Surreal numbers are used to represent values of positions in partizan games with perfect information and discrete outcomes (win/lose or win/draw/lose). The primary example is Blue-Red Hackenbush, but games such as chess also apply. In such real-world games played by humans, potential exists for mistakes, and positions are often valued in light of a player's propensity to win, or some related idea of "being less-far/farther ahead," or "winning by a narrow/wider margin." I want to know how well the magnitude of the surreal number value of Blue-Red Hackenbush encapsulates such an idea, and how well the connection generalizes to all combinatorial games grounded in the surreal numbers.
There are a lot of parameters to consider, some associated with one or more solution concepts in ordinary, non-combinatorial game theory, such as the trembling hand equilibrium. Mistakes might occur in the execution of a strategy, in the strategy itself, or either. Mistakes might occur with infinitesimal probability, real probability, or either. The probabilities might be constant, a function of the severity of the mistake, or either. There are surely many more factors, and an answer should feel free to incorporate the effects of others. Since surreal numbers represent values of positions, independent of the human playing them, some factors, such as the players having different probability distributions of making mistakes (per turn or per game?), and player psychology, may need to be excluded. It is unclear whether some concept of the difficulty of finding specific winnable plays (beyond just the ratio of winnable to non-winnable plays in that position) should be admissible, such as to model when players deviate from heuristics and sacrifice pieces in chess.
Despite all of these possible parameter combinations, experienced players of games like chess seem capable of intuitively approximating by how much a position favors a player, so perhaps many roads lead to the same conclusion the vast majority of the time. Can we make any sort of mathematically warranted statements in Blue-Red Hackenbush such as "optimizing the value of a position uniquely maximizes the ratio of winnable to non-winnable plays on each of your subsequent turns" or "it never matters which winnable play you make in terms of minimizing future mistakes"? If not, then can we at least show some statistical correlation between values of positions after a player's turn and her propensity to win in spite of potential mistakes?
Does the answer for Blue-Red Hackenbush generalize to all combinatorial games grounded in the surreal numbers, or does such an association have to be intentionally built into the way one assigns surreal numbers to a game's positions?