Geodesic
Weightedness for Simple Games with Less than 9 Voters
Contributions to game theory and management, Tome 2 (2009), pp. 63-71.

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In voting systems, game theory, switching functions, threshold logic, circuits and electrical engineering, coherent structures and systems' reliability and cryptography, among other fields, there is an important problem that consists in determining the weightedness of a binary voting system by means of trades among voters in sets of coalitions. (Taylor and Zwicker, 1995) construct a sequence of games $G_k$ based on magic squares which are $(k-1)$-trade robust but not $k$-trade robust, each one of these games $G_k$ has $k^2$ players. (Freixas and Molinero, 2008) propose a refinement on trade robustness called invariant-trade robustness and prove that as few as $2k+1$ voters are needed to find games being $k$-invariant trade robust but not $(k+1)$-invariant trade robust. In this work we classify all simple games with less than nine players according to the two criteria: invariant-trade robustness and trade robustness. The classification obtained in this work with eight players is new. Moreover, some new experiments establish new conjectures about the trade robustness of complete simple games.
Keywords: Simple games, Asummability, Trade robustness, Invariant-trade robustness. 
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Josep Freixas; Xavier Molinero. Weightedness for Simple Games with Less than 9 Voters. Contributions to game theory and management, Tome 2 (2009), pp. 63-71. http://geodesic.mathdoc.fr/item/CGTM_2009_2_a5/

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