Apportionment with parity constraints

Abstract

In the classic apportionment problem, the goal is to decide how many seats of a parliament should be allocated to each party as a result of an election. The divisor methods solve this problem by defining a notion of proportionality guided by some rounding rule. Motivated by recent challenges in the context of electoral apportionment, we consider the question of how to allocate the seats of a parliament under parity constraints between candidate types (e.g., an equal number of men and women elected) while at the same time satisfying party proportionality. We study two different approaches to solve this question. We first provide a theoretical analysis of a recently devised mechanism based on a greedy approach. We then propose and analyze a mechanism that follows the idea of biproportionality introduced by Balinski and Demange. In contrast with the classic biproportional method by Balinski and Demange, this mechanism is ruled by two levels of proportionality: Proportionality is satisfied at the level of parties by means of a divisor method, and then biproportionality is used to decide the number of candidates allocated to each type and party. A typical benchmark used in the context of two-dimensional apportionment is the fair share (a.k.a matrix scaling), which corresponds to an ideal fractional biproportional solution. We provide lower bounds on the distance between these two types of solutions, and we explore their consequences in the context of two-dimensional apportionment.

Appendix

Proof of Lemma 1

Properties (a–b) come directly from the definition of a divisor method. We now prove (c). Suppose there exist $\mathcal {J}\in \mathcal {A}_{\gamma }(\mathcal {Q},h)$ and $\mathcal {J}'\in \mathcal {A}_{\gamma }(\mathcal {Q}',h)$ such that $\mathcal {J}_{\textsf{p}}< \mathcal {J}'_{\textsf{p}}$. By the population monotonicity property [7, Appendix A, p. 117], since $\mathcal {Q}'_p/\mathcal {Q}'_i>\mathcal {Q}_p/\mathcal {Q}_i$ for every $i\ne \textsf{p}$ and $\mathcal {J}_{\textsf{p}}< \mathcal {J}'_{\textsf{p}}$, it holds that $\mathcal {J}'\le \mathcal {J}_i$ for every $i\ne \textsf{p}$. But then we have that $\sum _{j\in [n]}\mathcal {J}'_j<\sum _{j\in [n]}\mathcal {J}_j$, which is a contradiction since the house size is equal in both instances. That concludes (c).

We now prove (d). Consider the instance obtained from $(\mathcal {Q},h)$ and any $\mathcal {J}\in \mathcal {A}_{\gamma }(\mathcal {Q},h)$ as follows: The parties set is $[n]\setminus \{\textsf{p}\}$, the votes are given by $\mathcal {T}_j=\mathcal {Q}_j$ for each $j\ne \textsf{p}$ and the house size is $h-\mathcal {J}_{\textsf{p}}$. In particular, the restriction of $\mathcal {J}$ to the parties in $[n]\setminus \{\textsf{p}\}$ belongs to $\mathcal {A}_{\gamma }(\mathcal {T},h-\mathcal {J}_{\textsf{p}})$. Similarly, consider the instance obtained from $(\mathcal {Q}',h)$ and any $\mathcal {J}'\in \mathcal {A}_{\gamma }(\mathcal {Q}',h)$ as follows: The parties set is $[n]{\setminus } \{\textsf{p}\}$, the votes are given by $\mathcal {T}_j=\mathcal {Q}'_j=\mathcal {Q}_j$ for each $j\ne \textsf{p}$ and the house size is $h-\mathcal {J}'_{\textsf{p}}$. In particular, the restriction of $\mathcal {J}'$ to the parties in $[n]\setminus \{\textsf{p}\}$ belongs to $\mathcal {A}_{\gamma }(\mathcal {T},h-\mathcal {J}'_{\textsf{p}})$.

Observe that by property (c) we have that $\mathcal {J}_{\textsf{p}}\ge \mathcal {J}'_{\textsf{p}}$ for every $\mathcal {J}\in \mathcal {A}_{\gamma }(\mathcal {Q},h)$ and $\mathcal {J}'\in \mathcal {A}_{\gamma }(\mathcal {Q}',h)$. Fix a solution $\mathcal {J}\in \mathcal {A}_{\gamma }(\mathcal {Q},h)$ and fix a solution $\mathcal {J}'\in \mathcal {A}_{\gamma }(\mathcal {Q}',h)$. Let $\mathcal {G}$ be the vector with entries indexed by $[n]{\setminus } \{p\}$ such that $\mathcal {G}_j=\mathcal {J}_j$ for every $j\ne \textsf{p}$. In particular, $\mathcal {G}$ belongs to $\mathcal {A}_{\gamma }(\mathcal {T},h-\mathcal {J}_{\textsf{p}})$. By the house monotonicity property [7, Appendix A, p. 117] the following holds: There exists $\mathcal {S}\in \mathcal {A}_{\gamma }(\mathcal {T},h-\mathcal {J}'_{\textsf{p}})$ such that $\mathcal {J}_j=\mathcal {G}_{j}\le \mathcal {S}_{j}$ for every $j\ne \textsf{p}$. For any such solution $\mathcal {S}\in \mathcal {A}_{\gamma }(\mathcal {T},h-\mathcal {J}'_{\textsf{p}})$ consider the vector $\mathcal {H}$ defined as follows: $\mathcal {H}_{\textsf{p}}=\mathcal {J}'_{\textsf{p}}$ and $\mathcal {H}_{j}=\mathcal {S}_{j}$ for every $j\ne \textsf{p}$. By the uniformity property of divisor methods [4, 6] we have that $\mathcal {H}\in \mathcal {A}_{\gamma }(\mathcal {Q}',h)$ and it satisfies that $\mathcal {H}_j\ge \mathcal {J}_j$ for every $j\ne \textsf{p}$. This concludes (d).$\square$