Tyler C. Wunder | Searcharxiv

arXiv2026

Probability of Quota Violations in Divisor Apportionment Methods with Nonzero Allocations

Apportionment assigns indivisible items among groups. By the Balinski-Young theorem, no method can satisfy both house monotonicity and the quota rule. This paper investigates quota violations caused by nonzero allocation constraints, and derives exact probability formulas for their frequency. Such violations occur in systems like the U.S. House of Representatives, where each state is guaranteed at least one seat. We analyze the three-state case, introduce the $\tau$ statistic to parametrize population distributions, and prove an Asymptotic Quota Stabilization theorem: for fixed $\tau$, quota behavior stabilizes as populations grow, yielding probability results for quota violations determined by the set of ultimately violatory $\tau$ values. Applying this framework to the five classical divisor methods, we derive exact probability formulas. Additionally, we show that as the number of seats $M \to \infty$, these probabilities converge to method-specific constants. These results provide a precise, quantitative foundation for evaluating the fairness and frequency of quota violations in constrained apportionment systems.

math.PR↗

arXiv2025

Quantifying the Balinski-Young Theorem: Structure and Probability of Quota Violations in Divisor Methods for Three States

The apportionment problem asks how to assign representation to states based on their populations. That is, given census data and a fixed number of seats, how many seats should each state be assigned? Various algorithms exist to solve the apportionment problem, but by the Balinski-Young Impossibility Theorem, every such algorithm will be flawed in some way. This paper focuses on divisor methods of apportionment, where the possible flaws are known as quota violations. This paper presents a detailed analysis of quota violations that can arise under divisor methods for three states, By focusing on the three-state case, the paper makes the consequences of the Balinski-Young theorem particularly transparent and allows for a precise classification of quota violations that is difficult to obtain in more general formulations. The study focuses on quota violations in the Adams, Jefferson, Dean, and the Huntington-Hill methods when allocating M seats, but is expandable to a wider class of divisor functions. Theoretical results are proved about the behavior of these methods, particularly focusing on the types of quota violations that may occur, their frequency, and their structure and geometry. The key results of the paper are tests to detect quota violations which are employed to understand the geometry of violations and construct a probability function which calculates the likelihood of such violations occurring given an initial three state population vector whose components follow varying distributions.

math.PR↗