Cheap talk
This is in contrast to signalling, in which sending certain messages may be costly for the sender depending on the state of the world.
To give a formal definition, cheap talk is communication that is:[2] Therefore, an agent engaging in cheap talk could lie with impunity, but may choose in equilibrium not to do so.
Cheap talk can, in general, be added to any game and has the potential to enhance the set of possible equilibrium outcomes.
For example, one can add a round of cheap talk in the beginning of the Battle of the Sexes.
Each player announces whether they intend to go to the football game, or the opera.
Because the Battle of the Sexes is a coordination game, this initial round of communication may enable the players to select among multiple equilibria, thereby achieving higher payoffs than in the uncoordinated case.
The messages and strategies which yield this outcome are symmetric for each player.
It is not guaranteed, however, that cheap talk will have an effect on equilibrium payoffs.
It has been commonly argued that cheap talk will have no effect on the underlying structure of the game.
This general belief has been receiving some challenges (see work by Carl Bergstrom[3] and Brian Skyrms 2002, 2004).
In the basic form of the game, there are two players communicating, one sender S and one receiver R. Sender S gets knowledge of the state of the world or of his "type" t. Receiver R does not know t ; he has only ex-ante beliefs about it, and relies on a message from S to possibly improve the accuracy of his beliefs.
S decides to send message m. Message m may disclose full information, but it may also give limited, blurred information: it will typically say "The state of the world is between t1 and t2".
The form of the message does not matter, as long as there is mutual understanding, common interpretation.
It could be a general statement from a central bank's chairman, a political speech in any language, etc.
Whatever the form, it is eventually taken to mean "The state of the world is between t1 and t2".
The decision of S regarding the content of m is based on maximizing his utility, given what he expects R to do.
It can be financial profits, or non-financial satisfaction—for instance the extent to which the environment is protected.
The theory applies to more general forms of utility, but quadratic preferences makes exposition easier.
Parameter b is interpreted as conflict of interest between the two players, or alternatively as bias.UR is maximized when a = t, meaning that the receiver wants to take action that matches the state of the world, which he does not know in general.
Expectations get realized, and there is no incentive to deviate from this situation.
Crawford and Sobel characterize possible Nash equilibria.
When interests are aligned, then information is fully disclosed.
The model allowing for more subtle case when interests are close, but different and in these cases optimal behavior leads to some but not all information being disclosed, leading to various kinds of carefully worded sentences that we may observe.
This partition is shown on the top right segment of Figure 1.
The action function is now indirectly characterized by the fact that each value ai optimizes return for the R, knowing that t is between t1 and t2.
→ Quadratic utilities: Given that R knows that t is between ti-1 and ti, and in the special case quadratic utility where R wants action a to be as close to t as possible, we can show that quite intuitively the optimal action is the middle of the interval:
At t = ti, The sender has to be indifferent between sending either message mi-1 or mi.
With N = 1, we get the coarsest possible message, which does not give any information.
However, it remains quite coarse compared to full revelation, which would be the 45° line, but which is not a Nash equilibrium.
With a higher N, and a finer message, the blue area is more important.
