Game Theory - Correlated Equilibrium Example I struggle to understand the meaning of a correlated equilibrium. I found this example of a game on the internet:
$$
\begin{array}{c|c|c|c}
& L & C & R \\ \hline
T & 2, 2 & 0, 3 & 0, 0 \\ \hline
M & 1, 1 & -1, -1 & 1, 1 \\ \hline
B & 0, 0 & 0, 3 & 2, 2
\end{array}
$$
I want to check if playing $(T, L)$ and $(B, R)$ each with probability $0.5$ constitutes a correlated equilibrium. 
My reasoning is this: If Player 1 is assigned $T$, then he knows (does he actually?) that Player 2 must be assigned $L$. There is no incentive to deviate for 1. But 2 also knows that 1 is assigned $T$ and can change his strategy to $C$. So this is not an equlibrium.
Is this reasoning correct? Can a player deviate to every strategy or only to the two, i.e. $(T, L)$ and $(B, R)$?
Also, I hinted at this before: Do the players at least know the entire strategy tuples? E.g. if player 1 gets $T$ does he know that player 2 must have $L$?
 A: Your reasoning is correct, the player can deviate to any strategy. Thus, your suggested strategies and randomization device do not constitute a correlated equilibrium.
The way you try to construct it, the correlated equilibrium cannot get more than Nash equilibrium. The main point is that you observe your recommendation but not the recommendation to the other player. However, you construct it in such a way that the players should play either $(T,L)$ or $(B,R)$. Thus, if the a player is assigned a strategy, she knows the assigned strategy of the other player. Let me illustrate this point by constructing a correlated equilibrium in your example.
Consider the following recommendations. With probability $x_1=\frac{1}{4}$ the players are assigned the strategies $(T,L)$, with probability $x_2=\frac{3}{8}$ the strategies $(M,L)$, and with probability $x_3=\frac{3}{8}$ the strategies $(M,R)$.
Now, if Player 1 observes the recommendation $T$, she is  sure that player 2 was recommended $L$ and $T$ is a best reply. If player one observes the recommendation $M$, she is not sure whether player 2 is supposed to play $L$ or $R$. However, she knows that both strategies are equally likely given her own recommendation. Thus, she is indifferent between playing either of her strategies. In particular, $M$ is a best reply.
If Player 2 observes the recommendation $L$, she knows that Player 1 is recommended $T$ with probability $\frac{2}{5}$ and $M$ with probability $\frac{3}{5}$ (Bayesian updating). In this case, she is indifferent between playing $L$ or $C$, so $L$ is a best reply. If player 2 observes the recommendation $R$, she knows that player 1 is recommended $M$ and $R$ is a best reply. So overall you have a correlated equilibrium.
