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Kalai-Smorodinsky Balances for N-Tuples of Interfering Elements Read chapter Read first chapter

2017 | OriginalPaper | Chapter

Preorders in Simple Games

Authors : Josep Freixas, Montserrat Pons

Published in: Transactions on Computational Collective Intelligence XXVII

Publisher: Springer International Publishing

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Abstract

Any power index defines a total preorder in a simple game and, thus, induces a hierarchy among its players. The desirability relation, which is also a preorder, induces the same hierarchy as the Banzhaf and the Shapley indices on linear games, i.e., games in which the desirability relation is total. The desirability relation is a sub–preorder of another preorder, the weak desirability relation, and the class of weakly linear games, i.e., games for which the weak desirability relation is total, is larger than the class of linear games. The weak desirability relation induces the same hierarchy as the Banzhaf and the Shapley indices on weakly linear games. In this paper, we define a chain of preorders between the desirability and the weak desirability preorders. From them we obtain new classes of totally preordered games between linear and weakly linear games.

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previous chapter On Ordering a Set of Degressively Proportional Apportionments
next chapter Sub-Coalitional Approach to Values
Footnotes
1
For a detailed discussion of some issues raised by allowing abstentions, see Felsenthal and Machover [6] and for several levels of approval in input and output, see Freixas and Zwicker [9].
 
2
Linear games are also called complete, ordered or directed games in the literature, see Taylor and Zwicker [16] for references on these names.
 
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2.
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3.
Carreras, F., Freixas, J.: On ordinal equivalence of power measures given by regular semivalues. Math. Soc. Sci. 55(2), 221–234 (2008)MathSciNetCrossRefMATH
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Shapley, L.S.: A value for n-person games. In: Tucker, A.W., Kuhn, H.W. (eds.) Contributions to the Theory of Games II, pp. 307–317. Princeton University Press, Princeton (1953)
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Metadata
Title
Preorders in Simple Games
Authors
Josep Freixas
Montserrat Pons
Copyright Year
2017
Book
Transactions on Computational Collective Intelligence XXVII

Print ISBN: 978-3-319-70646-7

Electronic ISBN: 978-3-319-70647-4

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-70647-4_5

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