We study dynamic signalling when the sender does not see the signals that her actions generate. The sender then uses her past play to forecast what a receiver believes, in turn forcing the receiver to forecast the previous forecast, and so forth. We identify a class of linear-quadratic-Gaussian games where this endogenous higher-order uncertainty can be handled. The sender’s second-order belief is key: it is a private state that she controls, and it creates a new channel for information transmission. We examine the role of higher-order uncertainty and this new signalling channel in applications to macroeconomics, reputation, and trading: inflationary biases under discretion can be larger; career-concerned agents may benefit from not knowing their reputations; and informed trades can carry more price impact. We also introduce an existence method for boundary value problems that can be used in other dynamic games.

We study how an organization should dynamically screen an agent of uncertain loyalty whom it suspects of committing damaging acts of undermining. The organization controls the stakes of the relationship, while the agent strategically times undermining, which can occur repeatedly and is detected only stochastically. The optimal commitment stakes policy exhibits both discreteness and gradualism, with distinct “untrusted” and “trusted” phases featuring gradually rising stakes during the untrusted phase and a discrete gap in stakes between phases. This policy is also the equilibrium outcome when the organization cannot commit, and the agent’s equilibrium undermining policy exhibits variable, non-monotonic intensity.