Multiple Referrals and Multidimensional Cheap Talk

Cheap talk games have been widely used to analyze situations in which a policy maker needs expert advice. In previous work, agent 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, with more than one dimension full transmission of information in all states of nature is typically possible, provided a very simple and intuitive condition is satisfied. When utilities are quadratic and there are simultaneous reports, linear independence of senders' ideal points is a sufficient condition to guarantee full revelation; with sequential reports, linear independence and a simple condition on the gradients of senders' utilities at the receiver's ideal point are sufficient. In particular as an application of the theory we are able to explain an empirical puzzle related to informational theories of legislative organization. These theories predict that legislative committees (senders) should have strong alignment of preferences with the Floor; but this doesn't fit with empirical facts. We prove that what really matters in transmission of information is the local behavior of the utilities of the senders at the ideal point of the policy maker (receiver), not the distances between the ideal points of players. We interpret this as an argument in support of informational theories of legislative organizations.