Who Should Act After Silence? Exact Coordination Policies for Agents on Unreliable Relay Lines

Agentic pipelines increasingly rely on sequential communication that can be delayed or dropped. When an expected message is missing, should every agent in the pipeline respond in the same way? We study unanimous-action coordination on a finite one-way relay line with heterogeneous delivery probabilities, an informed participating root, and utilities that blend private and group-average payoffs. Within coordinating observation-monotone root-veto pure Bayesian Nash equilibria, every equilibrium consists of a message-responsive prefix followed by an unconditional (\emph{blind}) suffix, and three posterior-odds inequalities are necessary and sufficient for each cutoff. Heterogeneous links can support a hybrid equilibrium when both uniform policies fail; homogeneous links admit only the two extremes, and even permuting the same link reliabilities can change the feasible cutoff. The characterization yields $O(n)$ enumeration of feasible cutoffs and exact validity regions in the incentive-alignment parameter. We additionally evaluate LLM execution of the resulting local policies. Interfaces that must reconstruct the selected action err most on the one decision that separates the theory from generic intuition, unconditional action after silence; action-bearing compiled policies are executed with high fidelity.