Value dividends, the Harsanyi set and extensions, and the proportional Harsanyi payoff

Besner, Manfred (2019): Value dividends, the Harsanyi set and extensions, and the proportional Harsanyi payoff.

Abstract

A new concept for TU-values, called value dividends, is introduced. Similar to Harsanyi dividends, value dividends are defined recursively and provide new characterizations of values from the Harsanyi set. In addition, we generalize the Harsanyi set where each of the TU-values from this set is defined by the distribution of the Harsanyi dividends via sharing function systems and give an axiomatic characterization. As a TU value from the generalized Harsanyi set, we present the proportional Harsanyi payoff, a new proportional solution concept. As a side benefit, a new characterization of the Shapley value is proposed. None of our characterizations uses additivity.

• Item Type: MPRA Paper
• Language: English
• Keywords: TU-game · Value dividends · (Generalized) Harsanyi set · Weighted Shapley values · (Proportional) Harsanyi payoff · Sharing function systems
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C71 - Cooperative Games

References

Banzhaf, J.F. (1965). J.F. Banzhaf, Weighted voting does not work: a mathematical analysis, Rutgers Law Review 19 (1965) 317–343.

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

Besner, M. (2019). Axiomatizations of the proportional Shapley value. Theory and Decision https://doi.org/10.1007/s11238-019-09687-7

Béal, S., Ferrières, S., Rémila, E., & Solal, P. (2018) The proportional Shapley value and applications. Games and Economic Behavior 108, 93–112.

Billot, A., & Thisse, J. F. (2005). How to share when context matters: The Möbius value as a generalized solution for cooperative games. Journal of Mathematical Economics, 41(8), 1007–1029.

van den Brink, R., van der Laan, G., & Vasil’ev, V. A. (2014). Constrained core solutions for totally positive games with ordered players. International Journal of Game Theory, 43(2), 351–368.

van den Brink, R., Levínský, R., & Zelený, M. (2015). On proper Shapley values for monotone TU-games. International Journal of Game Theory, 44(2), 449–471.

Casajus, A. (2017). Weakly balanced contributions and solutions for cooperative games. Operations Research Letters, 45(6), 616–619.

Casajus, A. (2018). Symmetry, mutual dependence, and the weighted Shapley values. Journal of Economic Theory, 178, 105–123.

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

Dragan, I. C. (1992) Multiweighted Shapley values and random order values. University of Texas at Arlington.

Feldman, B. (1999) The proportional value of a cooperative game. Manuscript. Chicago: Scudder Kemper Investments.

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.

Megiddo, N. (1974). On the nonmonotonicity of the bargaining set, the kernel and the nucleolus of game. SIAM Journal on Applied Mathematics, 27(2), 355–358.

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, 9(3), 169–182.

Ortmann, K. M. (2000) The proportional value for positive cooperative games. Mathematical Methods of Operations Research, 51(2), 235–248.

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.

Vasil’ev, V., & van der Laan, G. (2002). The Harsanyi set for cooperative TU-games. Siberian Advances in Mathematics 12, 97–125.

Vasil’ev, V. A. (1975). The Shapley value for cooperative games of bounded polynomial variation. Optimizacija Vyp, 17, 5-27.

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

Vorob’ev, N.N., & Liapunov, A. (1998). The proper Shapley value. Game theory and applications, 4, 155–159.

Young, H. P. (1985). Monotonic solutions of Cooperative Games. International Journal of Game Theory, 14(2), 65–72.