Dynamic Cheap Talk without Feedback

Referee: [Proof of the main equivalence theorem] The uniform-equilibrium construction (main theorem) must verify that, for every state and every δ sufficiently close to 1, the continuation-value loss from any marginal-preserving deviation strictly exceeds the one-shot gain. Because the receiver observes only the public message sequence and the sender observes the current state, a deviation that leaves the unconditional message distribution unchanged can be statistically indistinguishable from equilibrium play in finite samples; the manuscript therefore needs to exhibit explicit history-dependent punishment strategies and derive the required lower bound on the continuation loss.

Authors: We agree that the proof of the main theorem must explicitly verify incentive compatibility for all marginal-preserving deviations, including a lower bound on continuation losses that dominates one-shot gains for δ close to 1. The current construction uses a block-based strategy in which the sender commits to a fixed signaling policy with the target marginal for T periods (with T large but finite), after which the receiver updates beliefs on the empirical message frequencies and applies a continuation equilibrium. To address the concern, we will add a dedicated lemma in the revised proof that specifies the history-dependent punishment: upon observing an empirical marginal deviating by more than ε from the target (detectable asymptotically via the law of large numbers, which applies uniformly because the message process is i.i.d. conditional on the marginal), the receiver switches permanently to a myopic best response that imposes a per-period loss L > 0 on the sender. The one-shot gain from any such deviation is bounded above by some G independent of δ; choosing δ > 1 - L/G then ensures the discounted continuation loss strictly exceeds the gain for every state. This bound holds because the Markov state evolution does not affect the message marginal under the deviation, preserving statistical distinguishability in the limit. revision: yes

Referee: [Section on convex combinations of message distributions] The claim that convex combinations across message distributions are sustainable requires that the receiver can credibly switch between distinct response rules while deterring the sender from choosing a different marginal in any period. The construction must show that the induced per-period payoff vector remains an equilibrium payoff of the partial-commitment benchmark for every convex weight; otherwise the convex-hull statement does not follow from the single-distribution result.

Authors: We thank the referee for highlighting the need to connect the convex-hull result directly to the partial-commitment benchmark. The manuscript achieves convex combinations via public randomization (or an initial public draw) that selects a weight λ and corresponding marginal at the beginning of each block; the players then play the single-distribution equilibrium associated with the realized marginal, with the receiver's action rule conditioned on the selected marginal. We will revise the section to include a supporting lemma proving that, for any convex weight λ, the resulting per-period payoff vector is itself an equilibrium payoff of the partial-commitment model: any unilateral deviation by the sender to a non-selected marginal triggers the same punishment as in the single-distribution case (switch to a punishing action), and the linearity of the benchmark payoffs in the marginal ensures that the mixture remains incentive-compatible. This establishes that the convex hull is attainable without requiring additional credibility constraints beyond those already verified for each component. revision: yes