Player splitting, players merging, the Shapley set value and the Harsanyi set value

Besner, Manfred (2018): Player splitting, players merging, the Shapley set value and the Harsanyi set value.

Abstract

Shapley (1953a) introduced the weighted Shapley values as a family of values, also known as Shapley set. For each exogenously given weight system exists a seperate TU-value. Shapley (1981) and Dehez (2011), in the context of cost allocation, and Radzik (2012), in general, presented a value for weighted TU-games that covers the hole family of weighted Shapley values all at once. To distinguish this value from a weighted Shapley value in TU-games we call it Shapley set value. This value coincides with a weighted Shapley value only on a subdomain and allows weights which can depend on coalition functions. Hammer (1977) and Vasil’ev (1978) introduced independently the Harsanyi set, also known as selectope (Derks, Haller and Peters, 2000), containing TU-values which are referred to as Harsanyi-payoffs. These values are obtained by distributing the dividends from all coalitions by a sharing system that is independent from the coalition function. In this paper we introduce the Harsanyi set value that, similar to the Shapley set value, covers the hole family of Harsanyi payoffs at once, allows not exogenously given share systems and coincides thus also with non linear values on some subdomains. We present some new axiomatizations of the Shapley set value and the Harsanyi set value containing a player splitting or a players merging property respectively as a main characterizing element that recommend these values for profit distribution and cost allocation.

• Item Type: MPRA Paper
• Language: English
• Keywords: Cost allocation · Profit distribution · Player splitting · Players merging · Shapley set value · Harsanyi set value
• 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

Amer, R., Carreras, F., & Maga˜na, A. (2007). Two Main Methods for Utility Sharing in Joint Business: A Discussion. Journal of Mathematics and Statistics 3(1), 28–35.

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

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

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

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

Besner, M. (2018). Proportional Shapley levels values.

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

Casajus, A. (2017). Weakly balanced contributions and solutions for cooperative games. Operations Research Letters, 45, 616–619.

Dehez, P. (2011). Allocation of fixed costs: characterization of the (dual) weighted Shapley value. International Game Theory Review, 13, 141-157.

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.

Derks, J., Haller, H., & Peters, H. (2000). The selectope for cooperative games. International Journal of Game Theory, 29(1), 23–38.

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

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.

Hart, S., & Mas-Colell, A. (1989). Potential, value, and consistency. Econometrica: Journal of the Econometric Society, 589–614.

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

Lehrer, E. (1988). An axiomatization of the Banzhaf value. International Journal of Game Theory, 17(2), 89–99.

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

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

Nowak, A. S., & Radzik, T. (1995). On axiomatizations of the weighted Shapley values. Games and Economic Behavior, 8(2), 389–405.

Radzik, T. (2012). A new look at the role of players’ weights in the weighted Shapley value. European Journal of Operational Research, 223(2), 407–416.

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.

Shapley, L. S. (1981). Discussion comments on ”Equity considerations in traditional full cost allocation practices: an axiomatic approach”, in: Moriarity S. (Ed.), Joint Cost Allocation, Proceeding of the University of Oklahoma Conference on Costs Allocations, April 1981, Centre for Economic and Management Research, University of Oklahoma, 131–136.

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