Online Proportional Apportionment

TL;DR

This work introduces an online framework for proportional apportionment in which seats are allocated sequentially with no knowledge of future votes. It distinguishes deterministic and randomized approaches: the greedy deterministic method achieves a tight linear-in-$n$ deviation from cumulative quotas and can satisfy global quota only for $n\le 3$, while randomized, flow-network-based constructions realize global quota and ex-ante proportionality for $n\le 3$ and provide a full characterization of such methods. The paper further connects these results to online dependent rounding and extends to multidimensional, multi-stage hypergraph covering, yielding online near-feasible solutions and cost-approximation guarantees under resource augmentation. Overall, the results illuminate fundamental limits of online proportionality and provide constructive mechanisms for both deterministic and randomized online allocations with practical implications for online decision-making and rounding in multidimensional settings.

Abstract

Traditionally, the problem of apportioning the seats of a legislative body has been viewed as a one-shot process with no dynamic considerations. While this approach is reasonable for some settings, dynamic aspects play an important role in many others. We initiate the study of apportionment problems in an online setting. Specifically, we introduce a framework for proportional apportionment with no information about the future. In this model, time is discrete and there are $n$ parties that receive a certain share of the votes at each time step. An online algorithm needs to irrevocably assign a prescribed number of seats at each time, ensuring that each party receives its fractional share rounded up or down, and that the cumulative number of seats allocated to each party remains close to its cumulative share up to that time. We study deterministic and randomized online apportionment methods. For deterministic methods, we construct a family of adversarial instances that yield a lower bound, linear in $n$, on the worst-case deviation between the seats allocated to a party and its cumulative share. We show that this bound is best possible and is matched by a natural greedy method. As a consequence, a method guaranteeing that the cumulative number of seats assigned to each party up to any step equals its cumulative share rounded up or down (global quota) exists if and only if $n\leq 3$. Then, we turn to randomized allocations and show that, for $n\leq 3$, we can randomize over methods satisfying global quota with the additional guarantee that each party receives, in expectation, its proportional share in every step. Our proof is constructive: Any method satisfying these properties can be obtained from a flow on a recursively constructed network. We showcase the applicability of our results to obtain approximate solutions in the context of online dependent rounding procedures.

Figures (6)

• Figure 1: Illustration of the flow network used in the proof of \ref{['prop:offline']} for $T=5$ steps and $n=3$ parties. A single arc label represents an upper capacity; labels below and above an arc correspond to lower and upper capacities, respectively.
• Figure 2: Illustration of adversarially constructed partial instances to induce surpluses and deficits close to $(n-1)/2$, for $n\in \{3,4\}$. At each step $t\geq 1$, an $(i,j)$-splitter is applied for parties $i,j$ whose surpluses then split by one unit at the next step, while keeping the average constant. When multiple parties have the same surplus at some step, they are depicted with a small horizontal offset for ease of understanding.
• Figure 3: The flow network $\normalfont\text{FN}(V^1,A^1,v^2,\pi)$ for $v^1=(0.6,0.3,0.1)$ and $v^2=(0.5,0.2,0.3)$. Edges with upper capacity zero are omitted. The upper capacities of the remaining edges are $c((o,\{i\}))=\pi(\{i\})=v_i^1$, $c((\{i\},j))=\pi(\{i\})/H^1=v_i^1$, and $c((i,d))=v_i^2$. An edge is highlighted in red if the lower capacity equals the upper capacity. The lower capacities of the remaining edges are zero. The labels indicate the unique flow of value $1$.
• Figure 4: Case 1 of the proof of \ref{['lem:flowpolytope']} with $H^{t+1}=1$. The upper capacity of an edge $e=(u,i)$ is $c(e)=\pi(u)$ if it is drawn and $c(e)=0$, otherwise. Red edges have lower capacities $\ell(e)=c(e)$. Grey edges carry no flow in the unique flow with value $1$.
• Figure 5: Case 2 of the proof of \ref{['lem:flowpolytope']} with $H^{t+1}=2$. The upper capacity of an edge $e=(u,i)$ is $c(e)=\pi(u)/2$ if it is drawn and $c(e)=0$, otherwise. Red edges have lower capacities $\ell(e)=c(e)$.

Theorems & Definitions (32)

• Proposition 1
• Theorem 1
• Corollary 1
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 2
• Lemma 4
• Lemma 5
• Proposition 2