Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Differentially Private Condorcet Voting

Zhechen Li, Ao Liu, Lirong Xia, Yongzhi Cao, Hanpin Wang

TL;DR

The paper tackles the challenge of designing Condorcet-based voting rules that protect voter privacy via differential privacy. It introduces the CM^{Rand}_λ family, instantiated as CM^{LAP}_λ, CM^{EXP}_λ, and CM^{RR}_λ, which noise pairwise comparisons to achieve DP. The authors establish DP guarantees (ε-eDP bounds), analyze axiom satisfaction (absolute monotonicity, lexi-participation, p-Pareto, p-Condorcet variants, and SD-strategyproofness), and reveal fundamental tradeoffs between privacy and core voting axioms (Condorcet and Pareto), including how DP interacts with these properties when λ and ε vary. They also discuss DP as a voting axiom, showing incompatibilities (e.g., no ε-DP rule can satisfy Condorcet or Pareto) and deriving bounds that connect privacy to probabilistic guarantees on outcomes. Overall, the work provides a principled framework for private Condorcet voting and sets direction for extending DP to other social-choice settings.

Abstract

Designing private voting rules is an important and pressing problem for trustworthy democracy. In this paper, under the framework of differential privacy, we propose a novel famliy of randomized voting rules based on the well-known Condorcet method, and focus on three classes of voting rules in this family: Laplacian Condorcet method ($\CMLAP_λ$), exponential Condorcet method ($\CMEXP_λ$), and randomized response Condorcet method ($\CMRR_λ$), where $λ$ represents the level of noise. We prove that all of our rules satisfy absolute monotonicity, lexi-participation, probabilistic Pareto efficiency, approximate probabilistic Condorcet criterion, and approximate SD-strategyproofness. In addition, $\CMRR_λ$ satisfies (non-approximate) probabilistic Condorcet criterion, while $\CMLAP_λ$ and $\CMEXP_λ$ satisfy strong lexi-participation. Finally, we regard differential privacy as a voting axiom, and discuss its relations to other axioms.

Differentially Private Condorcet Voting

TL;DR

The paper tackles the challenge of designing Condorcet-based voting rules that protect voter privacy via differential privacy. It introduces the CM^{Rand}_λ family, instantiated as CM^{LAP}_λ, CM^{EXP}_λ, and CM^{RR}_λ, which noise pairwise comparisons to achieve DP. The authors establish DP guarantees (ε-eDP bounds), analyze axiom satisfaction (absolute monotonicity, lexi-participation, p-Pareto, p-Condorcet variants, and SD-strategyproofness), and reveal fundamental tradeoffs between privacy and core voting axioms (Condorcet and Pareto), including how DP interacts with these properties when λ and ε vary. They also discuss DP as a voting axiom, showing incompatibilities (e.g., no ε-DP rule can satisfy Condorcet or Pareto) and deriving bounds that connect privacy to probabilistic guarantees on outcomes. Overall, the work provides a principled framework for private Condorcet voting and sets direction for extending DP to other social-choice settings.

Abstract

Designing private voting rules is an important and pressing problem for trustworthy democracy. In this paper, under the framework of differential privacy, we propose a novel famliy of randomized voting rules based on the well-known Condorcet method, and focus on three classes of voting rules in this family: Laplacian Condorcet method (), exponential Condorcet method (), and randomized response Condorcet method (), where represents the level of noise. We prove that all of our rules satisfy absolute monotonicity, lexi-participation, probabilistic Pareto efficiency, approximate probabilistic Condorcet criterion, and approximate SD-strategyproofness. In addition, satisfies (non-approximate) probabilistic Condorcet criterion, while and satisfy strong lexi-participation. Finally, we regard differential privacy as a voting axiom, and discuss its relations to other axioms.
Paper Structure (9 sections, 16 theorems, 102 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 9 sections, 16 theorems, 102 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. Related work and discussions.
  4. Preliminaries
  5. Axioms of voting.
  6. Differentially Private Condorcet Methods
  7. Axioms-Privacy Tradeoff
  8. Differential Privacy as a Voting Axiom
  9. Conclusion and Future Work

Key Result

Lemma 1

For any $\texttt{Rand}\in \{ \text{\rm LAP}, \text{\rm EXP}, \text{\rm RR} \}$ and $\lambda>0$, $\operatorname{CM}^{\texttt{Rand}}_\lambda$ can be sampled as follows:

Figures (3)

  • Figure 1: The lower and upper bounds of privacy budget (left: $m=5$, right: $m=20$).
  • Figure 2: The satisfaction of p-Condorcet with different $\lambda$.
  • Figure 3: Relations between $\epsilon$-DP and other axioms, where $X\Rightarrow Y$ indicates that $X$ implies $Y$, a solid line between $X$ and $Y$ indicates that $X,Y$ are compatible with some condition, and a dash line between $X$ and $Y$ means that $X,Y$ are incompatible.

Theorems & Definitions (41)

  • Definition 1: Differential privacy
  • Definition 2: Exact differential privacy Dwork06D
  • Remark
  • Definition 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Definition 4: Probabilistic Condorcet criterion
  • Theorem 2
  • ...and 31 more