This is a rather complex question; it may require nontrivial assumptions about human cognition. But, I'm interested in getting mathematicians' perspective.
With some finagling, you can associate many popular games, plus starting positions, with surreal numbers -- or more generally with games in Conway's sense. The number (or game) tells you which player has a winning strategy.
Some of these games have online clients with tens of thousands of users, accumulating huge amounts of raw data. In particular, we can look at the win percentage for player one versus player two. That should give us some information about the surreal number or game (Conway) of the game. In the former case, perhaps an estimate or a bound. In the latter case, perhaps whether or not the game is comparable with zero, and if so where it falls.
My question is: what kind of estimate can we get? How good would we expect it to be? And what other statistics might we consider looking at? I realize that in the case of e.g. standard chess, with a symmetric starting position, we can probably get a confidence estimate for whether the game is zero or incomparable in some obvious way. But I'm interested in the general case -- weird starting positions and so on. With advanced players this will lead to 100% win rates in all but the smallest (or incomparable) games, but we might still get some information about the larger games from beginner players, or more generally from statistics about win rate versus player rank.