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Optimal bounds for dissatisfaction in perpetual voting

Alexander Kozachinskiy, Alexander Shen, Tomasz Steifer

TL;DR

This work studies how to bound voter dissatisfaction in perpetual voting with $k$ options, $N$ agents, and $T$ rounds by introducing the conflict-number $C$. It shows that when $C = o(T)$ sublinear dissatisfaction is achievable and provides both an EW-based strategy with a sublinear bound and a non-constructive Kolmogorov-complexity lower bound indicating near-tightness, along with an explicit prediction-with-expert-advice–style method achieving similar guarantees. It also establishes lower bounds demonstrating that sublinear guarantees fail without the bounded-conflicts condition and that simple voting rules perform poorly in this regime. By connecting perpetual voting to prediction-theory methods and complexity theory, the paper provides both practical strategies for fair long-horizon decisions and fundamental limits on what can be guaranteed in dynamic voting environments.

Abstract

In perpetual voting, multiple decisions are made at different moments in time. Taking the history of previous decisions into account allows us to satisfy properties such as proportionality over periods of time. In this paper, we consider the following question: is there a perpetual approval voting method that guarantees that no voter is dissatisfied too many times? We identify a sufficient condition on voter behavior -- which we call 'bounded conflicts' condition -- under which a sublinear growth of dissatisfaction is possible. We provide a tight upper bound on the growth of dissatisfaction under bounded conflicts, using techniques from Kolmogorov complexity. We also observe that the approval voting with binary choices mimics the machine learning setting of prediction with expert advice. This allows us to present a voting method with sublinear guarantees on dissatisfaction under bounded conflicts, based on the standard techniques from prediction with expert advice.

Optimal bounds for dissatisfaction in perpetual voting

TL;DR

This work studies how to bound voter dissatisfaction in perpetual voting with options, agents, and rounds by introducing the conflict-number . It shows that when sublinear dissatisfaction is achievable and provides both an EW-based strategy with a sublinear bound and a non-constructive Kolmogorov-complexity lower bound indicating near-tightness, along with an explicit prediction-with-expert-advice–style method achieving similar guarantees. It also establishes lower bounds demonstrating that sublinear guarantees fail without the bounded-conflicts condition and that simple voting rules perform poorly in this regime. By connecting perpetual voting to prediction-theory methods and complexity theory, the paper provides both practical strategies for fair long-horizon decisions and fundamental limits on what can be guaranteed in dynamic voting environments.

Abstract

In perpetual voting, multiple decisions are made at different moments in time. Taking the history of previous decisions into account allows us to satisfy properties such as proportionality over periods of time. In this paper, we consider the following question: is there a perpetual approval voting method that guarantees that no voter is dissatisfied too many times? We identify a sufficient condition on voter behavior -- which we call 'bounded conflicts' condition -- under which a sublinear growth of dissatisfaction is possible. We provide a tight upper bound on the growth of dissatisfaction under bounded conflicts, using techniques from Kolmogorov complexity. We also observe that the approval voting with binary choices mimics the machine learning setting of prediction with expert advice. This allows us to present a voting method with sublinear guarantees on dissatisfaction under bounded conflicts, based on the standard techniques from prediction with expert advice.
Paper Structure (7 sections, 6 theorems, 33 equations)

This paper contains 7 sections, 6 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Our contribution.
  3. Formal setting and contributions
  4. Proof of Theorem \ref{['thm_combi']}
  5. Proof of Theorem \ref{['thm_kolmogorov']}
  6. Proof of Theorem \ref{['thm_lower']}
  7. Acknowledgments

Key Result

Theorem 1

For every $k$ there exists a constant $W > 0$ that for every $N, T, C$ there exists a strategy in the $C$-conflict perpetual voting with $k$ options, $N$ agents, and $T$ rounds that guarantees dissatisfaction at most $T^{1 - 1/k}\cdot C^{1/k} \cdot (\ln N\cdot \ln C)^{W}$.

Theorems & Definitions (10)

  • Theorem 1
  • Remark 1
  • Proposition 1
  • proof
  • Theorem 2
  • Theorem 3
  • Proposition 2
  • proof
  • Lemma 1
  • proof