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October 29, 2004

Sender-Receiver Games: Truthful Announcement, Cheap Talk, and Signalling

After a chat with Professor Harbaugh, I thought I'd collect my thoughts on communication games, thinking about revisions to my Games and Information. These notes won't mean much to non-economists, I'm afraid.

There are a variety of games in which one player, the Sender, tries to communicate something-- which we can call "his type"-- to another, the Receiver. The Sender is the informed player, so he is often an Agent; the Receiver is uninformed, and so is often a Principal.

I wonder if the games can usefully be divided into Truthful Announcement, Cheap Talk, and Signalling....

...In Truthful Announcement games, the Sender may be silent or send a message, but the message must be truthful if it is sent. There is no cost to sending the message, but it may induce the Receiver to take actions that affect the Sender. If the Receiver ignores the message, the Sender's payoff is unaffected by the message. The Sender's type varies from bad to good in these models usually.

An example of a Truthful Announcement game is when the Sender's ability A is uniformly distributed on [0,1], and the Sender can send a message Y such as "A>.5" or "A=.2".

In Cheap Talk games, the Sender's message is costless, but need not be truthful. If the Receiver ignores the message, the Sender's payoff is unaffected by the message. If the Receiver acts, though, that might affect the Sender. Usually, these are coordination games, where the Sender's preferred Receiver-action, given the true state of the world that he knows, is positively correlated with the Receiver's preferred Receiver-action.

An example of a Cheap Talk game is when the Sender and Receiver want to go to the same restaurant, either A or B, but only the Sender knows which restaurant is better. The Sender send a message-- "A" or "B"-- and if the Receiver ignores it, there is no cost to the Sender.

In Signalling games, the Sender's message is costly-- or at least a false message is-- but need not be truthful. The Sender's payoff is affected even if the Receiver ignores his message. The Sender's type varies from bad to good in these models usually. The "single-crossing property" is crucial-- that if the Sender's type is better, it is cheaper for him to send a message that his type is good.

An example is credentials. The Sender is dull or bright. If he is bright, it is easier for him to acquire credentials, which is his message to a Receiver employer.

In writing this up, some awkwardnesses strike me.

1. Zero-Cost Signals. A signalling game doesn't change its essential properties if sending the message of high quality is costless for the truly high quality type. It could even have negative cost for him-- that he gets a reward for truthfully declaring his type. What matters is that the same signal be too costly for a low quality type to think worth sending.

2. Lying Being what Is Costly. In the usual models, if a high signal is sent, that is more expensive than a low signal, especially for the low type of Sender. But I think the model would work out very much the same if what is expensive is not a high signal, but a false signal. The difference is that in the usual models, it is cheap for the High type to falsely signal that he is low, but in a truth-based model, it would be expensive for him to be modest.

3. Expensive-Talk Games. Imagine a cheap-talk game in which the signal is costly-- but the cost is the same for everyone, regardless of type. The usual sort of signalling won't work, because signalling high quality is no more expensive for the Low type than for the High type. But truthful communication might still work, for reasons more akin to those of the Cheap-Talk Game, if the High type Sender has a greater desire than the Low type for the Receiver to adopt a High response.

Thus, imagine that the Low Sender could make $100 as a salesman for himself and $100 for the Receiver if the Receiver hires him, and the High Sender could make $900 for himself and $900 for the Receiver. If messages are costless, both Senders would send the message "I am a High type" (not, I guess, "Hire me--I'm high"), and the message would be uninformative. If the message costs $200, only the truly High Sender would send the message. There is now an equilibrium in which the message is informative (there is also a pooling equilibrium, perhaps implausible, in which messages are still ignored).

People might think of this as a signalling game, applying the single-crossing property to the ultimate payoffs, but it is really more akin to the cheap-talk game, I think. It is like the PHD Admissions Game in Chapter 6 of my book. Perhaps it like Mechanism Design games too, which might be thought of as a form of cheap-talk games, since they have Senders and Receivers and costless messages, though in Mechanism Design games there is commitment to the mechanism.

Posted by erasmuse at October 29, 2004 09:10 PM

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