Chakraborty and Harbaugh (2010) prove the existence of influential cheap talk equilibria

in one sender one receiver games when the state is multidimensional and the preferences

of the sender are state-independent. We show that only the babbling equilibrium survives

the introduction of any small degree of uncertainty about the sender’s preferences in the

spirit of Harsanyi (1973). Introducing small costs of lying as in Kartik (2009), i.e. a small

preference for sending the actual state as the message, while removing some influential

equilibria, makes others robust to payoff uncertainty. Finally, modelling a small desire

to be truthful endogenously, i.e. by taking into account how the receiver interprets the

message, may make some influential equilibria robust, but may also remove all influential

equilibria.