  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.
