Mechanism design and game theory is often radically different when the state is multidimensional instead of unidimensional: finding the differences has been one of the most productive parts of economic theory over the past decade. This classic paper by Battaglini is the one to read when it comes to multidimensional cheap talk.
Consider a president listening to two expert advisors, or a median voter in Congress listening to two members of a committee. Everyone is biased. The experts know exactly the results of some policy, but the receiver does not: that is, the outcome x=y+a, where y is the policy chosen, and a is some noise whose realization is known only to experts. When the policy and state are unidimensional, a number of classic results (Gilligan & Krehbiel 1989, for example) note that cheap talk from the experts can only be influential in equilibrium if the biases of the expert are small. Even then, equilibrium existence relies on out-of-equilibrium beliefs (the solution concept is Perfect Bayesian Nash) that are in some sense crazy.
This turns out not to be true in a multidimensional world. Consider potential policies which will affect both global warming and unemployment, where these are mapped into utilities in two-dimensional Euclidean space. The two experts know exactly how these policies will affect the environment and the economy, while the receiver only knows the effect of policy y, and knows that the signal a has expected value of zero. In this case, it turns out full revelation is almost always possible, no matter what the biases are; this result does not rely on crazy out-of-equilibrium beliefs and it is robust to a specific form of collusion among the experts.
What magic is being used? The basic idea is to find dimensions upon which each agent has preferences that are aligned with those of the receiver, and ask agents only about those preferences. Intuitively, ask the environmentally-conscious guy about which policy is best given that the economy is affected in the optimal way, and ask the economically-minded guy about which policy is best for the environment given that the economy is affected in the optimal way. Mathematically, let the optimal outcome of the receiver be represented at the origin, and consider the vectors tangent to each expert’s indifference curves at the origin. Ask each expert only to reveal a dimension of the state he prefers along this line in two-dimensional space. By construction, if an expert has to choose from only that line, he will choose the origin. This intuition will always work as long as utility is quasiconcave and the gradients of each agent’s utilities are linearly independent at the origin.
This clears up some puzzles in political economy. For instance, the unidimensional result suggested that biased committees are uninformative, but committee members in Congress tend to be made up of Congressmen with the strongest biases. So why do such committees persist if they aren’t influential? Battaglini’s result shows that on multidimensional problems, committees are indeed useful, even when made up of very biased members, because they still transmit information to Congress at large in equilibrium.
A quick mathematical caveat: Ambrus and Takahashi note in a 2008 Theoretical Economics that Battaglini’s result is not just a dimensionality of state space argument, but also one that relies on the state space being the entire Euclidean space. When the state space is compact (say, the policy is spending on education and military, and there is a fixed budget), it is not true, under some robustness conditions, that information is always fully revealed. The trick is dealing with out-of-equilibrium cases that are “impossible”, such as when the strategies of the experts imply that the optimal spending is strictly greater than the budget. If you like Battaglini’s paper, it’s probably worth taking a look at Ambrus & Takahashi.
http://www.princeton.edu/~mbattagl/cheaptalk.pdf (Final WP – published in Econometrica 2002)