I agree that the leaf outcomes should not be independent. But ultimately the value at the root is determined by the value at the leaves, not the other way round.hgm wrote:To me it is generating random outcomes at the leaves that seems wierd. Like all preceding moves did not have any effect on the outcome at all. You play a number of moves, and at the end you flip a coin to determine the outcome, and the probability does depend in no way on what you played. The moves only serve to alter your luck. What kind of game is that?
Of course this has the effect that the leaf values depend conditionally on the root value (in a probabilistic sense). This is why generating from the root may make sense.
Your model did not include such an ordering!hgm wrote: As to the binomial distribution:
If you suppose all sibbling nodes are equivalent, i.e. that the frequency of ++-++ is equal to that of -++++, +-+++ etc., then a binomial distribution fixes the frequency of any individual outcome in a way to make them independent. Only if you assume non-equivalence you could still get a binomial outcome with dependent win probabilities. E.g. that a + implies all its 'older' sibblings must also be +.
Concerning the model: it seems to yield the counter intuitive conclusion that in tactical situations (lots of singular moves) one should not search at all (actually depth 1 to get a move) and in quiet positions one should search as deeply possible.
One can probably fix this by including hard wins/losses (E=0). I recall reading an article once which showed that enough hard scores also have a negative effect on pathology (this is intuitively clear, if all scores are hard then there is no pathology).