Random Apportionment: A Stochastic Solution to the Balinski-Young Impossibility

An apportionment paradox occurs when the rules for apportionment in a political system or distribution system produce results which seem to violate common sense. For example, The Alabama paradox occurs when the total number of seats increases but decreases the allocated number of a state and the population paradox occurs when the population of a state increases but its allocated number of seats decreases. The Balinski-Young impossibility theorem showed that there is no deterministic apportionment method that can avoid the violation of the quota rule and doesn’t have both the Alabama and the population paradoxes. In this paper, we propose a randomized apportionment method as a stochastic solution to the Balinski-Young impossibility.