Entropy-Regularized Optimal Transport in Information Design

TL;DR

The paper advances multidimensional information design by casting the moment Bayesian persuasion problem as an entropy-regularized, semi-discrete optimal transport task. Optimal information policies correspond to Lipschitz-exposed points of the convex set of fusions $F_\\nu$ and are numerically tractable via Laguerre (power) diagrams, with entropy regularization ensuring robust optimization. The authors establish existence and convergence results for the relaxed and penalized formulations, develop a spatial discretization, and demonstrate the method numerically. They apply the framework to a two-product monopolist, showing substantial revenue gains under unit-demand and additive valuations compared to full information and Lloyd benchmarks, thereby highlighting the practical impact for complex market design problems.

Abstract

In this paper, we explore a scenario where a sender provides an information policy and a receiver, upon observing a realization of this policy, decides whether to take a particular action, such as making a purchase. The sender's objective is to maximize her utility derived from the receiver's action, and she achieves this by careful selection of the information policy. Building on the work of Kleiner et al., our focus lies specifically on information policies that are associated with power diagram partitions of the underlying domain. To address this problem, we employ entropy-regularized optimal transport, which enables us to develop an efficient algorithm for finding the optimal solution. We present experimental numerical results that highlight the qualitative properties of the optimal configurations, providing valuable insights into their structure. Furthermore, we extend our numerical investigation to derive optimal information policies for monopolists dealing with multiple products, where the sender discloses information about product qualities.

Figures (7)

• Figure 1: Convergence of the optimal power diagram for different entropy parameters $\varepsilon=25,5,1,0.2.$ (four right-most plots), where the blur parameter values are given in units of $1/N$, for $N=128$. The function $\Phi$ is plotted on the left-most panel.
• Figure 2: Convergence of the optimal power diagram for a function $\Phi$ (column (i)) with global maxima at $(0.25, 0.75), (0.75,0.75)$ and $(0.5,0.25)$ for regularization parameter $\eta=1\mathrm{e}{-1}, 1\mathrm{e}{-2}, 1\mathrm{e}{-3}, 1\mathrm{e}{-4}, 1\mathrm{e}{-5}$ (columns (ii)-(vi)).
• Figure 3: Convergence of the optimal power diagram for a concave function $\Phi$ (left). One sees how the algorithm automatically pushes cells outside of the relevant unit square to enact the trivial solution. Plotted are the computed power diagrams after iterations $it=1, 2, 4, 8, 16$ (five right-most plots).
• Figure 4: Optimal configurations for the monopolist's problem with unit demand, with prices $p_1=1$ and $p_2=1, 1.25, 1.5, 1.75, 2$ (second row, from left to right) and quality boundaries $\underline{q}=0$ and $\overline{q}=2$. The respective revenue function $R$ for each case is plotted in the first row.
• Figure 5: Optimal configurations for the monopolist's problem with unit demand, prices $p_1=1$ and $p_2=1.25$, upper quality bound $\overline{q}=2$ and, from left to right, lower quality bounds $\underline{q}=0.25,0.5,0.75,1,1.25$ (second row). The respective revenue function $R$ (defined above) is plotted in the first row.

Theorems & Definitions (23)

• Definition 2.1: Shaked and Shanthikumar shaked, Cartier et al. Cartier and Phelps Phelps
• Theorem 2.2: Kleiner et al. Kleiner2
• Definition 2.3
• Theorem 2.4: Kleiner et al. Kleiner2
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Theorem 3.4