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Multi-Sender Persuasion: A Computational Perspective

Safwan Hossain, Tonghan Wang, Tao Lin, Yiling Chen, David C. Parkes, Haifeng Xu

TL;DR

It is proved that finding an equilibrium in general is PPAD-Hard; in fact, even computing a sender's best response is NP-Hard; and a novel differentiable neural network is proposed to approximate this game's non-linear and discontinuous utilities.

Abstract

We consider the multi-sender persuasion problem: multiple players with informational advantage signal to convince a single self-interested actor to take certain actions. This problem generalizes the seminal Bayesian Persuasion framework and is ubiquitous in computational economics, multi-agent learning, and multi-objective machine learning. The core solution concept here is the Nash equilibrium of senders' signaling policies. Theoretically, we prove that finding an equilibrium in general is PPAD-Hard; in fact, even computing a sender's best response is NP-Hard. Given these intrinsic difficulties, we turn to finding local Nash equilibria. We propose a novel differentiable neural network to approximate this game's non-linear and discontinuous utilities. Complementing this with the extra-gradient algorithm, we discover local equilibria that Pareto dominates full-revelation equilibria and those found by existing neural networks. Broadly, our theoretical and empirical contributions are of interest to a large class of economic problems.

Multi-Sender Persuasion: A Computational Perspective

TL;DR

It is proved that finding an equilibrium in general is PPAD-Hard; in fact, even computing a sender's best response is NP-Hard; and a novel differentiable neural network is proposed to approximate this game's non-linear and discontinuous utilities.

Abstract

We consider the multi-sender persuasion problem: multiple players with informational advantage signal to convince a single self-interested actor to take certain actions. This problem generalizes the seminal Bayesian Persuasion framework and is ubiquitous in computational economics, multi-agent learning, and multi-objective machine learning. The core solution concept here is the Nash equilibrium of senders' signaling policies. Theoretically, we prove that finding an equilibrium in general is PPAD-Hard; in fact, even computing a sender's best response is NP-Hard. Given these intrinsic difficulties, we turn to finding local Nash equilibria. We propose a novel differentiable neural network to approximate this game's non-linear and discontinuous utilities. Complementing this with the extra-gradient algorithm, we discover local equilibria that Pareto dominates full-revelation equilibria and those found by existing neural networks. Broadly, our theoretical and empirical contributions are of interest to a large class of economic problems.
Paper Structure (25 sections, 10 theorems, 62 equations, 8 figures, 1 table)

This paper contains 25 sections, 10 theorems, 62 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Additional Related Work
  3. Model
  4. Preliminaries
  5. Persuasion
  6. Theoretical Results
  7. Best Response
  8. Equilibrium Characterization
  9. Deep Learning for Local Equilibrium
  10. Method
  11. Empirical Results
  12. Didactic Example
  13. Synthetic Benchmark
  14. Real-World Scenarios
  15. Discussion
  16. ...and 10 more sections

Key Result

Proposition 1

The sender's utility function $\overline{u}_i(\bm \pi)$ is discontinuous and piecewise non-linear in $(\pi_1, \dots, \pi_n)$. Fixing $\bm \pi_{-i}$, $\overline{u}_i(\pi_i, \bm \pi_{-i})$ is discontinuous and piecewise linear in $\pi_i$.

Figures (8)

  • Figure 1: Discontinuous utility functions in a multi-sender persuasion game with 2 senders, 2 signals, 2 actions, and 2 states. In each subplot: the x-axis represents the probability of $\mathtt{Sender 1}$ transmitting $\mathtt{Signal 1}$ at $\mathtt{State 1}$, the y-axis shows the probability of $\mathtt{Sender 2}$ emitting $\mathtt{Signal 1}$ at $\mathtt{State 1}$, and the z-axis quantifies $\mathtt{Sender 2}$'s utility. Signaling strategies of both senders at $\mathtt{State 2}$ are set to $(0.5, 0.5)$ in the top row and to $(0.2, 0.8)$ and $(0.8, 0.2)$ in the bottom row. In each column, we show the groundtruth ex-ante utility, and the approximation results achieved by our method, ReLU, and DeLU wang2023deep networks, respectively.
  • Figure 2: Our method finds better $\epsilon$-local Nash equilibrium than the baseline DeLU wang2023deep and ReLU networks.
  • Figure 3: The $\epsilon$-local Nash equilibria found by our method typically Pareto dominate the full revelation equilibria and improve the random initial policies of extra-gradient by a large margin.
  • Figure 4: Our network achieves lower approximation errors compared to baseline network structures.
  • Figure 5: Our method achieves higher social welfare compared against baselines and full-revelation solutions in games with 4 senders.
  • ...and 3 more figures

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Proposition 1: Discontinuous Utility
  • Proposition 1: Best Response Program
  • Theorem 2: NP-hardness of Best Response
  • Theorem 3: Full-Revelation Equilibrium
  • Theorem 4: PPAD-Hardness
  • Definition 3: $\epsilon$-Local Nash Equilibrium
  • Definition 4: Activation Pattern
  • proof
  • ...and 23 more