John Maynard Keynes described the same thing 70 years ago. The phenomenon is sometimes called a Keynesian Beauty Contest. In choosing (or writing) the winning caption in the NYer caption contest, the object is not to write the best caption, but to write the caption that the average reader will believe that the average reader will choose as the best caption. Keynes used the examples of a contest to pick the photo of the prettiest girl from 100 photos:

“It is not a case of choosing those [faces] which, to the best of one’s judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees.” (J. M. Keynes, The General Theory of Employment Interest and Money, Chapter 12).

I know that you know that I know that you know that I know that you know that I know...