The problem with this game is that the only Nash equilibrium is 0. Given a prior distribution of answers from the other players, you should always guess lower. But if everyone does this, it changes the prior distribution until the "right" guess is again lower. It's more fun if you restrict play to [1,100] rather than [0,100].
In practice this isn't actually an exercise in mathematics, it's an exercise in psychology. The "obvious" average is of course 50. People who "understand" the puzzle will guess around 33. Of course, there are those who "understand" this first level of recursion and guess 22; and so on. The problem, for those who ACTUALLY understand the puzzle (because they studied it) is not to do the recursion, because it has no end. The problem is actually to guess: how well do you think your OPPONENTS understand recursion?
Experiments show that most people can figure out roughly 1.3 levels of recursion. So the typical average in this game is around 26-30, and you should guess 18-20. Repeating the experiment with different populations will give different results; in particular, people who have advanced training in recursion (computer programmers or some mathematicians, but in general NOT other kinds of engineers) will, on average, manage another level or so, making the "correct" play 12-13. But really, if all the players understand the game, the only correct play is 0.