A Note on Positions and Power of Players in Multicameral Voting Games

A multicameral simple game is an intersection of a number of simple games played by the same set of players: a coalition is winning in the multicameral game if and only if it is winning in all the individual games played. Examples include decision rules in multicameral parliaments where a bill must be passed in all the houses of the parliament, and voting rules in the European Union Council where a winning coalition of countries must satisfy two or three independent criteria. This paper is a preliminary study of relations between the positions and power indices of players in the “chamber” games and in the multicameral game obtained as the intersection. We demonstrate that for any power index satisfying a number of standard properties, the index of a player in the multicameral game can be smaller (or greater) than in all the chamber games; this can occur even when the players are ordered the same way by desirability relations in all the chamber games. We also observe some counterintuitive effects when comparing the positions and decisiveness of players. However, as expected, introducing an additional chamber with all the players equal (a one man - one vote majority game) to a complete simple game reduces all the differences between the Shapley-Shubik indices of players.