Quantum Game Theory, Revisited

My post on quantum game theory ignited something of a firestorm over on crookedtimber.org, where the badly confused Daniel Davies asserts (quite mistakenly) that the quantum mechanism I described allows players to communicate, making it unsurprising that they can beat a coordination game.

The followup posters are in some instances equally confused, though there are excellent responses from several, most notably Glenn Bridgman.

Here's the analogy that I hope will clarify all the issues:

Suppose you and I sit in separate rooms. Once per minute, we each receive a red or green tennis ball through our mailslots. When we look at them, they're always opposite colors. We know this, for example, because we each write down the sequence of colors we see and compare them afterward.

The same thing happens if I wear sunglasses. My vision appears to be affected not a whit, and we always see opposite colors.

Ditto if you wear the sunglasses.

But whenever we both wear sunglasses, we invariably see balls of the same color.

Something very like that happens in quantum mechanics. It works with electrons instead of tennis balls, and the correlations are less than 100%, but in every essential aspect, this is the story.

Notice that we can use this mechanism to win the dog/cat game. (The game again: We are each asked one question: "Do you like dogs?" or "Do you like cats?" . We win if our answers differ, unless we were both asked about dogs, in which case we win if our answers match.) All we have to do is agree to leave off the sunglasses when we're asked about cats, put them on when we're asked about dogs, answer yes when we see a red ball, and answer no when we see a green.

Now it is Daniel Davies's position that there's nothing extraordinary about this; of course we can win--- because we're exchanging information.

But we're not. And for those who think we are, here is my question: You take off your sunglasses (or put them on, as you prefer). A ball comes through the slot. You notice it's red. What information has been exchanged?