In previous work on cheap talk, uncertainty has
almost always been modeled using a single-dimensional state variable.
In this paper we prove that the dimensionality of the uncertain
variable has an important
qualitative impact on results and yields interesting insights into the
``mechanics'' of information transmission. Contrary to the
unidimensional case, if there is more then one sender, full revelation
of information in all states of nature is generically possible, even
when the conflict
of interest is arbitrarily large. What really matters in transmission
information is the local behavior of senders' indifference curves at
ideal point of the receiver, not the proximity of players' ideal point.
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