New Combinatorial Insights for Monotone Apportionment

Abstract

The apportionment problem constitutes a fundamental problem in democratic societies: How to distribute a fixed number of seats among a set of states in proportion to the states' populations? This--seemingly simple--task has led to a rich literature and has become well known in the context of the US House of Representatives. In this paper, we connect the design of monotone apportionment methods to classic problems from discrete geometry and combinatorial optimization and explore the extent to which randomization can enhance proportionality. We first focus on the well-studied family of stationary divisor methods, which satisfy the strong population monotonicity property, and show that this family produces only a slightly superlinear number of different outputs as a function of the number of states. While our upper and lower bounds leave a small gap, we show that--surprisingly--closing this gap would solve a long-standing open problem from discrete geometry, known as the complexity of $k$-levels in line arrangements. The main downside of divisor methods is their violation of the quota axiom, i.e., every state should receive $\lfloor q_i\rfloor$ or $\lceil q_i\rceil$ seats, where $q_i$ is the proportional share of the state. As we show that randomizing over divisor methods can only partially overcome this issue, we propose a relaxed version of divisor methods in which the total number of seats may slightly deviate from the house size. By randomizing over them, we can simultaneously satisfy population monotonicity, quota, and ex-ante proportionality. Finally, we turn our attention to quota-compliant methods that are house-monotone, i.e., no state may lose a seat when the house size is increased. We provide a polyhedral characterization based on network flows, which implies a simple description of all ex-ante proportional randomized methods that are house-monotone and quota-compliant.

Figures (5)

• Figure 1: Illustration of the outputs and breaking points of apportionment instances. Linear functions in $\mathcal{L}$ corresponding to state $1$ ($2$,$3$, respectively) are illustrated by blue (yellow, red, respectively) lines. The function $\lambda_H$ corresponding to the $(H-1)$-level is illustrated by thick light gray segments. Filled circles correspond to breaking points of $(p,H)$; unfilled circles correspond to vertices of $\lambda_H$ that are not breaking points of $(p,H)$. Outputs for each value of $\delta\in [0,1]$ are shown below the plots.
• Figure 2: Illustration of the set $\hat{\mathcal{L}}(\delta)$ from the proof of the upper bound in \ref{['thm:main']} via the same example as given in \ref{['fig:line-arrangement-b']}. For each $\delta$, $\hat{\mathcal{L}}(\delta)$ are those functions from $\mathcal{L}$ for which there exists a function with higher index that is included in $\mathcal{L}_{\leq H}(\delta')$ for some $\delta' \in [0,\delta]$. We illustrate $\hat{\mathcal{L}}(\delta)$ by dashed lines. The important property of this set is that once a line is included for some $\delta$, it will not intersect with the $(H-1)$-level for any $\delta' \geq \delta$.
• Figure 3: Illustration of the construction of a line arrangement corresponding to an apportionment instance from an arbitrary line arrangement satisfying the conditions in \ref{['lem:wlog-form-of-arrangement']}, such that the $k$-level of the original arrangement corresponds to the $(H-1)$-level of the new one. For illustration purposes, one of the slopes has been chosen out of the range specified in \ref{['lem:wlog-form-of-arrangement']} (smaller than $1$).
• Figure 4: Example of a randomized fixed-divisor method with three states, populations $p=(50,30,20)$, and house size $H=13$. The quotas are $(6.5, 3.9, 2.6)$; the apportionment for this realization is $(7,4,3)$, with a deviation of one seat from $H=13$. Realizations of the signposts are denoted with red dots.
• Figure 5: An example of the network flow formulation with $n=3$ states and $H=6$ seats. The two rhombuses represent the source and the sink. The source sends exactly one unit to each of the square nodes representing each of the $H=6$ seats, for which purpose we fix a lower and an upper capacity of exactly $1$ for the dashed edges. Then, for each of the three states, the circular nodes on the vertical layer corresponding to seat $t$ keep track of how many seats the state has been allocated up to that point. The edges connecting the squared nodes to the circular nodes of the same vertical layer have a lower capacity of zero and an upper capacity of one, and every square node sends precisely one unit of flow to those circular nodes (representing the assignment of the seat to a state). The edge connecting two consecutive circular nodes in the same horizontal layer (representing the states) have lower and upper capacities of $\lfloor p_it/P\rfloor$ and $\lceil p_it/P\rceil$ for each state $i$ and seat $t$, respectively, to make sure that the allocation is quota-compliant.

Theorems & Definitions (60)

• Definition 1
• Theorem 1
• Corollary 1
• Proposition 1
• Proposition 2
• proof : Proof of the upper bound in \ref{['thm:main']}
• Lemma 1
• proof : Proof of the lower bound in \ref{['thm:main']}.
• Proposition 3
• Proposition 4