The fundamental theorem of voting schemes (Q1906123)

A coalition \(A\subseteq V\) is said to be blocking (in \(S\)) if \(\overline A\not\in S\). The set of all blocking coalitions is denoted \(S^*\), and is called the dual of \(S\), since \((S^*)^*= S\). A game \(S\) is simple if \(A\in S\) (\(A\) wins) implies \(\overline A\not\in S\) (\(A\) blocks); i.e. \(S\subseteq S^*\). Conversely, a game \(S\) is strong if \(\overline A\not\in S\) (\(A\) blocks) implies \(A\in S\) (\(A\) wins); i.e. \(S^*\subseteq S\). The main result of this paper is the proof of the ``Fundamental Theorem'' of strong simple games, namely, all full strong simple games have unique decompositions into prime strong-simple-games.