Hi Kenneth,
you are right. It is more important the horizontal of action (the exponent)
than the branching factor (the base) for that exponentiation formula. But I
believe that the formula num_turns^branch_fact is not a reliable measure
for the depth of a game.
I have two arguments:
1. We should count the number of important decisions per turn, not the
general number of move per turn. Take this stupid game played on a goban:
in every turn the player drop a single stone, the winning player is the
last dropping player. This game is as deep as Go with that formula,
actually it has zero depth.
2. If we have an opaque and/or a highly tactical game (with zero strategy)
that has an average of 300 turn, those turns don't add extra depth to the
game.
About the first point, we usually can ignore it. But I believe the second
point is important for designing a game.
If we want true deep games, difficult for AI, we have to focus on big
branching factor and extremely stratigical games. In my view a strategical
game is a game in which is "easy" to look ahead and that this looking
ahead is the central skill for winning the game.
I’m trying to follow this way designing my cv Kingdrops:
http://www.chessvariants.org/index/displaycomment.php?commentid=29503