Power index rankings in bicameral legislatures and the US legislative system

In this paper we study rankings induced by power indices of players in simple game models of bicameral legislatures. For a bicameral legislature where bills are passed with a simple majority vote in each house we give a condition involving the size of each chamber which guarantees that a member of the smaller house has more power than a member of the larger house, regardless of the power index used. The only case for which this does not apply is when the smaller house has an odd number of players, the larger house has an even number of players, and the larger house is less than twice the size of the smaller house. We explore what can happen in this exceptional case. These results generalize to multi-cameral legislatures. Using a standard model of the US legislative system as a simple game, we use our results to study power index rankings of the four types of players—the president, the vice president, senators, and representatives. We prove that a senator is always ranked above a representative and ranked the same as or above the vice president. We also show that the president is always ranked above the other players. We show that for most power index rankings, including the Banzhaf and Shapley–Shubik power indices, the vice president is ranked above a representative, however, there exist power indices ranking a representative above the vice president.