My question is about an unsourced sentence in Wikipedia: https://en.wikipedia.org/wiki/Correlated_equilibrium
(emphasis mine)
The following correlated equilibrium has an even higher payoff to both players: Recommend (C, C) with probability 1/2, and (D, C) and (C, D) with probability 1/4 each. Then when a player is recommended to play C, she knows that the other player will play D with (conditional) probability 1/3 and C with probability 2/3, and gets expected payoff 14/3, which is equal to (not less than) the expected payoff when she plays D. In this correlated equilibrium, both players get 5.25 in expectation. It can be shown that this is the correlated equilibrium with maximal sum of expected payoffs to the two players.
Why does this correlated equilibrium have the maximal sum of expected payoffs? Why is it not the correlated equilibrium with (D,C) probability 1/2 and (C,D) probability 1/2. Isn't that a correlated equilibrium (neither player will deviate if they know their opponent's action)? And isn't the sum of payoffs = 9 for that?
Nevermind, I am daft. In the 1/2, 1/4, 1/4 CE, each player gets an EV of 5.25. With the 0, 1/2, 1/2 CE, each player gets an EV of 4.5.