Explainable Information Design

TL;DR

This paper introduces explainable information design for linear information design with a continuous state in $[0,1]$ and studies $K$-partitional signaling schemes that deterministically map each state-interval to a single signal. It proves a tight worst-case price of explainability of exactly $1/2$, meaning any $K$-partitional policy attains at least half the performance of an unrestricted scheme, and shows this bound is tight. On the algorithmic side, exact optimization is NP-hard, but a polynomial-time $ ext{FPTAS}$ exists for Lipschitz utility functions, and a $1/2$-approximation is achievable for piecewise-constant/Lipschitz utilities by converting an optimal unrestricted policy to a $K$-partition policy via a bi-pooling→partition conversion. The analysis relies on reformulating signaling via mean-preserving contractions, characterizing extreme points as bi-pooling MPCs, and providing a constructive conversion algorithm that preserves a constant fraction of the objective. The results offer a principled trade-off between explainability and performance, with practical implications for designing auditable and transparent information schemes in economics, recommender systems, and policy design. The work also outlines extensions to high-dimensional state spaces and discusses open questions about conditions under which PoE remains 1 and how to handle mass points and discrete states.

Abstract

The optimal signaling schemes in information design (Bayesian persuasion) problems often involve non-explainable randomization or disconnected partitions of state space, which are too intricate to be audited or communicated. We propose explainable information design in the context of information design with a continuous state space, restricting the information designer to use $K$-partitional signaling schemes defined by deterministic and monotone partitions of the state space, where a unique signal is sent for all states in each part. We first prove that the price of explainability (PoE) -- the ratio between the performances of the optimal explainable signaling scheme and unrestricted signaling scheme -- is exactly $1/2$ in the worst case, meaning that partitional signaling schemes are never worse than arbitrary signaling schemes by a factor of 2. We then study the complexity of computing optimal explainable signaling schemes. We show that the exact optimization problem is NP-hard in general. But for Lipschitz utility functions, an $\varepsilon$-approximately optimal explainable signaling scheme can be computed in polynomial time. And for piecewise constant utility functions, we provide an efficient algorithm to find an explainable signaling scheme that provides a $1/2$ approximation to the optimal unrestricted signaling scheme, which matches the worst-case PoE bound. A technical tool we develop is a conversion from any optimal signaling scheme (which satisfies a bi-pooling property) to a partitional signaling scheme that achieves $1/2$ fraction of the expected utility of the former. We use this tool in the proofs of both our PoE result and algorithmic result.

Figures (5)

• Figure 1: Illustration of (non-explainable) optimal information policies in \ref{['example:bi-pooling']}. The rectangle in the top axis represents the state space with uniform prior, and the two rectangles in the bottom axis represent the induced posterior means.
• Figure 2: Illustration of partitional information policies in Example \ref{['ex:continued']}. Left: an optimal $(K = 2)$-partitional information policy. Right: an optimal $(K = 4)$-partitional information policy.
• Figure 3: Illustration of the instance in Section \ref{['sec:proof-PoE-upper-bound']} where $\mathsf{OPT}^{\mathsf{Part}}_{\mathcal{I}}\!\left({K}\right) \le (1/2+\varepsilon) \mathsf{OPT}_{\mathcal{I}}\!\left({K}\right)$. The prior is a discrete distribution on $\{0, 1/2, 1\}$ and the interim utility is $1$ only at posterior means $\mu_1, \mu_2$. Left: an optimal signaling scheme (bi-pooling) that achieves expected utility $1$. Right: an optimal partitional policy that achieves expected utility $1-p = 1/2+\varepsilon$.
• Figure 4: Illustration of the reduction from Partition with $c = (1, 2, 3)$. The black points at the bottom are the elements of the set $X$. The leftmost of these points must be chosen as the centerpoint of an interval, in addition to one point from each pair of the subsequent pairs, corresponding to choosing each $b_j \in \{-1, +1\}$. Once the set of centerpoints has been chosen, the $(n + 1)$-partitional information policy is uniquely determined, covering the entire interval $[0, 1]$ exactly if and only if $b$ is a solution to the Partition instance $c$.
• Figure 5: Illustration of \ref{['ex:high-D']}. Here the gray circles are the point masses of the prior probability distribution $F$, and blue circles are the points where the interim utility function $u$ is strictly positive.

Theorems & Definitions (30)

• Example 1.1
• Example 2.1: Adapted from arieli_optimal_2023
• Definition 2.1: Partitional information policy
• Example 2.2: \ref{['example:bi-pooling']} continued
• Definition 2.2: Price of explainability
• Proposition 3.1: PoE for special utility functions
• Theorem 3.2: PoE for general utility functions
• Lemma 3.3
• Definition 3.1: Bi-pooling arieli_optimal_2023
• Lemma 3.4