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Almost-Bayesian Quadratic Persuasion (Extended Version)

Olivier Massicot, Cédric Langbort

Abstract

In this article, we relax the Bayesianity assumption in the now-traditional model of Bayesian Persuasion introduced by Kamenica & Gentzkow. Unlike preexisting approaches -- which have tackled the possibility of the receiver (Bob) being non-Bayesian by considering that his thought process is not Bayesian yet known to the sender (Alice), possibly up to a parameter -- we let Alice merely assume that Bob behaves 'almost like' a Bayesian agent, in some sense, without resorting to any specific model. Under this assumption, we study Alice's strategy when both utilities are quadratic and the prior is isotropic. We show that, contrary to the Bayesian case, Alice's optimal response may not be linear anymore. This fact is unfortunate as linear policies remain the only ones for which the induced belief distribution is known. What is more, evaluating linear policies proves difficult except in particular cases, let alone finding an optimal one. Nonetheless, we derive bounds that prove linear policies are near-optimal and allow Alice to compute a near-optimal linear policy numerically. With this solution in hand, we show that Alice shares less information with Bob as he departs more from Bayesianity, much to his detriment.

Almost-Bayesian Quadratic Persuasion (Extended Version)

Abstract

In this article, we relax the Bayesianity assumption in the now-traditional model of Bayesian Persuasion introduced by Kamenica & Gentzkow. Unlike preexisting approaches -- which have tackled the possibility of the receiver (Bob) being non-Bayesian by considering that his thought process is not Bayesian yet known to the sender (Alice), possibly up to a parameter -- we let Alice merely assume that Bob behaves 'almost like' a Bayesian agent, in some sense, without resorting to any specific model. Under this assumption, we study Alice's strategy when both utilities are quadratic and the prior is isotropic. We show that, contrary to the Bayesian case, Alice's optimal response may not be linear anymore. This fact is unfortunate as linear policies remain the only ones for which the induced belief distribution is known. What is more, evaluating linear policies proves difficult except in particular cases, let alone finding an optimal one. Nonetheless, we derive bounds that prove linear policies are near-optimal and allow Alice to compute a near-optimal linear policy numerically. With this solution in hand, we show that Alice shares less information with Bob as he departs more from Bayesianity, much to his detriment.
Paper Structure (48 sections, 37 theorems, 293 equations, 4 figures, 2 tables)

This paper contains 48 sections, 37 theorems, 293 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A tractable example
  3. A simple strategic communication problem
  4. When Bob is not Bayesian
  5. Linear policies with noise
  6. An outperforming radius-threshold policy
  7. Discussion and a preview of things to come
  8. General problem of interest
  9. Review of Kamenica & Gentzkow's setup
  10. Approximate Bayesianity
  11. Linear-quadratic persuasion
  12. Approximating Alice's program under an ellipsoidal hypothesis
  13. Tractable robust hypotheses
  14. Rewriting the program under an ellipsoidal hypothesis
  15. The framing programs
  16. ...and 33 more sections

Key Result

Lemma 1

The linear policy achieving the lowest value of eq:exobj (i.e., Alice's "optimal linear policy") is either no- or full-information, with value where $(.)^- = \min(.,0)$. When $k>1/2$ is different than $1$ and $\epsilon$ is large enough, this amounts to $k^2n+\epsilon^2$. For all these $k$, there exists a radius-threshold policy whose value is strictly better.

Figures (4)

  • Figure 1: Plot of all the objectives.
  • Figure 2: Plot of the true cost of the optimal solutions to each program.
  • Figure 3: Plot of the rank of the solution to the pessimistic program.
  • Figure 4: Plot of the various bounds.

Theorems & Definitions (70)

  • Lemma 1
  • Lemma 2: from Tamura
  • Lemma 3
  • Theorem 1
  • Definition 1
  • Lemma 4
  • Lemma 5
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • ...and 60 more