Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Differentially Private Equilibrium Finding in Polymatrix Games

Mingyang Liu, Gabriele Farina, Asuman Ozdaglar

TL;DR

The paper tackles the problem of computing equilibria in polymatrix games under differential privacy constraints. It proves that, under strong adversaries or when accuracy is measured by Euclidean distance to the equilibrium, high accuracy and vanishing DP budget cannot co-occur, establishing fundamental limits. Focusing on a realistic setting where the adversary has a constant number of channels, it introduces a distributed algorithm that combines noisy strategy sharing with proximity-based proximal updates and an adaptive regularizer tied to the graph degrees. The analysis shows convergence to a coarse correlated equilibrium with diminishing exploitability and a DP budget that improves as the number of players grows, with distinct behavior for dense and sparse graphs. Experiments corroborate the theoretical predictions, highlighting the practical viability of achieving privacy-conscious equilibrium computation in large polymatrix games, especially in dense graphs.

Abstract

We study equilibrium finding in polymatrix games under differential privacy constraints. To start, we show that high accuracy and asymptotically vanishing differential privacy budget (as the number of players goes to infinity) cannot be achieved simultaneously under either of the two settings: (i) We seek to establish equilibrium approximation guarantees in terms of Euclidean distance to the equilibrium set, and (ii) the adversary has access to all communication channels. Then, assuming the adversary has access to a constant number of communication channels, we develop a novel distributed algorithm that recovers strategies with simultaneously vanishing Nash gap (in expected utility, also referred to as exploitability and privacy budget as the number of players increases.

Differentially Private Equilibrium Finding in Polymatrix Games

TL;DR

The paper tackles the problem of computing equilibria in polymatrix games under differential privacy constraints. It proves that, under strong adversaries or when accuracy is measured by Euclidean distance to the equilibrium, high accuracy and vanishing DP budget cannot co-occur, establishing fundamental limits. Focusing on a realistic setting where the adversary has a constant number of channels, it introduces a distributed algorithm that combines noisy strategy sharing with proximity-based proximal updates and an adaptive regularizer tied to the graph degrees. The analysis shows convergence to a coarse correlated equilibrium with diminishing exploitability and a DP budget that improves as the number of players grows, with distinct behavior for dense and sparse graphs. Experiments corroborate the theoretical predictions, highlighting the practical viability of achieving privacy-conscious equilibrium computation in large polymatrix games, especially in dense graphs.

Abstract

We study equilibrium finding in polymatrix games under differential privacy constraints. To start, we show that high accuracy and asymptotically vanishing differential privacy budget (as the number of players goes to infinity) cannot be achieved simultaneously under either of the two settings: (i) We seek to establish equilibrium approximation guarantees in terms of Euclidean distance to the equilibrium set, and (ii) the adversary has access to all communication channels. Then, assuming the adversary has access to a constant number of communication channels, we develop a novel distributed algorithm that recovers strategies with simultaneously vanishing Nash gap (in expected utility, also referred to as exploitability and privacy budget as the number of players increases.

Paper Structure

This paper contains 34 sections, 12 theorems, 83 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Online Learning with Differential Privacy.
  4. Differential Privacy in Games.
  5. Preliminaries
  6. Basics.
  7. Polymatrix Games.
  8. Differential Privacy.
  9. Necessary Conditions for Achieving High Accuracy and Differential Privacy Simultaneously
  10. Methods
  11. Convergence Analysis
  12. Proof Sketch
  13. Analysis of Differential Privacy
  14. Proof Sketch: the Case of Dense Graphs ($\clubsuit$)
  15. Proof Sketch: the Case of Sparse Graphs ($\spadesuit$)
  16. ...and 19 more sections

Key Result

lemma 1

If an algorithm satisfies $(\alpha, \epsilon)$- DP for $\alpha>1$, then for any $\delta\in (0, 1)$, the algorithm also satisfies $\rbr{\epsilon + \frac{\log\rbr{1/\delta}}{\alpha-1}, \delta}$-DP. Moreover, $(\infty, \epsilon)$- DP is equivalent to $(\epsilon, 0)$-DP.

Figures (2)

  • Figure 1: Experiment results of dense graphs. Each node (player) in the graph will be connected to another node with probability $p$, the connection is sampled i.i.d. for each node. Then, the duplicate edges will be removed. The action set sizes of all players are set to $A$. We can see from the result that the exploitability and the privacy budget both decrease as the number of players increases.
  • Figure 2: Experiment results of sparse graphs. We will randomly generate $cN$ edges in total, with each node appearing $c$ times. Then, duplicate edges and self-loops will be removed. The action set sizes of all players are set to $A$. The result shows that while the exploitability remains unchanged, the exploitability decreases as the number of players increases.

Theorems & Definitions (25)

  • definition 1: Game adjacency
  • definition 2: $(\epsilon, \delta)$-Differential Privacy
  • definition 3: $(\alpha, \epsilon)$- Differential Privacy
  • lemma 1
  • lemma 2
  • lemma 3
  • proposition 1
  • remark 1
  • theorem 1
  • theorem 2
  • ...and 15 more