Privacy Guarantee for Nash Equilibrium Computation of Aggregative Games Based on Pointwise Maximal Leakage

TL;DR

This work introduces a Pointwise Maximal Leakage (PML) framework to quantify privacy leakage when computing Nash equilibria in aggregative games, leveraging prior knowledge about players' cost functions. It derives a computable upper bound on privacy leakage for iterative NE algorithms under Laplace noise and shows that PML can offer tighter guarantees than Differential Privacy (DP) in the presence of correlations. The paper provides both entire-view and individual-view analyses: in the former, DP implies a weaker PML bound, while in the latter, carefully constructed correlated datasets can yield PML leakage exceeding DP bounds. Numerical experiments on a contagion-vaccination game validate that PML better captures real privacy risks under correlation and prior information. Collectively, the results establish PML as a more appropriate privacy metric for NE computation in correlated, prior-aware settings and offer practical guidance for privacy-preserving algorithm design.

Abstract

Privacy preservation has served as a key metric in designing Nash equilibrium (NE) computation algorithms. Although differential privacy (DP) has been widely employed for privacy guarantees, it does not exploit prior distributional knowledge of datasets and is ineffective in assessing information leakage for correlated datasets. To address these concerns, we establish a pointwise maximal leakage (PML) framework when computing NE in aggregative games. By incorporating prior knowledge of players' cost function datasets, we obtain a precise and computable upper bound of privacy leakage with PML guarantees. In the entire view, we show PML refines DP by offering a tighter privacy guarantee, enabling flexibility in designing NE computation. Also, in the individual view, we reveal that the lower bound of PML can exceed the upper bound of DP by constructing specific correlated datasets. The results emphasize that PML is a more proper privacy measure than DP since the latter fails to adequately capture privacy leakage in correlated datasets. Moreover, we conduct experiments with adversaries who attempt to infer players' private information to illustrate the effectiveness of our framework.

Figures (3)

• Figure 1: Value of the gain function $g(\theta,\hat{\theta})$ under different noise levels. The blue line, red line, yellow line, and purple line represent the adversary's gain values when the privacy-preserving algorithm incorporates no noise, low noise ($M^k=1+0.5k$), medium noise ($M^k=1+k$), and high noise ($M^k=1+1.5k$), respectively.
• Figure 2: Comparison between PML and DP in the entire view. The orange line and the red line show the upper bound of the PML guarantees and the DP guarantees for groups, respectively. The blue line denotes the real information leakage when facing the adversary in zhu2019deep with the gain function (\ref{['eq::gain_exp']}).
• Figure 3: Comparison between PML and DP in the individual view. The red line and the green dot line represent an upper bound and a lower bound of PML guarantees in the individual view, according to Theorem \ref{['th::compare_DP_PML_lower_boundd']}. The red line describes the upper bound of the DP guarantees that remains constant, while correlation increases with the number of players. The black dotted line shows the upper bound of the privacy leakage, $\epsilon_{\max }\left(F_1\right)$, which is approached by the upper and lower PML bounds. The blue line denotes the real information leakage of the $1$st player when facing the adversary in zhu2019deep with the gain function (\ref{['eq::gain_exp']}). This experiment shows that DP is not able to capture the effect of correlations on the privacy risk.

Theorems & Definitions (19)

• Definition 1: Nash equilibrium
• Remark 1
• Definition 2: Pointwise maximal lemakage
• Proposition 1: saeidian2023pointwisegeneral
• Definition 3: $\epsilon$-pointwise maximal leakage
• Definition 4: $\epsilon$-differential privacy
• Definition 5
• Proposition 2: dwork2014algorithmic
• Lemma 1
• Lemma 2