Endgame: How to Divide by 2, 4, 8?

[RobertJasiek]

Some people have claimed that a local gote endgame could be evaluated by developing the settled followers. For the count of each follower reached after 1, 2, 3... plays, divide by 2, 4, 8... Except for the fun with large fractions, this sounds convincing until you try to actually do it...

Let me simplify by assuming that each follower is a local gote (none of the followers is a local sente, ko or something else). Hint: settled positions can contain black and white territories. We might use Black's value perspective, but maybe you want to suggest to sometimes use White's.

What is the resulting general method and correct calculation of the initial local gote endgame's move value and count? You have already made all those aforementioned divisions. Now, how to add or subtract the values to derive move value and count?

In my endgame theory, I have avoided such by proceding step by step, one division by 2 after another (or any other operation for other types of local endgames and their followers), determination of one count after another.

However, I would like to know: is division by 2, 4, 8, 16, 32, 64, 128... just a nightmare myth or a well worked out alternative method for people preferring divisions by large numbers?

[RobertJasiek]

Since nobody can defend or explain such a method, it must have been a myth. Divisions by 16 have been fear mongering.

[dhu163]

If you are talking about my ideas, my problem with starting from the bottom of the tree is that the tree is too big an complicated for a real game, and the details don't matter for a game. If you want perfection, I agree the only way is to start from the bottom and divide by 2 at a merge.

However, in a real game, approximations work very well (and miai counting can help justify this). The detail is how you approximate the count of followers without working them out perfectly. In a real game, since endgames are also fixed to space, moves tend to also crawl like a tree on the board (and that is only 2d not like infinity d possible for general endgames). Hence, the goal is to just say the top count is the average of the count of the two followers without doing much more calculation than that.

This is part of my idea behind
https://www.lifein19x19.com/viewtopic.p ... 60#p273160

[RobertJasiek]

"the top count is the average of the count of the two followers"

Which two followers?

What do you mean by top count?

[dhu163]

I mean the count of the position without any moves played.

[RobertJasiek]

In practice, only considering the initial position's two children is often not good enough. A realistic approximation is after three moves in every relevant branch and by using multiples of 4 there if a gote follower.

Of course, local endgames elsewhere determine how roughly one can approximate or how accurate calculations need to be. Some comparisons are "roughly 7 versus roughly 4" while other comparison are "two values that are roughly 4" and need accurate calculations.

[dhu163]

Ok, I think I understand what I meant.

The point is that like the difference between pointwise and uniform convergence, if you are only given the tree of scores, then starting from the bottom is the only way to link the top count with given scores.

However, we also have the board graph, which allows intuition at least to make associations. There are discrete intersections on the board and the effects of moves are primarily local with follow ups being near to preceding moves. We think that a move will shift the control of some intersection from say 40% to 90%. So the question of detail is how to work out the association between moves with board control.