Online Proportional Apportionment

2025-10-16

AI Summary

This paper studies online proportional seat allocation under dynamic voting: seats must be assigned in real time as vote streams arrive, without future information, to minimize the deviation between each party’s cumulative seat count and its cumulative vote share. We establish the first theoretical framework for online proportional allocation, proving a linear lower bound on the competitive ratio for deterministic algorithms. We further show that deterministic quota-satisfying allocations exist only when the number of parties is at most three. To overcome this limitation, we propose a novel randomized rounding mechanism—integrating stochasticity with network flow modeling—that achieves both global quota satisfaction and proportional fairness in expectation. We design a greedy algorithm matching the lower bound and provide a recursive network flow construction that generalizes to arbitrary numbers of parties.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Fair Division
• Machine Learning: Online Learning & Bandits

Application Category

• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $nleq 3$. Then, we turn to randomized allocations and show that, for $nleq 3$, 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.

Problem

• Studying online proportional apportionment with no future information
• Analyzing deterministic and randomized seat allocation methods
• Establishing deviation bounds and feasibility conditions for global quota

Innovation

• Online proportional apportionment with no future information
• Deterministic greedy method matching worst-case deviation bounds
• Randomized allocations using flow networks for global quota