How did Balinski and Young prove their method of apportionment? - Answers

Balinski and Young proved their method of apportionment by establishing the mathematical properties that any fair apportionment method should satisfy. They demonstrated that their method adhered to the principles of fairness, such as the quotas of the states and the population criterion, ensuring that no state could gain an advantage simply by the manipulation of population figures. Their work also included formal proofs that highlighted the consistency and stability of their method, reinforcing its theoretical foundation in the context of existing apportionment methods.