Stochastic Apportionment

Abstract. The problem of how to allocate to states the seats in the US House of Representatives is the most studied instance of what is termed the ‘apportionment problem’. We propose a new method of apportionment which is stochastic, which meets the quota condition, and which is fair in the sense of expectations. Two sources of systematic unfairness are identified, firstly the lower bound condition (every state shall receive at least one seat), and secondly the lower quota condition (every state shall receive at least the integer part of its quota). 1. The problem of apportionment Ten goats are to be assigned between three brothers in numbers proportional to the ages (in years) of the recipients. Given the integral nature of a goat, it is not generally possible to meet exactly the condition of proportionality, and the resulting ‘apportionment problem ’ is a classic of operational research. The associated literature is extensive, and includes on the one hand discussions of criteria to be used in assessing different schemes, and on the other hand accounts of the properties of specific classes of scheme. A point of especial focus has been the apportionment of the seats in the House of Representatives between the states of the USA. There are currently 50 states (excluding the District of Columbia) and 435 seats, which are to be divided between the states according to the US Constitution Article I, Section 2 of 1787 thus: “Representatives... shall be apportioned among the several States... according to their respective Numbers...” No scheme is proposed in the Constitution, and the Article therein permits a spectrum of interpretation of the phrase ‘according to their respective numbers’. Politicians, lawyers, mathematicians and others have been involved since in the cyclical debate of how to apportion the seats. Although there is little in this note which is specific to the US Congress, we shall, for ease of exposition, use the terminology of the last problem. Our targets here are to survey the general area, and to propose a new method of apportionment which, in a certain way to be made more precise, meets all the usual criteria for such schemes. This new method is a lottery scheme whose implementation uses (pseudo-)random numbers. The scheme is fair so long as no minimal number of