Quotatone Apportionment Methods

The problem of apportionment is that of allocating an integer number of seats “proportionally” among a set of states or regions as a fraction of their populations. An apportionment method satisfies quota if it accords to each state the exactly proportional (rational) number of seats due it rounded up or rounded down. A method is house monotone if no state’s allocation goes down when the total number of seats to be distributed goes up. This paper gives a simple characterization of all house monotone methods satisfying quota. Further, a manner of exposition is formulated which unites several key house monotone apportionment methods, thus showing clearly their connections.