In This Apportionment Lottery, the House Always Wins

TL;DR

The paper tackles the problem of apportioning $h$ seats among $n$ states in proportion to population while preserving quota and monotonicity properties. It introduces a randomized framework that achieves ex ante proportionality and quota while also ensuring house monotonicity via a novel cumulative rounding technique, a generalization of dependent rounding applied across multiple copies of a bipartite graph. A key theoretical contribution is showing an incompatibility between population monotonicity and quota (even under randomized, ex ante proportional schemes), alongside a population-monotone, ex ante proportional construction that does not enforce quota. The main constructive result proves the existence of a house-monotone, quota-compliant, and ex ante proportional apportionment method, with cumulative rounding offering a broader toolkit and potential applications in fair division and scheduling. The work advances both the theory and practice of randomized allocation, offering principled, implementable mechanisms with strong fairness guarantees and insights into deterministic method characterizations via geometric and polyhedral perspectives.

Abstract

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 $q_i$ of seats in expectation (ex ante proportionality) and receives either $\lfloor q_i \rfloor$ or $\lceil q_i \rceil$ 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.

Figures (4)

• Figure 1: Illustration of cumulative rounding. Dashed lines indicate edges $e \in E$ in the bipartite graph $(V,E)$, which are labeled with weights $w_e^t$. The red lines indicate a possible random outcome of cumulative rounding.
• Figure 2: Illustration of the population-monotone method in \ref{['thm:proppopmon']}.
• Figure 3: Interpretation of each edge being rounded up in the constructed graph, for arbitrary nodes $v, v' \in A \cup B$ and $1 \leq t \leq T$. The correctness of this characterization will be shown along the proof of \ref{['thm:cumulative']}, specifically in the sections on degree preservation and cumulative degree preservation.
• Figure 4: Illustration of the counting argument for proving cumulative degree preservation. Edges in the figure are edges from the constructed graph, a superset of the edges in the rounded graph. Node color and shape indicate the set that a node belongs to, as indicated in the legend.

Theorems & Definitions (27)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3: GKP+06
• Theorem 4
• Lemma 1
• Theorem 5
• Theorem 6
• Lemma 2