Differentially Private Approval-Based Committee Voting

TL;DR

This work studies how differential privacy ($\epsilon$-DP) constrains approval-based committee voting axioms (JR, PJR, EJR, PE, CC). It first proves incompatibility between DP and exact satisfaction of these axioms, then derives upper and lower bounds for two-way tradeoffs with approximate versions (e.g., $\theta$-JR, $\rho$-PJR, $\kappa$-EJR, $\beta$-PE, $\eta$-CC). It then analyzes three-way tradeoffs among DP and pairs of axioms, showing DP adds further limitations even among axioms that are compatible without DP, while highlighting the special incompatibility between Condorcet and JR-family axioms regardless of DP. The results provide quantitative guidance for designing DP-robust ABC rules and motivate future work on tighter bounds and efficient, private mechanisms. Overall, the paper advances understanding of the privacy-axiom frontier in multiwinner voting and informs practical rule design under privacy constraints.

Abstract

In this paper, we investigate tradeoffs among differential privacy (DP) and several representative axioms for approval-based committee voting, including justified representation, proportional justified representation, extended justified representation, Pareto efficiency, and Condorcet criterion. Without surprise, we demonstrate that all of these axioms are incompatible with DP, and thus establish both upper and lower bounds for their two-way tradeoffs with DP. Furthermore, we provide upper and lower bounds for three-way tradeoffs among DP and every pairwise combination of such axioms, revealing that although these axioms are compatible without DP, their optimal levels under DP cannot be simultaneously achieved. Our results quantify the effect of DP on the satisfaction and compatibility of the axioms in approval-based committee voting, which can provide insights for designing voting rules that possess both privacy and axiomatic properties.

Figures (4)

• Figure 1: Two-way tradeoffs between DP and axioms.
• Figure 2: Three-way tradeoffs among DP and axioms (the dash line indicates that standard Condorcet criterion and justified representation are incompatible, so there is no 3-way tradeoff for them).
• Figure 3: Diagram corresponding to the profile $P$.
• Figure 4: Compatibility between axioms, where a solid line indicates that the axioms are compatible, while a dash line indicates that the axioms are incompatible.

Theorems & Definitions (24)

• Definition 1: $\epsilon$-Differential privacy, $\epsilon$-DP for short Dwork06D
• Lemma 1
• Corollary 1
• Definition 2: $\theta$-Justified representation, $\theta$-JR for short
• Theorem 1: $\theta$-JR, upper bound
• Definition 3: $\rho$-Proportional justified representation, $\rho$- PJR for short
• Theorem 2: $\rho$-PJR, upper bound
• Definition 4: $\kappa$-Extended justified representation, $\kappa$-EJR for short
• Theorem 3: $\kappa$-EJR, upper bound
• Proposition 1