Editing Cheap talk (section)

==Crawford and Sobel's definition==

===Setting===
In the basic form of the game, there are two players communicating, one sender ''S'' and one receiver ''R''.

====Type====
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.

====Message====
''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 ''t<sub>1</sub>'' and ''t<sub>2</sub>''". It may give no information at all.

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 ''t<sub>1</sub>'' and ''t<sub>2</sub>''".

====Action====
Receiver ''R'' receives message ''m''. ''R'' updates his beliefs about the state of the world given new information that he might get, using [[Bayes' rule|Bayes's rule]]. ''R'' decides to take action ''a''. This action impacts both his own utility and the sender's utility.

====Utility====
The decision of ''S'' regarding the content of ''m'' is based on maximizing his utility, given what he expects ''R'' to do. Utility is a way to quantify satisfaction or wishes. It can be financial profits, or non-financial satisfaction—for instance the extent to which the environment is protected.
''→ Quadratic utilities:''
The respective utilities of ''S'' and ''R'' can be specified by the following:
<math display="block">U^S(a, t) = -(a-t-b)^2</math>
<math display="block">U^R(a,t)=-(a-t)^2</math>

The theory applies to more general forms of utility, but quadratic preferences makes exposition easier. Thus ''S'' and ''R'' have different objectives if ''b ≠ 0''. Parameter ''b'' is interpreted as ''conflict of interest'' between the two players, or alternatively as bias.''U<sup>R</sup>'' 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. ''U<sup>S</sup>'' is maximized when ''a = t + b'', meaning that ''S'' wants a slightly higher action to be taken, if ''b > 0''. Since ''S'' does not control action, ''S'' must obtain the desired action by choosing what information to reveal. Each player's utility depends on the state of the world and on both players' decisions that eventually lead to action ''a''.

====Nash equilibrium====
We look for an equilibrium where each player decides optimally, assuming that the other player also decides optimally. Players are rational, although ''R'' has only limited information. Expectations get realized, and there is no incentive to deviate from this situation.

===Theorem===
[[File:Cheap talk.png|thumb|upright=2|'''Figure 1:''' Cheap talk communication setting]]

Crawford and Sobel characterize possible [[Nash equilibria]].

* There are typically '''multiple equilibria''', but in a finite number.
* '''Separating''', which means full information revelation, is not a Nash equilibrium.
* '''Babbling''', which means no information transmitted, is always an equilibrium outcome.

When interests are aligned, then information is fully disclosed. When conflict of interest is very large, all information is kept hidden. These are extreme cases. 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.

More generally:
*There exists ''N<sup>*</sup> > 0'' such that for all ''N'' with ''1 ≤ N ≤ N<sup>*</sup>'',
*there exists at least an equilibrium in which the set of induced actions has cardinality ''N''; and moreover
*there is no equilibrium that induces more than ''N<sup>*</sup>'' actions.

====Messages====
While messages could ex-ante assume an infinite number of possible values ''μ(t)'' for the infinite number of possible states of the world ''t'', actually they may take only a finite number of values ''(m<sub>1</sub>, m<sub>2</sub>, . . . , m<sub>N</sub>)''.

Thus an equilibrium may be characterized by a partition ''(t<sub>0</sub>(N), t<sub>1</sub>(N). . . t<sub>N</sub>(N))'' of the set of types [0, 1],
where ''0 = t<sub>0</sub>(N) < t<sub>1</sub>(N) < . . . < t<sub>N</sub>(N) = 1''. This partition is shown on the top right segment of Figure 1.

The ''t<sub>i</sub>(N)''{{'}}s are the bounds of intervals where the messages are constant: for ''t<sub>i-1</sub>(N) < t < t<sub>i</sub>(N), μ(t) = m<sub>i</sub>''.

====Actions====
Since actions are functions of messages, actions are also constant over these intervals:
for ''t<sub>i-1</sub>(N) < t < t<sub>i</sub>(N)'', ''α(t) = α(m<sub>i</sub>) = a<sub>i</sub>''.

The action function is now indirectly characterized by the fact that each value ''a<sub>i</sub>'' optimizes return for the ''R'', knowing that ''t'' is between ''t<sub>1</sub>'' and ''t<sub>2</sub>''. Mathematically (assuming that ''t'' is uniformly distributed over [0, 1]),

<math>a_i = \bar{a}(t_{i-1}, t_i) = \mathrm{arg}\max_a \int_{t_{i-1}}^{t_i} U^R(a, t) dt</math>

→ ''Quadratic utilities:''

Given that ''R'' knows that ''t'' is between ''t<sub>i-1</sub>'' and ''t<sub>i</sub>'', 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:
<math display="block">a_i = \frac{t_{i-1} + t_i}{2}</math>
====Indifference condition====
At {{math|''t {{=}} t<sub>i</sub>''}}, The sender has to be indifferent between sending either message {{math|''m<sub>i-1</sub>''}} or {{math|''m<sub>i</sub>''}}.
<math>U^S (a_i, t_i) = U^S (a_{i+1}, t_i)</math> &nbsp; &nbsp; &nbsp; ''1 ≤ i≤ N-1''

This gives information about ''N'' and the ''t<sub>i</sub>''.

''→ Practically:''
We consider a partition of size ''N''.
One can show that <math display="block">t_i = t_1 i + 2 b i (i-1) \qquad t_1 = \frac{1-2 b N (N-1)}{N}</math>

''N'' must be small enough so that the numerator is positive. This determines the maximum allowed value <math display="block">N^* = \langle -\frac{1}{2}+\frac{1}{2} \sqrt{1+\frac{2}{b}} \rangle</math> where <math>\langle Z \rangle</math> is the ceiling of <math>Z</math>, i.e. the smallest positive integer greater or equal to <math>Z</math>. 
Example: We assume that ''b = 1/20''. Then ''N<sup>*</sup> = 3''. We now describe all the equilibria for ''N=1'', ''2'', or ''3'' (see Figure 2).

[[File:Crawford Sobel.png|thumb|upright=3|center|'''Figure 2:''' Message and utilities for conflict of interest ''b = 1/20'', for ''N=1'', ''2'', and ''3'']]

'''''N = 1:''''' This is the babbling equilibrium. ''t<sub>0</sub> = 0, t<sub>1</sub> = 1''; ''a<sub>1</sub> = 1/2 = 0.5''.

'''''N = 2:''''' ''t<sub>0</sub> = 0, t<sub>1</sub> = 2/5 = 0.4, t<sub>2</sub> = 1''; ''a<sub>1</sub> = 1/5 = 0.2, a<sub>2</sub> = 7/10 = 0.7''. 

'''''N = N<sup>*</sup> = 3:''''' ''t<sub>0</sub> = 0, t<sub>1</sub> = 2/15, t<sub>2</sub> = 7/15, t<sub>3</sub> = 1''; ''a<sub>1</sub> = 1/15, a<sub>2</sub> = 3/10 = 0.3, a<sub>3</sub> = 11/15''.

With ''N = 1'', we get the ''coarsest'' possible message, which does not give any information. So everything is red on the top left panel. With ''N = 3'', the message is ''finer''. 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. This implies higher utility. Disclosing more information benefits both parties.