  We initiate the study of Bayesian conversations, which model interactive
communication between two strategic agents without a mediator. We compare this
to communication through a mediator and investigate the settings in which
mediation can expand the range of implementable outcomes.
  In the first part of the paper, we ask whether the distributions of posterior
beliefs that can be induced by a mediator protocol can also be induced by a
(unmediated) Bayesian conversation. We show this is not possible -- mediator
protocols can ``correlate'' the posteriors in a way that unmediated
conversations cannot. Additionally, we provide characterizations of which
distributions over posteriors are achievable via mediator protocols and
Bayesian conversations.
  In the second part of the paper, we delve deeper into the eventual outcome of
two-player games after interactive communication. We focus on games where only
one agent has a non-trivial action and examine the performance of communication
protocols that are individually rational (IR) for both parties. We consider
different levels of IR including ex-ante, interim, and ex-post; and we impose
different restrictions on how Alice and Bob can deviate from the protocol: the
players are committed/non-committed. Our key findings reveal that, in the cases
of ex-ante and interim IR, the expected utilities achievable through a mediator
are equivalent to those achievable through unmediated Bayesian conversations.
However, in the models of ex-post IR and non-committed interim IR, we observe a
separation in the achievable outcomes.
