In This Apportionment Lottery, the House Always Wins

Apportionment is the problem of distributing h indivisible seats across states in proportion to the states' populations. In the context of the US House of Representatives, this problem has a rich history and is a prime example of interactions between mathematical analysis and political practice. Grimmett [2004] suggested to apportion seats in a randomized way such that each state receives exactly their proportional share qi of seats in expectation (ex ante proportionality) and receives either ↾qi↿ or ⇂qi⇃ many seats ex post (quota). However, there is a vast space of randomized apportionment methods satisfying these two axioms, and so we additionally consider prominent axioms from the apportionment literature. Our main result is a randomized method satisfying quota, ex ante proportionality and house monotonicity — a property that prevents paradoxes when the number of seats changes and which we require to hold ex post. This result is based on a generalization of dependent rounding on bipartite graphs, which we call cumulative rounding and which might be of independent interest, as we demonstrate via applications beyond apportionment. The full version of this paper is available at \urlhttps://arxiv.org/pdf/2202.11061.pdf.