This is a nice paper from the recent AER whose result is simple enough that the reader surely wonders, why didn’t I think of this first?
Consider someone – a lawyer, a vendor, whatever – who has state-independent preferences on the actions of another. No matter how good a product is, a salesman wants you to buy it. No matter how guilty a criminal is, a lawyer wants you to acquit. Talk is cheap in these situations: no matter what the salesman says, I know he wants me to buy the product, so I just don’t believe him. The standard ways to get around this problem are to make talk costly (Spence signaling) or to make claims sometimes verifiable (we sometimes call these “persuasion games”).
Chakraborty and Harbaugh describe another situation where cheap talk can have real effects: multidimensional games. Let there be two products a customer might buy. Let a customer have value of not buying anything of x distributed uniform on [0,1] and known only to the buyer. Let each good confer some benefit v1 (and v2) each distributed uniform[0,1] i.i.d., with the exact value known only to the seller. The customer buys if his expected benefit from buying is more than his value x. Since both goods are identical to the buyer, he expects both of them to confer benefit .5, and he buys (arbitrarily) one of the goods if x<.5, so a sale is made half the time. The seller does not care which good he wants to sell, so he can credibly (in equilibrium) claim that one of the goods is better than the other. The better good, from the perspective of the customer, now has expected value 2/3, and a sale is now made 2/3 of the time. That is, the seller is made better off by credibly talking up one of the products and talking down the other one.
More generally, for any utility functions and any distribution of prior beliefs about values unknown to one of the agents, there is always an information-revealing equilibrium in a multidimensional problem; this is true even if the agent with private information wants to increase the other agent’s estimate of one dimension and decrease his estimate of a second dimension. Both agents are strictly better off if the information-revealer has strictly quasiconvex utility. The intuition is simply that influential communication will spread out the decision maker’s estimates of each dimension, and quasiconvexity ensures that this spread is better for the communicator.
Beyond the salesman example, this logic explains a number of other phenomena. If a jury requires unanimity, and half the voters will say “guilty” only if they think the crime actually happened, and half the voters will say “guilty” only if they think the criminal was immoral, then a defense lawyer can (credibly!) spend time discussing only one of culpability or morality. If the lawyer discusses morality, then all jurors will think the potential criminal more likely to have committed the crime, but they will also all consider him more moral; since only one juror needs to vote “innocent”, this spread is good for the defense attorney. Applications to auctions and advertising are also found in the paper – this might be the rare paper that gets to AER on the strength of its examples alone.
http://www.bus.indiana.edu/riharbau/PersuasionByCheapTalk.pdf (final version, published in December 2010 AER)
A Fine Theorem 