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Persuading a Behavioral Agent: Approximately Best Responding and Learning

Yiling Chen, Tao Lin

TL;DR

The paper investigates Bayesian persuasion when the receiver can approximately best respond, introducing a robustification framework to bound the sender's utility. Under a key assumption that each action is uniquely optimal in some state with a positive gap, the authors show that the sender can attain a utility close to the classic optimal value against any gamma-delta approximately best-responding receiver, while no signaling scheme can significantly exceed the classic benchmark even when the receiver responds advantageously. The core technique, robustification, perturbs optimal schemes to encourage obedient, well-behaved responses with minimal loss to the sender's payoff. The work extends to learning receivers in repeated settings, showing that the sender can perform nearly as well as in the non-learning model, and, in some learning scenarios, can even outperform the classic baseline, highlighting nuances when agents learn and adapt.

Abstract

The classic Bayesian persuasion model assumes a Bayesian and best-responding receiver. We study a relaxation of the Bayesian persuasion model where the receiver can approximately best respond to the sender's signaling scheme. We show that, under natural assumptions, (1) the sender can find a signaling scheme that guarantees itself an expected utility almost as good as its optimal utility in the classic model, no matter what approximately best-responding strategy the receiver uses; (2) on the other hand, there is no signaling scheme that gives the sender much more utility than its optimal utility in the classic model, even if the receiver uses the approximately best-responding strategy that is best for the sender. Together, (1) and (2) imply that the approximately best-responding behavior of the receiver does not affect the sender's maximal achievable utility a lot in the Bayesian persuasion problem. The proofs of both results rely on the idea of robustification of a Bayesian persuasion scheme: given a pair of the sender's signaling scheme and the receiver's strategy, we can construct another signaling scheme such that the receiver prefers to use that strategy in the new scheme more than in the original scheme, and the two schemes give the sender similar utilities. As an application of our main result (1), we show that, in a repeated Bayesian persuasion model where the receiver learns to respond to the sender by some algorithms, the sender can do almost as well as in the classic model. Interestingly, unlike (2), with a learning receiver the sender can sometimes do much better than in the classic model.

Persuading a Behavioral Agent: Approximately Best Responding and Learning

TL;DR

The paper investigates Bayesian persuasion when the receiver can approximately best respond, introducing a robustification framework to bound the sender's utility. Under a key assumption that each action is uniquely optimal in some state with a positive gap, the authors show that the sender can attain a utility close to the classic optimal value against any gamma-delta approximately best-responding receiver, while no signaling scheme can significantly exceed the classic benchmark even when the receiver responds advantageously. The core technique, robustification, perturbs optimal schemes to encourage obedient, well-behaved responses with minimal loss to the sender's payoff. The work extends to learning receivers in repeated settings, showing that the sender can perform nearly as well as in the non-learning model, and, in some learning scenarios, can even outperform the classic baseline, highlighting nuances when agents learn and adapt.

Abstract

The classic Bayesian persuasion model assumes a Bayesian and best-responding receiver. We study a relaxation of the Bayesian persuasion model where the receiver can approximately best respond to the sender's signaling scheme. We show that, under natural assumptions, (1) the sender can find a signaling scheme that guarantees itself an expected utility almost as good as its optimal utility in the classic model, no matter what approximately best-responding strategy the receiver uses; (2) on the other hand, there is no signaling scheme that gives the sender much more utility than its optimal utility in the classic model, even if the receiver uses the approximately best-responding strategy that is best for the sender. Together, (1) and (2) imply that the approximately best-responding behavior of the receiver does not affect the sender's maximal achievable utility a lot in the Bayesian persuasion problem. The proofs of both results rely on the idea of robustification of a Bayesian persuasion scheme: given a pair of the sender's signaling scheme and the receiver's strategy, we can construct another signaling scheme such that the receiver prefers to use that strategy in the new scheme more than in the original scheme, and the two schemes give the sender similar utilities. As an application of our main result (1), we show that, in a repeated Bayesian persuasion model where the receiver learns to respond to the sender by some algorithms, the sender can do almost as well as in the classic model. Interestingly, unlike (2), with a learning receiver the sender can sometimes do much better than in the classic model.
Paper Structure (30 sections, 7 theorems, 70 equations, 1 figure)

This paper contains 30 sections, 7 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The worst way of approximately best responding
  3. The best way of approximately best responding
  4. Application: persuading a learning receiver
  5. Related Works
  6. Model: Approximately Best Responding Agent
  7. Receiver's behavior
  8. Sender's objective
  9. Restrict attentions to $\gamma$-best-responding strategies
  10. Direct-revelation schemes, obedient strategy, and "advantage"
  11. Main Result: Persuading an Approximately Best Responding Agent
  12. Overview of the Proof of Theorem \ref{['thm:BP-robustness']}
  13. Main Idea: Robustification
  14. Lower Bounding: Proof of Lemma \ref{['lem:lower_bound']}
  15. Upper Bounding: Proof of Lemma \ref{['lem:upper_bound']}
  16. ...and 15 more sections

Key Result

Lemma 2.1

Fix a signaling scheme $\pi$. For any $(\gamma, \delta)$-best-responding receiver strategy $\rho$, there exists a $\gamma$-best-responding receiver strategy $\tilde{\rho}$ such that $| U_\mu(\pi, \rho) - U_\mu(\pi, \tilde{\rho}) | \le \delta$.

Figures (1)

  • Figure 1: Markov chains

Theorems & Definitions (23)

  • Example 1.1
  • Example 1.2
  • Definition 2.1
  • Example 2.1
  • Lemma 2.1
  • Definition 2.2
  • Theorem 3.1
  • Claim 3.2
  • Lemma 3.3: Lower bound
  • Lemma 3.4: Upper bound
  • ...and 13 more