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Computational Aspects of Bayesian Persuasion under Approximate Best Response

Kunhe Yang, Hanrui Zhang

TL;DR

This paper studies Bayesian persuasion under approximate best-response behavior, focusing on robustness to suboptimal receivers and the resulting computational challenges. It shows that the revelation principle fails in this setting, and develops an LP-based framework to compute robust signaling schemes efficiently when either the action space or the state space is small, plus a quasi-polynomial-time approximation scheme for the general case. The authors prove NP-hardness of exact optimization in the general setting and provide a detailed algorithmic toolkit, including a connected-subset enumeration approach for small-state cases and a QPTAS that discretizes posterior utilities into cells. The work advances robust information design by delivering practical algorithms and hardness results, with potential applicability to principal-agent problems where robustness to behavior is essential.

Abstract

We study Bayesian persuasion under approximate best response, where the receiver may choose any action that is not too much suboptimal given their posterior belief upon receiving the signal. We focus on the computational aspects of the problem, aiming to design algorithms that efficiently compute (almost) optimal strategies for the sender. Despite the absence of the revelation principle -- which has been one of the most powerful tools in Bayesian persuasion -- we design polynomial-time exact algorithms for the problem when either the state space or the action space is small, as well as a quasi-polynomial-time approximation scheme (QPTAS) for the general problem. On the negative side, we show there is no polynomial-time exact algorithm for the general problem unless $\mathsf{P} = \mathsf{NP}$. Our results build on several new algorithmic ideas, which might be useful in other principal-agent problems where robustness is desired.

Computational Aspects of Bayesian Persuasion under Approximate Best Response

TL;DR

This paper studies Bayesian persuasion under approximate best-response behavior, focusing on robustness to suboptimal receivers and the resulting computational challenges. It shows that the revelation principle fails in this setting, and develops an LP-based framework to compute robust signaling schemes efficiently when either the action space or the state space is small, plus a quasi-polynomial-time approximation scheme for the general case. The authors prove NP-hardness of exact optimization in the general setting and provide a detailed algorithmic toolkit, including a connected-subset enumeration approach for small-state cases and a QPTAS that discretizes posterior utilities into cells. The work advances robust information design by delivering practical algorithms and hardness results, with potential applicability to principal-agent problems where robustness to behavior is essential.

Abstract

We study Bayesian persuasion under approximate best response, where the receiver may choose any action that is not too much suboptimal given their posterior belief upon receiving the signal. We focus on the computational aspects of the problem, aiming to design algorithms that efficiently compute (almost) optimal strategies for the sender. Despite the absence of the revelation principle -- which has been one of the most powerful tools in Bayesian persuasion -- we design polynomial-time exact algorithms for the problem when either the state space or the action space is small, as well as a quasi-polynomial-time approximation scheme (QPTAS) for the general problem. On the negative side, we show there is no polynomial-time exact algorithm for the general problem unless . Our results build on several new algorithmic ideas, which might be useful in other principal-agent problems where robustness is desired.
Paper Structure (35 sections, 9 theorems, 59 equations, 4 figures, 1 table, 2 algorithms)

This paper contains 35 sections, 9 theorems, 59 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Results and Techniques
  3. Direct-revalation signaling schemes are suboptimal.
  4. LP formulation, and efficient algorithm with small action spaces.
  5. Efficient algorithm with small state spaces.
  6. Hardness of exact computation of the general problem.
  7. Approximation in quasi-polynomial time.
  8. Related Works
  9. Bayesian persuasion
  10. Computational aspects of Bayesian persuasion and information design
  11. Robust Bayesian persuasion
  12. Preliminaries
  13. Bayesian persuasion: the classical model
  14. Basic setting and notations.
  15. Signaling scheme.
  16. ...and 20 more sections

Key Result

Proposition 3.1

[prop]prop:direct-fails There exists a sequence of Bayesian persuasion instances with a robustness level $\delta=\Theta(1)$, such that the following holds in the limit: any direct-revelation scheme $\varphi$ is suboptimal, at least by a factor of $2$ or an additive gap of $\frac{1}{2}$. That is, for

Figures (4)

  • Figure 1: A program for the optimal robust signaling scheme supported on $\Sigma=\{(A,\tilde{a})\}$.
  • Figure 2: Relaxed LP for the optimal robust signaling scheme
  • Figure 3: QPTAS for computing an $\varepsilon$-approximate robust signaling scheme, where $\varepsilon'=\frac{\varepsilon}{5}$.
  • Figure 4: LP for checking the feasibility of $(A,\tilde{a})\in\Sigma$. The tuple is feasible if and only if $\boldsymbol{\varepsilon}^\star>0$.

Theorems & Definitions (28)

  • Example 1.1
  • Definition 2.1: $\delta$-best response receiver strategies
  • Remark 2.2: worst-case $\delta$-BR strategy
  • Remark 2.3: Strict inequality in definition of $\textsf{BR}_\delta$ set
  • Proposition 3.1: Suboptimality of direct-revelation schemes
  • Example 3.2
  • Lemma 4.1
  • proof
  • Proposition 4.2: Efficient algorithm for small action space
  • Definition 5.1: Feasible subset-action tuple
  • ...and 18 more