Parallel axiomatizations of weighted and multiweighted Shapley values, random order values, and the Harsanyi set

Besner, Manfred (2019): Parallel axiomatizations of weighted and multiweighted Shapley values, random order values, and the Harsanyi set. Published in: Social Choice and Welfare 55.1 (2020): 193-212 , Vol. 55, No. 1 (2020): pp. 193-212.

Abstract

We present new axiomatic characterizations of five classes of TU-values, the classes of the weighted, positively weighted, and multiweighted Shapley values, random order values, and the Harsanyi set. The axiomatizations are given in parallel, i.e., they differ only in one axiom. In conjunction with marginality, a new property, called coalitional differential dependence, is the key that allows us to dispense with additivity. In addition, we propose new axiomatizations of the above five classes, in which, in part new, different versions of monotonicity, associated with the strong monotonicity in Young (1985), are decisive.

• Item Type: MPRA Paper
• Language: English
• Keywords: Cooperative game; Marginality; Strong monotonicity; Coalitional differential dependence; Weighted Shapley values · Harsanyi set
• 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

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

van den Brink, R., Funaki, Y., & Ju, Y. (2013). Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values. Social Choice and Welfare, 40(3), 693–714.

Casajus, A., & Huettner, F. (2008) Marginality is equivalent to coalitional strategic equivalence. Working paper.

Casajus, A., & Huettner, F. (2014) Weakly monotonic solutions for cooperative games. Journal of Economic Theory, 154, 162–172.

Casajus, A. (2019). Relaxations of symmetry and the weighted Shapley values. Economics Letters, 176, 75–78.

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.

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

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

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. 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.

Vasil’ev, V. A. (1981). On a class of imputations in cooperative games, Soviet Mathematics Dokladi 23 53–57.

Weber, R. J. (1988). Probabilistic values for games. In A.E. Roth (Ed.), The Shapley value, essays in honor of L.S. Shapley (pp. 101–119). Cambridge: Cambridge University Press.

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