In This Apportionment Lottery, the House Always Wins

Randomized Apportionment: A Fairer Distribution of Seats The question of how to apportion the seats of the U.S. House of Representatives to states has fueled century-long political debates and sparked mathematical theory. Traditional deterministic methods, such as the Hamilton method or the currently used Huntington–Hill method, may result in paradoxes or substantially deviate from proportionality. In their paper “In This Apportionment Lottery, the House Always Wins,” Gölz, Peters, and Procaccia propose a randomized approach that ensures each state receives its exact proportional share of seats in expectation and its proportional share, up to rounding, ex post. By incorporating randomization, the authors argue, the system can better adhere to the principle of proportional representation, minimizing the impact of small counting errors and ensuring fairness over time. In addition, their approach achieves house monotonicity, a property that prevents counterintuitive outcomes when the total number of seats changes. This is achieved through a novel cumulative rounding technique, a generalization of dependent rounding on bipartite graphs with potential applications beyond apportionment, including EU commission nominations and resource allocation.