Large Sender Pools Require Punishing Biased Consensus

When many interested parties each hold a private, independent signal, aggregation is difficult for a reason that voting-style intuition can obscure: no sender can verify anybody else’s report. The receiver must decide whether to accept or reject, while senders share a bias toward acceptance. With only a few senders and a small bias, the paper finds that private conference can align reports well enough to implement the sender-preferred allocation. That conclusion reverses as the population grows. The receiver can gain from keeping information divided rather than allowing consensus to form unchecked.

Core Contribution

The paper’s central contribution is incentive compatibility in the large (ICL), a limit concept for mechanisms with $N \to \infty$ senders. Rather than solve every finite-$N$ reporting game directly, it asks which limiting outcome rules can be approximated by incentive-compatible mechanisms in large finite economies.

Theorem 1 characterizes ICL mechanisms through two restrictions: an envelope condition, labeled ENV, and a monotonicity condition, MON. This matters because each sender’s information about aggregate reports vanishes with population size, yet a report can still have a first-order effect on the sender’s payoff. Simply treating senders as uninformed would therefore discard the incentive discipline that survives in the limit.

The resulting design problem is much smaller than the original mechanism-design problem. In the baseline model, the sender’s payoff is $(\omega+b)a$ and the receiver’s payoff is $(\omega+r)a$, where $a$ is the binary accept decision. The authors show that the receiver-optimal limiting rule depends only on the accept-payoff statistic rather than the full vector of reported signal frequencies.

Technical Approach

The mechanism uses the informational divide as an incentive instrument. A sender’s common bias makes coordinated optimistic reports suspicious: excessive agreement in the direction of that bias triggers a lower acceptance probability. These are surplus-burning punishments. They do not transfer surplus to the receiver; instead, they sometimes withhold acceptance even when acceptance would be efficient under full information.

That feature distinguishes the paper from settings where a principal can use transfers to enforce truth telling. Here, the binary action is the principal’s limited instrument. The envelope restriction captures how a marginal change in a sender’s reported type affects expected utility, while monotonicity rules out allocations that reward strategically lower reports in the wrong direction. The proof strategy then converts these constraints into an optimization problem over the aggregate accept-payoff statistic.

The extensions preserve this logic. With observable heterogeneous bias classes, Proposition 1 shows that the limiting problem depends on one aggregate statistic for each bias class. Under more general sender preferences, Proposition 2 derives the analogous limiting incentive constraints using derivatives of sender utility. The framework therefore is not tied solely to identical linear biases, though those extensions retain finite signal supports and the large-population asymptotics.

Results and Analysis

The paper’s main evidence is analytical rather than empirical. It establishes that, in large economies, optimal mechanisms converge to rules that penalize excessive consensus in the direction of the common sender bias, and that receiver payoffs remain bounded away from the first-best. The gap is not presented as a computational defect: it is the cost of eliciting dispersed information when no sender can check another sender’s statement.

Figure 5 provides a numerical illustration for a two-sender small economy with a relatively large bias, $b=2/3$, receiver parameter $r=1/4$, and outside option $U \in [-1,1]$. The authors discretize that square into 200 subintervals per axis, solve a linear program over 40,000 decision values with MATLAB linprog, and plot the resulting rule. Yellow regions correspond to acceptance, while blue regions correspond to rejection. The curved blue region cutting into otherwise favorable territory visualizes the same mechanism-design logic: reports that are too mutually supportive can be rejected despite looking favorable to both senders.

This is a useful corrective to the claim that more independent reports necessarily make information aggregation easier. More reports improve statistical information, but they also create a mechanism-design problem in which common bias can be expressed through broad agreement. The paper supplies a formal way to isolate that trade-off. Its conclusions are most relevant for committees, expert panels, mediated recommendations, and other institutions that must aggregate strategically reported information without verification or transfers.

Limits of the Result

The paper does not provide field, laboratory, or agent-based evidence that real groups produce the modeled consensus patterns. Its numerical exercise is an illustration of a small, discretized instance rather than a test of the large-economy characterization. The baseline also assumes unconditionally independent signals, a binary action, and a specific receiver–sender payoff structure. Those assumptions make the formal result clean, but they limit direct claims about environments with correlated evidence, repeated interaction, or richer institutional tools.