On the Voting Power of an Alliance and the Subsequent Power of its Members

Even, and in fact chiefly, if two or more players in a voting gamehave on a binary issue independent opinions, they may haveinterest to form a single voting alliance giving an average gainof influence for all of them. Here, assuming the usualindependence of votes, we first study the alliance voting powerand obtain new results in the so-called asymptotic limit for whichthe number of players is large enough and the alliance weightremains a small fraction of the total of the weights. Then, wepropose to replace the voting game inside the alliance by a randomgame which allows new possibilities. The validity of theasymptotic limit and the possibility of new alliances are examinedby considering the decision process in the Council of Ministers ofthe European Union.