Editing Cheap talk (section)

====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.