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Sequential Apportionment from Stationary Divisor Methods

Michael A. Jones, Brittany Ohlinger, Jennifer Wilson

TL;DR

This work analyzes sequential apportionment under stationary divisor methods, showing that with integer votes the seat-allocation sequence is periodic and determined by a cut point $c\in[0,1]$. It provides a complete characterization of two-party sequences, including explicit formulas for the run-lengths $k_i$, the $c$-intervals that yield a given sequence, and the special cases that recover Adams and d'Hondt methods; it then extends to $n$ parties via a lifting algorithm that combines all pairwise two-party sequences. The authors further develop a counting framework that expresses the number of distinct $n$-party sequences in terms of gcd-derived endpoint sets from all pairs, including a concrete example with $(25,17,13,5)$. Together, these results offer a refined view of large-party bias through sequence timing, and provide practical tools for analyzing sequential apportionment in political and JIT contexts.

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.

Sequential Apportionment from Stationary Divisor Methods

TL;DR

This work analyzes sequential apportionment under stationary divisor methods, showing that with integer votes the seat-allocation sequence is periodic and determined by a cut point . It provides a complete characterization of two-party sequences, including explicit formulas for the run-lengths , the -intervals that yield a given sequence, and the special cases that recover Adams and d'Hondt methods; it then extends to parties via a lifting algorithm that combines all pairwise two-party sequences. The authors further develop a counting framework that expresses the number of distinct -party sequences in terms of gcd-derived endpoint sets from all pairs, including a concrete example with . Together, these results offer a refined view of large-party bias through sequence timing, and provide practical tools for analyzing sequential apportionment in political and JIT contexts.

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 . 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 . We then show how sequences for all pairs of parties can be systematically extended to the -party setting. Further, we determine the number of distinct sequences in the -party problem for all . 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.
Paper Structure (8 sections, 21 theorems, 51 equations, 5 tables, 1 algorithm)

This paper contains 8 sections, 21 theorems, 51 equations, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Sequential Apportionment
  3. A Refinement of the Notion of Bias
  4. Periodic Behavior
  5. Apportionment Sequences for Two Parties
  6. Apportionment Sequences for More Than Two Parties
  7. Counting the number of distinct apportionment sequences for $n$ parties
  8. Conclusion

Key Result

Theorem 1

For every $\mathbf p$, if $S_1(\mathbf p) = (s_1, s_2, \dots)$ is the sequential apportionment under D'Hondt's method then the sequential apportionment under Adams' method is $S_0(\mathbf p) = (1, 2, \dots, n, s_1, s_2, s_3, \dots)$.

Theorems & Definitions (46)

  • Example 1
  • Definition 1
  • Example 2
  • Definition 2
  • Theorem 1
  • proof
  • Definition 3
  • Proposition 2
  • proof
  • Example 3
  • ...and 36 more