1), 9–20.

Vasil’ev, V. A. (1978). Support function of the core of a convex game. Optimizacija Vyp, 21, 30-35.

Winter, E. (1989). A value for cooperative games with levels structure of cooperation. International Journal of Game Theory, 18(2), 227–240.