Bayesian Conversations

Abstract

We study Bayesian conversations, a model of interactive information disclosure between two privately informed agents. Unlike Bayesian persuasion, information flows in both directions: agents alternately send messages, with each message depending only on the sender's private type and the public transcript. We study welfare-maximizing conversations subject to ex-post individual rationality, focusing on settings with a single action-taker. When both agents have binary types, we give a polynomial-time algorithm for computing the optimal $T$-round Bayesian conversation. Our approach combines alternating concavification over the belief space with an exact finite discretization whose size is polynomial in the action set and independent of $T$. We further establish a strict separation between Bayesian conversations and mediator protocols in terms of achievable social welfare under the same ex-post IR requirement. Finally, in the binary-type and binary-action setting, we prove a sharp dichotomy: optimal welfare is either attained within six rounds or approached only asymptotically over infinitely many rounds, with linear convergence. Both regimes can occur and can be distinguished in polynomial time.