Ordinal equivalence of power notions in voting games
The authors consider \textit{voting games} in the form \((N,W)\) where \(N\) is a finite set of \textit{voters} and \(W\) is a class of \textit{winning coalitions} in \(N\). With each voting game, a cooperative game \((N,v)\) is associated by putting: \(v(S)=1\) for all winning coalitions \(S\subset N\) and \(v(S)=0\) for others. In such a way, any value \(\Phi = (\Phi_1,\ldots ,\Phi_n)\) of a game \((N,v)\) induces a preordering ''\(\succeq\)'' for which \(i\succeq j\) iff \(\Phi_i(v) \geq \Phi_j(v)\) for all \(i,j\in N\). The main purpose of the paper is to analyze the three preorderings on the set of voters \(N\): two induced by the Shapley-Shubik's index (SS) and by Banzhaf-Coleman's absolute index (BC), respectively, and the preordering (T) called the \textit{influence relation} and defined in the way: \(i\succeq j\) iff one of the following two conditions holds: (1) for any coalition \(S\subset N\) with \(i,j\notin S\), \(S\cup \{i\} \in W \Leftrightarrow S\cup \{j\} \in W\); (2) for any \(S\subset N\) with \(i,j\notin S\), \(S\cup \{i\} \in W \Rightarrow S\cup \{j\} \in W\), and there is a \(T\subset N\) such that \(i,j\notin S\), \(T\cup \{i\} \in W\) and \(T\cup \{j\} \notin W\). The main result concerns a characterization of a clas of voting games for which the three preorderings (SS), (BC) and (T) coincide. It is shown that this happens if and only if the game is a \textit{swap robust}, which is a subclass of voting games containg the \textit{weighted majority games}. This generalizes an earlier result of Tomiyama. The second result of the paper is related with the \textit{veto-holder extension} of a voting game which is constructed by adding an extra ''veto-player'' to game. It is proved that the (T) preordering keeps itself in all the veto-holder extensions of voting games. As a consequence of this result is that (SS) preordering also keeps itself in such extensions of any swap robust voting game.
scientific article; zbMATH DE number 1968048