Separation between coarse correlated equilibria and correlated equilibria
Fix M>>1>>e and look at the following two player coordination game (both players get the same utility):

M   | 1+e  | 2e   |  e

1+e |  1   |  e   |  0

2e  |  e   |  M   | 1+e

e   |  0   | 1+e  | 1


The second and fourth row and column are strictly dominated so any correlated equilibrium cannot have them in its support, thus it would be on the sub-game:

M  |  2e

2e |  M


for which every correlated equilibrium would give each player more than M/2 utility.

On the other hand, consider the joint probability distribution giving probability 1/2 to each of the 1's, and thus utility 1 to each player.  The claim is that this is a coarse equilibrium.  In a coarse equilibrium the possible deviations of the row player are to one of the pure strategies independently of the outcome of the joint distribution.  Now if it is only known that the column player is mixing evenly between the 2nd and 4th column, then the maximum utility the row player can get is 0.5+e &lt; 1, so deviation is not profitable.