Voting power and at-large representation

Abstract Suppose one wishes to design a legislative body that contains both representatives from equipopulous districts and some at-large representatives. If the total number of seats in the legislature is considered fixed, how should one decide how many at-large representatives to allocate? This paper will present one answer to that question: if one wishes to maximize the total Banzhaf power of the voters and if we assume that the at-large representatives will vote as a bloc, then the number of at-large representatives should be roughly the square root of the total number of legislators.