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

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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:

enter image description here

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.

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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.

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  • $\begingroup$ This is great, thanks for the reference! Are you just incredibly familiar with the literature? $\endgroup$ – Пафну́тий Apr 19 '18 at 23:12
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    $\begingroup$ I had the impression that some results like that should restrict the role of correlation in zero-sum games and did an internet search for papers on correlated equilibria in zero-sum games. $\endgroup$ – Michael Greinecker Apr 19 '18 at 23:15
  • $\begingroup$ @Michael, this question may be of interest. $\endgroup$ – Theoretical Economist Apr 20 '18 at 8:31
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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.

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