Malcolm Gladwell's article this past May examined the "I-know-they-know-I-know-they-know" regress as it relates to spying and national intelligence. If country X knows that country Y is intercepting their communications, isn't country X likely to communicate intentionally wrong information? It's an interesting read.
On a more personal level, social situations where I know the other person knows something about me but they are not aware that I know that they know, or variations thereof, are always intriguing and challenging. Interactions bulging with meta data.
Andre Aciman, in a long, interesting essay in The American Scholar, touches on similar themes when, as an aside, he talks about his favorite French novels which have sentences or paragraphs like:
Her lover knew, by the way she showed every conceivable proof of love for him, that she was determined to say no to him.
Or:
Her future husband could tell, by the way she blushed whenever they were alone together, that she felt neither love, nor passion, nor desire for him; her blushes came from exaggerated modesty, which in her coy, girlish way she was pleased to mistake for love. The very means meant to conceal her blushes is precisely what gave them away. Her husband guessed by how happy his wife was when she heard that their friend was not going to join them on their trip to Spain that he was the one with whom she’d have betrayed him if only she had the courage.
Or:
The frown with which she seemed to dismiss the man she wished she didn’t love told him everything he longed to know. Even the abrupt, rude manner with which she snapped at him as soon as they were alone was a good sign: she was more in love with him than he had ever hoped.
Or:
I thought that if anything could rekindle your feelings for me, it was to let you see that mine too had changed, but to let you see this by feigning to wish to conceal it from you, as if I lacked the courage to acknowledge it to you.
You’re obviously interested in what has been dogging me for years! It goes back to a non-Cartesian but very 17th-century view of consciousness (I’ve written about it). Unfortunately, a simpleton’s version of the twists and turns of consciousness is now being touted as “Theory of Mind,” see: http://www.nytimes.com/2010/04/01/books/01lit.html
As I said, it is a simpleton’s view of the matter. Gladwell’s example was well chosen it threw irony into what was already a pretty chaotic situation: Irony + Chaos yields what, in good hands, could frequently turn out to be literature. I think.
Have you read Cryptonomicon (by Neal Stephenson)? They deal with this issue from the Information Theory point of view. In the framework of WWII, they ask how the allies must behave in order to not show that they have break the Enigma code.
Here is an excerpt:
““Let us recast this in information theory terms,” says the don. “Information flows from Germany to us, through the Ultra system at Bletchley Park. That information comes to us as seemingly random Morse code transmissions on the wireless. But because we have very bright people who can discover order in what is seemingly random, we can extract information that is crucial to our endeavors. Now, the Germans have not broken our important cyphers. But they can observe our actions—the routing of our convoys in the North Atlantic, the deployment of our air forces. If the convoys always avoid the U-boats, if the air forces always go straight to the German convoys, then it is clear to the Germans—I’m speaking of a very bright sort of German here, a German of the professor type—that there is not randomness here. This German can find correlations. He can see that we know more than we should. In other words, there is a certain point at which information begins to flow from us back to the Germans.” “We need to know where that point is,” says the Main Guy. “Exactly where it is. We need then to stay on the right side of it. To develop the appearance of randomness.” “Yes,” Waterhouse says, “and it has to be a kind of randomness that would convince someone like Rudolf von Hacklheber.””
I haven't, but I should. Thanks.
http://en.wikipedia.org/wiki/Metagaming