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Probabilistic Voting Models with Varying Speeds of Correlation Decay

Gabor Toth

TL;DR

The paper develops a probabilistic framework for static, multi-group binary voting using de Finetti representations, with sequences of de Finetti measures $\mu_n$ that converge to $\delta_0$ to model waning social cohesion as population size grows. It identifies a phase transition controlled by the contraction speed: fast (supercritical) yields a universal $N(0,I_M)$ limit for normalized group margins, critical yields a convolution of a contracted $\mu$ with Gaussian noise, and slow (subcritical) yields a limit given by $\mu$ under an appropriate scaling. The results include a local limit theorem under fast convergence and a detailed analysis of correlations, showing decay to zero under weak convergence. The framework accommodates real-world data by selecting subcritical contraction rates (e.g., $n^{-\\alpha}$ with $0.1\le\alpha\le0.2$), offering a flexible tool for studying two-tier voting weights and social cohesion dynamics.

Abstract

We model voting behaviour in the multi-group setting of a two-tier voting system using sequences of de Finetti measures. Our model is defined by using the de Finetti representation of a probability measure (i.e. as a mixture of conditionally independent probability measures) describing voting behaviour. The de Finetti measure describes the interaction between voters and possible outside influences on them. We assume that for each population size there is a (potentially) different de Finetti measure, and as the population grows, the sequence of de Finetti measures converges weakly to the Dirac measure at the origin, representing a tendency toward weakening social cohesion as the population grows large. The resulting model covers a wide variety of behaviour, ranging from independent voting in the limit under fast convergence, a critical convergence speed with its own pattern of behaviour, to a subcritical convergence speed which yields a model in line with empirical evidence of real-world voting data, contrary to previous probabilistic models used in the study of voting. These models can be used, e.g., to study the problem of optimal voting weights in two-tier voting systems.

Probabilistic Voting Models with Varying Speeds of Correlation Decay

TL;DR

The paper develops a probabilistic framework for static, multi-group binary voting using de Finetti representations, with sequences of de Finetti measures that converge to to model waning social cohesion as population size grows. It identifies a phase transition controlled by the contraction speed: fast (supercritical) yields a universal limit for normalized group margins, critical yields a convolution of a contracted with Gaussian noise, and slow (subcritical) yields a limit given by under an appropriate scaling. The results include a local limit theorem under fast convergence and a detailed analysis of correlations, showing decay to zero under weak convergence. The framework accommodates real-world data by selecting subcritical contraction rates (e.g., with ), offering a flexible tool for studying two-tier voting weights and social cohesion dynamics.

Abstract

We model voting behaviour in the multi-group setting of a two-tier voting system using sequences of de Finetti measures. Our model is defined by using the de Finetti representation of a probability measure (i.e. as a mixture of conditionally independent probability measures) describing voting behaviour. The de Finetti measure describes the interaction between voters and possible outside influences on them. We assume that for each population size there is a (potentially) different de Finetti measure, and as the population grows, the sequence of de Finetti measures converges weakly to the Dirac measure at the origin, representing a tendency toward weakening social cohesion as the population grows large. The resulting model covers a wide variety of behaviour, ranging from independent voting in the limit under fast convergence, a critical convergence speed with its own pattern of behaviour, to a subcritical convergence speed which yields a model in line with empirical evidence of real-world voting data, contrary to previous probabilistic models used in the study of voting. These models can be used, e.g., to study the problem of optimal voting weights in two-tier voting systems.
Paper Structure (13 sections, 17 theorems, 120 equations, 1 table)

This paper contains 13 sections, 17 theorems, 120 equations, 1 table.

Table of Contents

  1. Introduction
  2. Results
  3. Proofs
  4. Proof of Lemma \ref{['lem:weak_conv']}
  5. $(i)\implies(ii)$
  6. $(ii)\implies(iii)$
  7. $(iii)\implies(i)$
  8. Proof of Theorem \ref{['thm:fast_conv']}
  9. Proof of Theorem \ref{['thm:phase_transition']}
  10. Proof of Proposition \ref{['prop:corr']}
  11. Proof of Theorem \ref{['thm:LLT']}
  12. Examples of de Finetti Voting Models
  13. Conclusion

Key Result

Lemma 9

Let $\left(\mu_{n}\right)_{n}$ be a sequence of probability measures on $\mathbb{R}^{M}$. Then the following statements are equivalent:

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Definition 4
  • Remark 5
  • Remark 6
  • Definition 7
  • Lemma 9
  • Theorem 11
  • Theorem 13
  • Corollary 14
  • ...and 24 more