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2019 | OriginalPaper | Chapter

Orders of Criticality in Graph Connection Games

Authors : Marco Dall’Aglio, Vito Fragnelli, Stefano Moretti

Published in: Transactions on Computational Collective Intelligence XXXIV

Publisher: Springer Berlin Heidelberg

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Abstract

The order of criticality of a player in a simple game and two indices inspired by the reasoning à la Shapley and à la Banzhaf were introduced in two previous papers [3] and [4], respectively, mainly having in mind voting situations. Here, we devote our attention to graph connection games, and to the computation of the order of criticality of a player. The indices introduced in [4] may be used as centrality measures of the edges in preserving the connection of a graph.

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Footnotes
1
This property is called monotonicity.
 
2
In other terms, \(v(M \setminus T )=0\) or \(v(M \setminus (T \cup \{i\} ))=1\) for any \(T \subset M \setminus \{i\}\) with \(|T|< k\).
 
Literature
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go back to reference Banzhaf, J.F.: Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev. 19, 317–343 (1965) Banzhaf, J.F.: Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev. 19, 317–343 (1965)
3.
go back to reference Dall’Aglio, M., Fragnelli, V., Moretti, S.: Orders of criticality in voting games. Oper. Res. Decis. 26, 53–67 (2016)MathSciNetMATH Dall’Aglio, M., Fragnelli, V., Moretti, S.: Orders of criticality in voting games. Oper. Res. Decis. 26, 53–67 (2016)MathSciNetMATH
4.
go back to reference Dall’Aglio, M., Fragnelli, V., Moretti, S.: Indices of criticality in simple games. Int. Game Theory Rev. 21(01), 1–21 (2019)MathSciNetCrossRef Dall’Aglio, M., Fragnelli, V., Moretti, S.: Indices of criticality in simple games. Int. Game Theory Rev. 21(01), 1–21 (2019)MathSciNetCrossRef
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8.
go back to reference Shapley, L.S., Shubik, M.: A method for evaluating the distribution of power in a committee system. Am. Polit. Sci. Rev. 48, 787–792 (1954)CrossRef Shapley, L.S., Shubik, M.: A method for evaluating the distribution of power in a committee system. Am. Polit. Sci. Rev. 48, 787–792 (1954)CrossRef
Metadata
Title
Orders of Criticality in Graph Connection Games
Authors
Marco Dall’Aglio
Vito Fragnelli
Stefano Moretti
Copyright Year
2019
Publisher
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-662-60555-4_3