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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.

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  • $\begingroup$ How do you plan to deal with Left/Right or 1st/2nd issues? And why would looking at beginners give you a better answer than advanced players? $\endgroup$ – Henry Feb 22 '14 at 20:04
  • $\begingroup$ Let's take chess as an example. Left/Right correspond to White/Black. The standard starting position has to be either equivalent or incomparable to $0$, by symmetry (I'd expect incomparable). But what if you remove all of Black's pieces except their king? Now you have a starting position which is certainly positive as a surreal number: no matter who goes first, White has a winning strategy. But we're not just interested in whether it's positive or negative -- we want its actual value. (Cont'd...) $\endgroup$ – Skatche Feb 22 '14 at 22:40
  • $\begingroup$ Now take a more subtle example: a difficult chess problem, but where Black maybe has a winning strategy regardless of who goes first. Give this as a starting position to advanced players, and they simply solve it; you get a 100% win rate for Black, no interesting statistics can be done. Only one bit of information on the specific surreal value of the problem. Give the game to beginners, however, and they probably make a lot of mistakes; you get to explore a lot more of the game tree. $\endgroup$ – Skatche Feb 22 '14 at 22:43
  • $\begingroup$ On further reflection, I may have misunderstood the structure of the surreals; there may not be an immediate connection between values of gameplay statistics and surreal values of games. But maybe someone will surprise me. $\endgroup$ – Skatche Feb 23 '14 at 6:01
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The utility of the game values in Combinatorial Game Theory (and in particular, of the Surreal numbers as they show up in that context) is not really related to how easy it is to win, but merely related to the value in sums with other games. For an extreme example, both {99|} (value 100) and {0|} (value 1) are games in which Left wins by making their only available move, and Right has no moves available at all!

I don't think general win rates will help you figure this out. Rather, if I wanted to try to do my best to empirically figure out the value of a game, I would add it to (place it side by side with; person who has no move in either game loses) games whose values I knew, and put the world's best players in front of the game (because they have the best hope of finding the winning plays when they exist).

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  • $\begingroup$ Interesting -- so you'd be looking at which of the two games your expert players are choosing to play in on each turn? What kind of information would that give you about the unknown game? $\endgroup$ – Skatche Mar 11 '14 at 20:36
  • $\begingroup$ In the examples you gave, Right loses if they have to play first, but don't forget Left may blunder -- we could imagine a game that goes $\lbrace 99\vert\rbrace\to 99\sim\lbrace 98\vert 100\rbrace\to 100\sim\lbrace -10,99\vert 101\rbrace\to -10$, and now Right has a winning strategy. So for optimal players any game greater than 0 may be interchangeable with any other, but for suboptimal players you might get more interesting effects. I suppose surreal numbers aren't the right objects though: taking the equivalence relation destroys a lot of the information about suboptimal play. $\endgroup$ – Skatche Mar 11 '14 at 20:44
  • $\begingroup$ @Skatche To your first comment: No; I would simply set up things like $G+99$ and $G+100$ and if Right often won the former and Left the latter, I might have grounds to bound the value of $G$ between $-99$ and $-100$. Second comment: I had intended the numbers to be in canonical form, so that right could have no moves at any subposition (for the most extreme case), but you bring up a good point: looking at surreal values won't distinguish between $\lbrace 99\mid101\rbrace$ and, say, $\lbrace 99,-1,-2,-3,-23897,-5897\mid101,100\rbrace$, so that the values don't account for chances to slip up. $\endgroup$ – Mark S. Mar 11 '14 at 23:32

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