Withholding Verifiable Information

Technical Summary: Withholding Verifiable Information

Problem Statement
This paper investigates disclosure games where a sender possesses verifiable information about a state $\omega \in [0, 1]$ and communicates with a receiver who chooses from a finite set of actions $N = \{1, \dots, n\}$. The sender's preferences are state-independent and strictly increasing in the receiver's action, while the receiver's optimal action depends monotonically on the posterior mean of the state.

The canonical prediction in such disclosure games (e.g., Grossman, 1981; Milgrom, 1981) is "unraveling," leading to full revelation. However, this result relies on the assumption that the receiver's action space is sufficiently flexible (continuous) to respond to marginal belief changes. This paper addresses the gap in understanding when the receiver's action space is discrete. In such settings, small changes in beliefs may not alter the receiver's action, potentially allowing the sender to withhold information (pooling states) without triggering a deviation to a less favorable action. The central question is: What is the structure of equilibrium partitions in these finite-action games, and under what conditions can the sender achieve the payoff attainable under commitment (information design)?

Methodology
The authors analyze Perfect Bayesian Equilibria (PBE) in a disclosure game with a continuous state space and finite actions. The analysis relies on characterizing the set of equilibrium partitions and comparing them to the commitment solution (the optimal information structure if the sender could commit to an experiment).

Key methodological steps include:

1. Partition Characterization: The authors define a partition as obedient if the receiver's posterior mean for each partition element falls within the action's optimality interval, and revelation-proof if no sender type prefers to fully reveal their state to induce a higher action.
2. Laminar Structure: The paper introduces the concept of laminar partitions. A partition is laminar if, for any element $B_i$, its convex hull either contains no other elements or nests lower-indexed elements ($B_j$ where $j < i$) in a specific hierarchical manner. This generalizes the "bi-pooling" property found in linear persuasion literature (e.g., Kleiner et al., 2021; Arieli et al., 2023).
3. Transformation Argument: A core technical contribution is showing that any equilibrium partition can be transformed into a laminar equilibrium partition that induces the same posterior mean distribution (PMD) and yields the same sender payoff. Crucially, laminar partitions are shown to be the "most revelation-proof" among all obedient partitions.
4. Optimization: The sender's problem is reduced to maximizing expected payoff over the set of obedient and revelation-proof laminar partitions. The authors utilize tools from convex analysis and mean-preserving contractions to characterize the solution.

Key Contributions and Results

1. Characterization of Equilibrium Payoffs (Theorem 1):

   • The set of sender equilibrium payoffs is fully characterized by laminar partitional equilibria.
   • There exists a sender-worst equilibrium associated with the fully informative partition (yielding the lowest payoff).
   • There exists a sender-preferred equilibrium associated with an obedient and revelation-proof laminar partition.
   • Any payoff between these extremes is achievable via a laminar equilibrium.

2. Structure of Sender-Preferred Equilibria (Theorem 2):

   • Sender-preferred laminar equilibria are barely obedient: the posterior means of all but the lowest unskipped action are exactly at the lower cutoff of the corresponding action interval.
   • The complexity of these partitions is bounded: action $i$ is recommended on at most $\max\{i-1, 1\}$ intervals.
   • Revelation proofness for barely obedient laminar partitions simplifies to a local condition: it suffices to check that the maximum state in $B_i$ does not exceed the upper cutoff $\gamma_{i+1}$ for specific configurations (when action $i+1$ is skipped or nests $i$).

3. Attainability of the Commitment Payoff:

   • Proposition 1: A commitment solution (optimal under commitment) is implementable in the disclosure game if and only if the associated bi-pooling partition is revelation-proof.
   • Corollary 1: Provides a verifiable condition for implementability based on the maximum state in partition elements.
   • Proposition 2 & Corollary 2: The paper identifies sufficient conditions on model primitives (specifically, the convexity of the sender's utility relative to the action cutoffs and the shape of the prior density) under which the commitment payoff is always attainable. If the sender's utility increases sufficiently fast in the expected state, the commitment constraint is non-binding.
   • Proposition 3: For the specific case of three actions, the paper provides an explicit solution for the sender-preferred equilibrium, showing that if the commitment solution is not implementable, the equilibrium involves pooling non-adjacent states such that the "middle" action is recommended as frequently as revelation proofness allows.

Applications
The authors apply their theoretical framework to two specific settings:

1. Selling with Quality Disclosure: Extending Milgrom (1981) to a setting where the buyer purchases integer units (indivisible product). The authors show that if the buyer's utility satisfies specific risk aversion and prudence conditions ($PU > 2AU$) and the prior is increasing, the seller can achieve the commitment payoff by withholding information, contrasting with the full revelation result in the divisible case.
2. Influencing Voters: An expert discloses information to a voter choosing among three alternatives (status quo, amended bill, original bill). The paper demonstrates a counter-intuitive result: if no commitment solution is implementable, an increase in the voter's inclination toward the expert's preferred alternative (e.g., higher sensitivity to the state for the bill) can actually decrease the expert's equilibrium payoff. This occurs because the tightening of the revelation-proofness constraint on the "middle" action outweighs the direct benefit of the voter's shifted preferences.

Significance and Claims
The paper claims to resolve the ambiguity regarding equilibrium outcomes in disclosure games with finite actions. It establishes that full revelation is not the unique relevant outcome; instead, a rich set of simple equilibria exists, organized by laminar structures. These equilibria allow for the pooling of non-adjacent states, a feature previously associated primarily with commitment or receiver private information.

The authors argue that their results explain the variety of disclosure policies observed in practice, where voluntary disclosure may lead to significant pooling of low and high states, potentially harming the receiver. Furthermore, the paper provides a bridge between information design (commitment) and strategic disclosure (no commitment), identifying specific conditions under which the assumption of commitment is unnecessary because the commitment payoff is naturally achieved in equilibrium. Finally, the analysis suggests that mandatory disclosure can improve receiver welfare by eliminating sender-preferred pooling equilibria, even in environments where a fully revealing equilibrium exists.