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Irrationality as a mean of regularization in Bayesian Persuasion

Romain Duboscq, Frédéric de Gournay

TL;DR

This work extends Bayesian Persuasion by introducing a divergence-based regularization in the receiver’s decision problem, modeling irrational behavior and smoothing the optimization landscape. The authors show that the regularized problem admits unique, tractable solutions and that, under mild assumptions, these solutions converge to those of the original problem as the regularization parameter goes to zero, while enabling efficient computation via quasi-Newton methods and Softmax-based parameterizations. They develop a concavification framework and a representation via measure mixtures, establish existence and structure results, and illustrate the approach with analytical and numerical examples, supported by the BASIL Python library for reproducible signaling experiments. The methodology provides practical tools for designing signaling schemes in complex environments where the classical formulation is intractable or ill-posed, with clear guidance on choosing regularization parameters and message counts through a regulatory revelation principle.

Abstract

We study a regularized variant of the Bayesian Persuasion problem, where the receiver's decision process includes a divergence-based penalty that accounts for deviations from perfect rationality. This modification smooths the underlying optimization landscape and mitigates key theoretical issues, such as measurability and ill-posedness, commonly encountered in the classical formulation. It also enables the use of scalable second-order optimization methods to compute numerically the optimal signaling scheme in a setting known to be NP-hard. We present theoretical results comparing the regularized and original models, including convergence guarantees and structural properties of optimal signaling schemes. Analytical examples and numerical simulations illustrate how this framework accommodates complex environments while remaining tractable and robust. A companion Python library, BASIL, makes use of all the practical insights from this article.

Irrationality as a mean of regularization in Bayesian Persuasion

TL;DR

This work extends Bayesian Persuasion by introducing a divergence-based regularization in the receiver’s decision problem, modeling irrational behavior and smoothing the optimization landscape. The authors show that the regularized problem admits unique, tractable solutions and that, under mild assumptions, these solutions converge to those of the original problem as the regularization parameter goes to zero, while enabling efficient computation via quasi-Newton methods and Softmax-based parameterizations. They develop a concavification framework and a representation via measure mixtures, establish existence and structure results, and illustrate the approach with analytical and numerical examples, supported by the BASIL Python library for reproducible signaling experiments. The methodology provides practical tools for designing signaling schemes in complex environments where the classical formulation is intractable or ill-posed, with clear guidance on choosing regularization parameters and message counts through a regulatory revelation principle.

Abstract

We study a regularized variant of the Bayesian Persuasion problem, where the receiver's decision process includes a divergence-based penalty that accounts for deviations from perfect rationality. This modification smooths the underlying optimization landscape and mitigates key theoretical issues, such as measurability and ill-posedness, commonly encountered in the classical formulation. It also enables the use of scalable second-order optimization methods to compute numerically the optimal signaling scheme in a setting known to be NP-hard. We present theoretical results comparing the regularized and original models, including convergence guarantees and structural properties of optimal signaling schemes. Analytical examples and numerical simulations illustrate how this framework accommodates complex environments while remaining tractable and robust. A companion Python library, BASIL, makes use of all the practical insights from this article.

Paper Structure

This paper contains 42 sections, 10 theorems, 128 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Notations and preliminaries
  4. Agents and variables
  5. The receivers' problem : choosing the action
  6. The sender's problem : choosing the message
  7. Limitations of the model
  8. The concavification of the sender's problem and the revelation principle
  9. Regularization by strict convexity
  10. Problems with the standard formulation
  11. Regularization
  12. Main results
  13. On the concavification
  14. On the regularization
  15. Numerical considerations
  16. ...and 27 more sections

Key Result

Lemma 1

Let $M\in\mathbb{N}^*$. The spaces $\mathsf{T}_{\mu}$ and $\mathsf{T}_{M,\mu}$ are compact.

Figures (7)

  • Figure 1: The effects of regularization on the standard example of the judge and prosecutor for $\varepsilon = 0.1$ and different irrational strategies.
  • Figure 2: The voting problem. A point in the triangle correspond to an element of $\mathcal{P}(\mathcal{S})$, the pure states are the vertices. The coloring corresponds to the different strategies of the receivers (voters) for a given state. The black diamond represents $\mu$ and the three stars represents the different $\nu$ for the three messages. Each star is located in one of the small triangle where the majority of voters approve the bill. In these triangles, the utility of the sender is $1$, it is $0$ elsewhere.
  • Figure 3: The voting problem. On the left we display the evolution of $\nu_m$ through the iterations, the starting point is a diamond and the ending point is a circle. On the right, we display the value of $p_m$, the probability that the message is sent. When the said probability gets under $1\%$, the curves are dashed and the circle of the corresponding ending $\nu$ is not displayed. The number of messages is respectively 3,6,9 from the top to the bottom and the optimal number of messages is $3$. On the middle, the three extra messages are not used whereas in the bottom, the algorithm ends up with more messages than needed, the gold, pink and brown messages are the same. The algorithm always end with the correct solution (dirac masses on each middle of the faces).
  • Figure 4: Numerical convergence of the minimizer with respect to $\varepsilon$. For different $\varepsilon$, we compute the optimal signals which are represented as pairs of stars connected by a dashed line (which passes through the prior, represented by a black diamond). Their color depends on the value of $\varepsilon$: the larger the $\varepsilon$, the darker the color.
  • Figure 5: Prior for the hunter problem. The number in the upper-half (resp. the lower-half) of each square is the probability (rounded up) of the presence of a single deer (resp. two deers). Special attention must be given to the upper right cell which is a protected area, close to the cabin and with a high probability of deers.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Remark 3
  • Lemma 3
  • Corollary 1
  • Theorem 1
  • Lemma 4
  • Corollary 2
  • ...and 4 more