The notions of k-monotonicity and superadditivity for non-additive measures (e.g., capacity and cooperative games) are used as indices to measure the complementarity of criteria/coalitions in decision-making involving multiple criteria and/or cooperative game theory. To avoid exponential complexity in capacity-based multicriteria decision-making models, k-additive capacities and/or 𝒞-decomposable capacities are often adopted. While, in cooperative game theory, under communication-restricted situations, some coalitions cannot generally be formed. This paper investigates the inheritance of complementary relationships/effects in non-additive measures with restricted domains (or under bounded interactions).

1. [1] R. Myerson, “Graphs and cooperation in games,” Mathematics of Operations Research, Vol.2, pp. 225-229, 1977.
2. [2] R. Myerson, “Conference structures and fair allocation rules,” Int. J. of Game Theory, Vol.9, pp. 169-182, 1980.
3. [3] A. van den Nouweland, P. Borm, and S. Tijs, “Allocation rules for hypergraph communication situations,” Int. J. of Game Theory, Vol.20, pp. 255-268, 1992.
4. [4] M. Sugeno, K. Fujimoto, and T. Murofushi, “A hierarchical decomposition of Choquet integral model,” Int. J. of Uncertainty, Fuzziness and Knowledge-based Systems, Vol.3, Issue 1, pp. 1-15, 1995.
5. [5] M. Grabisch, “k-order additive discrete fuzzy measures,” Proc. of 6th Int. Conf. on Information Proc. and Management of Uncertainty in Knowledge-Based Systems (IPMU), pp. 1345-1350, 1996.
6. [6] M. Grabisch, “k-Order additive discrete fuzzy measures and their representation,” Fuzzy Sets and Systems, Vol.92, pp. 167-189, 1997.
7. [7] P. Miranda, M. Grabisch, and P. Gil, “Axiomatic structure of k-additive capacities,” Math. Soc. Sci., Vol.49, pp. 153-178, 2005.
8. [8] G. Choquet, “Theory of capacities,” Annals. Inst. Fourier (Grenoble), Vol.5, pp. 131-295, 1953.
9. [9] A. P. Dempster, “Upper and lower probabilities induced by a multi-valued mapping,” Annals of Mathematical Statistics, Vol.38, pp. 325-339, 1967.
10. [10] G. Shafer, “A mathematical theory of evidence,” Princeton University Press, 1976.
11. [11] I. Gilboa and E. Lehrer, “Global Games,” Int. J. of Game Theory, Vol.20, pp. 129-147, 1991.
12. [12] M. Marinacci, “Decomposition and representation of coalitional games,” Mathematics of Operations Research, Vol.21, pp. 1000-1015, 1996.
13. [13] I. Gilboa and D. Schmeidler, “Canonical representation of set functions,” Mathematics of Operations Research, Vol.20, pp. 197-212, 1995.
14. [14] A. Chateauneuf and J.-Y. Jaffray, “Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion,” Mathematical Social Sciences, Vol.17, pp. 263-283, 1989.
15. [15] T. Asano and H. Kojima, “Modularity and monotonicity of games,” Mathematical Methods of Operations Research, Vol.80, Issue 1, pp. 29-46, 2014.
16. [16] E. Ben-Porath and I. Gilboa, “Linear measures, the Gini index and the income-equality trade-off,” J. of Economic Theory, Vol.64, pp. 443-467, 1994.
17. [17] S. Hart and Mas-A. Colell, “Potential, value, and consistency,” Econometrica, Vol.57, pp. 589-614, 1989.
18. [18] K. Fujimoto, I. Kojadinovic, and J.-L. Marichal, “Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices,” Games and Economic Behavior, Vol.55, Issue 1, pp. 72-99, 2006.
19. [19] G. C. Rota, “On the foundations of combinatorial theory – I. Theory of Möbius functions,” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, pp. 340-368, 1964.
20. [20] J. C. Harsanyi, “A bargaining model for the cooperative n-person game,” In A. W. Tucker and D. R. Luce (Eds.), Contributions to the theory of games, Vol.4, Princeton University Press, Princeton, pp. 325-355, 1959.
21. [21] N. Jorzik, “Allocation Rules for Hypergraph Games,” Social Science Research Network, DOI: http://dx.doi.org/10.2139/ssrn.2125357, 2012.
22. [22] M. Slikker and A. van den Nouweland, “Social and Economic Networks in Cooperative Game Theory,” Kluwer, 2001.