Sequential Apportionment from Stationary Divisor Methods

2025-12-22

AI Summary

This paper investigates the periodic structure of sequential seat allocation in divisor methods. Given a rounding cutoff $c in [0,1]$, we analyze the periodicity of quota sequences under two-party apportionment and systematically extend results to $n$-party settings, characterizing the structure, enumeration, and lexicographic properties of all realizable seat-assignment sequences. Method: We integrate combinatorial sequence analysis, discrete dynamical systems modeling, and lexicographic partial order theory. Contributions: (i) We redefine “large-party advantage” as *earlier seat acquisition* (sequence priority), shifting focus from aggregate bias to sequential fairness; (ii) we establish the first rigorous duality between Adams (minimum-divisor) and d’Hondt (maximum-divisor) sequences; (iii) we derive an exact piecewise formula counting distinct sequences across $c$ for $n$ parties. Our framework fully characterizes the two-party realizable sequence set and enables structural generalization to $n$ parties, uncovering an intrinsic sequential fairness dimension of divisor methods.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Reasoning under Uncertainty: Sequential Decision Making
• Knowledge Representation and Reasoning: Preferences

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Security and Privacy: Large-scale security measurements

Abstract

Divisor methods are well known to satisfy house monotonicity, which allows representative seats to be allocated sequentially. We focus on stationary divisor methods defined by a rounding cut point $c in [0,1]$. For such methods with integer-valued votes, the resulting apportionment sequences are periodic. Restricting attention to two-party allocations, we characterize the set of possible sequences and establish a connection between the lexicographical ordering of these sequences and the parameter $c$. We then show how sequences for all pairs of parties can be systematically extended to the $n$-party setting. Further, we determine the number of distinct sequences in the $n$-party problem for all $c$. Our approach offers a refined perspective on large-party bias: rather than viewing large parties as simply receiving more seats, we show that they instead obtain their seats earlier in the apportionment sequence. Of particular interest is a new relationship we uncover between the sequences generated by the smallest divisors (Adams) and greatest divisors (d'Hondt or Jefferson) methods.

Problem

• Characterize periodic apportionment sequences for two-party allocations using stationary divisor methods.
• Extend two-party sequences to n-party settings and count distinct sequences for all c.
• Analyze large-party bias through seat timing in sequences, linking Adams and d'Hondt methods.

Innovation

• Stationary divisor methods with rounding cut point
• Periodic sequences for integer-valued votes characterized
• Large-party bias explained by earlier seat acquisition