Correlated Equilibrium for Rock Paper Scissors

Question

Consider the game of Rock, Paper, Scissors (RPS), with payoffs given as follows:

Is there a correlated equilibrium in this game?

Consider, for example the signal given to both players not to play the third strategy. In this case, the game (conditional upon one's opponent following the signal) becomes:

Which has a NE of playing the second strategy for both. This seems like a correlated equilibrium? On the other hand, if you know that your opponent follows the signal and plays the second strategy, you should respond by playing the third strategy. Hence, intuitively, as the signal reveals information on your opponent's play (conditional upon her following it) you should be able to exploit that by violating correlated equilibrium.

Answer 1

No, the unique Nash equilibrium is the unique correlated equilibrium by a general property of two-player zero-sum games pointed out in:

Forges, Françoise. "Correlated equilibrium in two-person zero-sum games." Econometrica (1986-1998) 58.2 (1990): 515.,

For every action of player 2 that is played with positive probability, the conditional distribution over player 1's actions is an optimal strategy for player 1 and vice versa. Since optimal strategies are unique in RPS, the conclusion follows.

Answer 2

Is there a correlated equilibrium in this game?

Michael has a great answer, but I just want to answer the question I quote above more generally.

Any Nash equilibrium is a correlated equilibrium, so a correlated equilibrium always exists whenever a Nash equilibrium exists. (And we know NE always exist in finite games like RPS.)

Of course, a game with no Nash equilibria may have a correlated equilibrium, but I'm not aware of any simple examples where this is the case.