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Monotone Randomized Apportionment

José Correa, Paul Gölz, Ulrike Schmidt-Kraepelin, Jamie Tucker-Foltz, Victor Verdugo

TL;DR

This work tackles the problem of designing monotone randomized apportionment methods that preserve quota and ex-ante proportionality while avoiding paradoxical incentives. It introduces a family of higher-order monotonicity axioms for joint representations and analyzes rounding rules under these axioms. The main technical result establishes that Sampford rounding satisfies selection monotonicity (and a two-coalition variant), whereas common rules like systematic, pipage, and conditional Poisson fail, with Grimmett’s method viable for two-party coalitions. The paper also develops a Lipschitz continuity guarantee for selection-monotone rounding rules and discusses broader implications for dependent randomized rounding and mechanism design, including several impossibility results for threshold-based monotonicity and raw-vote-count monotonicity. Overall, this work clarifies when randomized rounding can avoid coalition-based paradoxes and points to threshold monotonicity as a promising avenue for achieving strategic robustness in apportionment.

Abstract

Apportionment is the act of distributing the seats of a legislature among political parties (or states) in proportion to their vote shares (or populations). A famous impossibility by Balinski and Young (2001) shows that no apportionment method can be proportional up to one seat (quota) while also responding monotonically to changes in the votes (population monotonicity). Grimmett (2004) proposed to overcome this impossibility by randomizing the apportionment, which can achieve quota as well as perfect proportionality and monotonicity -- at least in terms of the expected number of seats awarded to each party. Still, the correlations between the seats awarded to different parties may exhibit bizarre non-monotonicities. When parties or voters care about joint events, such as whether a coalition of parties reaches a majority, these non-monotonicities can cause paradoxes, including incentives for strategic voting. In this paper, we propose monotonicity axioms ruling out these paradoxes, and study which of them can be satisfied jointly with Grimmett's axioms. Essentially, we require that, if a set of parties all receive more votes, the probability of those parties jointly receiving more seats should increase. Our work draws on a rich literature on unequal probability sampling in statistics (studied as dependent randomized rounding in computer science). Our main result shows that a sampling scheme due to Sampford (1967) satisfies Grimmett's axioms and a notion of higher-order correlation monotonicity.

Monotone Randomized Apportionment

TL;DR

This work tackles the problem of designing monotone randomized apportionment methods that preserve quota and ex-ante proportionality while avoiding paradoxical incentives. It introduces a family of higher-order monotonicity axioms for joint representations and analyzes rounding rules under these axioms. The main technical result establishes that Sampford rounding satisfies selection monotonicity (and a two-coalition variant), whereas common rules like systematic, pipage, and conditional Poisson fail, with Grimmett’s method viable for two-party coalitions. The paper also develops a Lipschitz continuity guarantee for selection-monotone rounding rules and discusses broader implications for dependent randomized rounding and mechanism design, including several impossibility results for threshold-based monotonicity and raw-vote-count monotonicity. Overall, this work clarifies when randomized rounding can avoid coalition-based paradoxes and points to threshold monotonicity as a promising avenue for achieving strategic robustness in apportionment.

Abstract

Apportionment is the act of distributing the seats of a legislature among political parties (or states) in proportion to their vote shares (or populations). A famous impossibility by Balinski and Young (2001) shows that no apportionment method can be proportional up to one seat (quota) while also responding monotonically to changes in the votes (population monotonicity). Grimmett (2004) proposed to overcome this impossibility by randomizing the apportionment, which can achieve quota as well as perfect proportionality and monotonicity -- at least in terms of the expected number of seats awarded to each party. Still, the correlations between the seats awarded to different parties may exhibit bizarre non-monotonicities. When parties or voters care about joint events, such as whether a coalition of parties reaches a majority, these non-monotonicities can cause paradoxes, including incentives for strategic voting. In this paper, we propose monotonicity axioms ruling out these paradoxes, and study which of them can be satisfied jointly with Grimmett's axioms. Essentially, we require that, if a set of parties all receive more votes, the probability of those parties jointly receiving more seats should increase. Our work draws on a rich literature on unequal probability sampling in statistics (studied as dependent randomized rounding in computer science). Our main result shows that a sampling scheme due to Sampford (1967) satisfies Grimmett's axioms and a notion of higher-order correlation monotonicity.
Paper Structure (24 sections, 21 theorems, 15 equations, 1 figure, 1 table)

This paper contains 24 sections, 21 theorems, 15 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. A Motivating Example
  3. Our Approach and Results
  4. Related Work
  5. Preliminaries
  6. Monotonicity of Rounding Rules
  7. Axioms and Rounding Rules
  8. Rules that Violate Selection Monotonicity
  9. Sampford Rounding is Monotone
  10. Potential Implications for Dependent Randomized Rounding
  11. Selection Monotonicity Implies Continuity
  12. Monotonicity of Apportionment Methods
  13. Axioms and Apportionment Methods
  14. Conjecture and $|T|=2$ Possibility for Threshold Monotonicity
  15. Impossibility of Pairwise Threshold Monotonicity
  16. ...and 9 more sections

Key Result

theorem 1

Let $\vec{p}, \vec{p}' \in [0, 1)^n$ be two residue vectors, summing up to the same integer $k$. Let $T$ be a set of $k$ parties such that $p_i' \geq p_i$ for all parties $i \in T$ and $p_i' \leq p_i$ for all $i \notin T$. Then, the probability that Sampford sampling rounds up the set $T$ is at leas

Figures (1)

  • Figure 1: Illustration of Grimmett's method Grimmett04 for the two elections from \ref{['tbl:apportia']}. Intervals corresponding to left-wing parties are gray. Dashed lines indicate the position of integers before the random shift.

Theorems & Definitions (26)

  • theorem 1: Sampford rounding satisfies selection monotonicity
  • definition 1: Selection monotonicity
  • definition 2: Pairwise selection monotonicity
  • proposition 1
  • proposition 2
  • theorem 2
  • proposition 3
  • lemma 1
  • lemma 2
  • lemma 3
  • ...and 16 more