If you wanted to measure the strength of, say, a chess player, the best way would involve knowing the true value of each position: then you could compute the frequency $W$ with which the player finds a winning move in a won position, and $D$ of finding a drawing move in a drawn position.

Even without a perfect evaluation algorithm, perhaps mathematics offers the possibility of saying something about a player's $W$ and $D$? So I ask, do there exists tools in probability theory, if not for chess then at least for some class of idealized games (only the morphology of the game tree would matter) that would would allow prediction of one player's winning percentage over another given just the two players' $W$ and $D$ frequencies?

If yes, then the distribution of winning percentages in a population of players might serve as data for an inverse problem allowing the statistical estimation of $W$ and $D$ frequencies (or at least associated derived quantities, or relative quantities).

Also welcome: thoughts about refining the model in the second paragraph to get results more realistic for real world games like chess (e.g., separate frequencies for opening, middle game and ending).

Edit: While I appreciate critiques of my model, I hope the weakness of the model doesn't distract from the purely mathematical question of the 2nd paragraph: for suitable idealized games, can one compute dominance in the game globally from the players' $W$ and $D$ frequencies? If that probability question turns out intractable then my whole project sinks; if it has a positive answer, I can hope to refine the models.

I'm not attached to $W$ and $D$ as the ultimate measure of game playing strength. I am interested in the mathematical challenge of estimating these frequencies in the absence of an evaluation oracle.

Also, is it enough merely to point out the naivete of my model? Shouldn't the critic argue that my distortion has significant numerical effect on the dominance calculation?

doesdistort the nature of these "grades". Back later! :) – sleepless in beantown Dec 20 '10 at 1:29averagesthat may vary by changing the sample space. Available data will reflect how players perform in practical positions. The meanings of $W$ and $D$ should be conditioned on the same (fuzzily defined) sample space. – David Feldman Dec 20 '10 at 2:37