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I feel like what we really want is a per-position model that takes the rating of the player into account. Rather than saying "what move wins this position" you try to predict "what would an 1800 do in this position". And then you can have a second model of "which move would most likely lead the current player to a victory, if the position was being played out by two players of rating X". It seems like you should be able to train these effectively given a large dataset of games with attached ratings. Maybe you don't even need to do any tree search, just modern neural net techniques.

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