Divide and Confer: Aggregating Information without Verification

Problem Statement
This paper investigates the design of information aggregation mechanisms in a setting where a decision-maker (receiver) must make a binary accept/reject decision based on private information held by a large population of senders. The core challenge is "informational division": the aggregate information about an unknown state is divided among $N$ senders, such that as the population grows, the information held by any single sender becomes negligible, yet the total information remains constant. Crucially, the signals observed by senders are unconditionally independent. This independence renders cross-verification impossible; no sender's report can be verified against another's. Furthermore, senders share a common bias toward "acceptance," creating a conflict of interest with the receiver, who prefers "reject" in an intermediate range of states (the "disagreement region"). The authors ask how a receiver can elicit truthful information and maximize her payoff when cross-checking is impossible and information is divided.

Methodology
The authors introduce a novel analytical framework termed "Incentive Compatibility in the Large" (ICL). Rather than solving for optimal mechanisms for finite $N$ (which are generally complex and depend on the full profile of reports), the paper studies the limit as $N \to \infty$.

1. Frequency Domain Reformulation: The authors reframe the mechanism design problem in terms of the normalized empirical frequencies (NEF) of signal realizations. This transforms the domain of the mechanism from the exponentially growing signal space $S^N$ to a fixed-dimensional space of frequency deviations, $h \in \mathbb{R}^K_0$.
2. Limit Characterization: Using the Multidimensional Central Limit Theorem, they show that the NEF converges to a multivariate normal distribution. They define a mechanism as ICL if its outcome distribution can be approximated arbitrarily well by a sequence of finite-$N$ incentive-compatible mechanisms.
3. Limiting Constraints: The paper derives two finite sets of linear constraints that fully characterize the set of ICL mechanisms:
   • (ENV): An envelope condition ensuring that an essentially uninformed sender (in the limit) has no incentive to "nudge" the distribution of reports in any direction.
   • (MON): A monotonicity condition ensuring the probability of acceptance is weakly increasing in the sender's report.
     These conditions serve as the limiting analogues of standard incentive compatibility and interim monotonicity constraints.

Key Results

1. Optimality of Simple Mechanisms: A central finding is that in the large-population limit, it is without loss of optimality to restrict attention to simple mechanisms—those that depend only on the aggregate reported state (the sample mean), rather than the full vector of individual reports. This contrasts with finite populations (even $N=2$), where complex mechanisms utilizing the full report profile can strictly outperform simple ones.
2. The Interval Mechanism: The optimal ICL mechanism is characterized as an interval mechanism. It recommends "accept" if the aggregate state lies within a specific bounded interval $[\underline{\omega}, \overline{\omega}]$ and "reject" otherwise.
   • The lower bound $\underline{\omega}$ is set higher than the sender's preferred threshold to reduce acceptance in the disagreement region.
   • The upper bound $\overline{\omega}$ introduces a "surplus-burning" region where the mechanism rejects even when the state is very high (where all parties prefer acceptance).
3. Surplus Burning as Punishment: The rejection at the upper threshold acts as a punishment for over-reporting. Because senders are essentially uninformed, a lie (over-reporting) shifts the aggregate state slightly. This shift is pivotal only at the thresholds. The mechanism trades off a small probability of rejecting a highly desirable state (low likelihood under truth-telling) to deter senders from lying to induce acceptance in the disagreement region.
4. Bounded Inefficiency: Despite the ability to improve upon the sender-preferred outcome, the optimal mechanism is strictly bounded away from the first-best (receiver-preferred) outcome. This inefficiency persists even as $N \to \infty$ because the informational division prevents the conflict of interest from vanishing. In standard additive information models, a vanishing fraction of the population can eventually identify the state, allowing the first best to be achieved; here, the conflict remains non-trivial.

Significance and Claims
The paper claims to provide a new methodology for studying information aggregation by directly examining the large-population limit through the lens of informational division.

• Distinct from Standard Models: The authors emphasize that their results differ fundamentally from literature on information aggregation with correlated signals or additive information (e.g., Feddersen and Pesendorfer, 1997; Gerardi et al., 2009). In those models, increasing the population size adds information, eventually resolving conflicts of interest. In this model, increasing $N$ divides the same information, preserving the conflict and the impossibility of cross-verification.
• Mechanism Design Insight: The paper demonstrates that "informational division" allows a receiver to benefit from the division itself by using the structure of incentives in the limit. By punishing consensus in the direction of the common bias (via surplus burning at the top), the receiver can extract more information than would be possible with a single sender or in an additive information setting.
• Applicability: The authors suggest that the concept of ICL and the insights regarding informational division can be applied to re-examine broader literature on voting, auctions, markets, and collective action, particularly in environments where information is dispersed and unverifiable (e.g., social media, polling).

The paper concludes that while cross-checking is impossible, the division of information creates a specific incentive structure where simple, non-monotonic mechanisms (interval rules) are optimal, though they inevitably fall short of the first-best due to the persistent nature of the informational division.