Mutual Reputation and Trust in a Repeated Sender–Receiver Game

We study a repeated sender–receiver game where inspections are public but the sender's action is hidden unless inspected. A detected deception ends the relationship or triggers a finite punishment. We show the public state is low-dimensional and prove existence of a stationary equilibrium with cutoff inspection and monotone deception. The sender's mixing pins down a closed-form total inspection probability at the cutoff, and a finite punishment phase implements the same cutoffs as termination. We extend to noisy checks, silent audits, and rare public alarms, preserving the Markov structure and continuity as transparency vanishes or becomes full. The model yields testable implications for auditing, certification, and platform governance: tapering inspections with reputation, bunching of terminations after inspection spurts, and sharper cutoffs as temptation rises relative to costs.