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 Post subject: Re: Principles from Basic Endgame Trees (Daniel Hu)
Post #21 Posted: Fri Apr 23, 2021 10:04 am 
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Cassandra wrote:
RobertJasiek wrote:
As is KataGo's? However, it may simply be that nobody has studied their endgame play carefully enough to verify it against modern theory. I may do it some time if I find the necessary time.

KataGo does   N O T   care about points!

In this sense, it might be a useless tool for you (during "play" / "analyse"), if you ever wanted to use its assistance for the verification of some parts of your "modern theory".
If KataGo has never encountered an endgame position in question (nor anything comparable), it is quite likely that it will be unable to find the "correct" (your understanding) endgame sequence, no matter how powerful the machine is it runs at.


One thing that I have noticed with the Elf annotated files is that Elf will frequently judge the human response more favorably than its top choice, especially when the human response was not on Elf's radar. Usually this may be considered noise, but it yields the possibility of coming up with alternative lines of play that may be better than the original ones produced by Elf. I don't know if that happens with KataGo, but I would try setting the komi for KataGo so that the winrate estimate is approximately 50% and seeing what happens. I am not talking about rare or unusual positions, but fairly common and everyday ones. :)

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 Post subject: Re: Principles from Basic Endgame Trees (Daniel Hu)
Post #22 Posted: Fri Apr 23, 2021 10:50 am 
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Bill Spight wrote:
One thing that I have noticed with the Elf annotated files is that Elf will frequently judge the human response more favorably than its top choice, especially when the human response was not on Elf's radar. Usually this may be considered noise, but it yields the possibility of coming up with alternative lines of play that may be better than the original ones produced by Elf. I don't know if that happens with KataGo, but I would try setting the komi for KataGo so that the winrate estimate is approximately 50% and seeing what happens. I am not talking about rare or unusual positions, but fairly common and everyday ones. :)

Dear Bill,

I am sorry, but I can only report on my experiences with the combination of KataGo and Igo Hatsuyôron 120, and here below especially for positions, for which we assume that KataGo has not sufficiently encountered these during the selfplay training. Simply because these positions are quite interesting for further comperative analysis, but "hidden" behind several mistakes of both sides (so there is no reason for KataGo to visit especially these positions during training). And yes, Igo Hatsuyôron 120 is kind of special, just because some decisive effects in the longish sequences seem to be far behind KataGo's event horizon, and so unavailable during "play" / "analysis".

You are right with emphasising the need to vary the komi during analysis. There will be cases where you will have to do so per move.
In my experience, it is especially important to achieve a win rate level (around 50 % likely) that stops KataGo from considering / favouring "desperate measures" for the side that seems to be too large behind for winning the game. These "desperate moves" are likely to result in an even worse result at the end of the game (human understanding, in points). On the other heand, you cannot trust that the side that is largely ahead (in win rate) favours the "best" move (human understanding, in points).
During this process, you must forget what you know about the position / result. It seems to be best to let KataGo believe -- based on its OWN analysis, not the absolut truth -- that the current position in nearly balanced.

However, you will encounter positions / sequences that are extremely sensible with varying the komi. In these cases, you will have to go back and forth some times, until you have a better understanding of what is going on.

There will be also positions, which ensuing sequences have several "local" maxima (human understanding, in points => several lines for winning the game, but with different scores at the end). If you vary the komi in these cases, it is likely that KataGo's favourite moves / sequences differ, dependent on which "local" maximum (being enough for winning the game with the current komi) is the nearest.

With Igo Hatsuyôron 120, we have the large advantage that we (believe to) have a quite good understanding what is going on, and indeed can find moves (sometimes) that are better (human understanding, in points) than what KataGo recommended, based on the current knowledge of the special net.
Based on this understanding, we can provide the selfplay training with suitable material for the database of starting positions to be examined (besides much more that KataGo chooses automatically). It needs a long time until KataGo has had so many selfplay games started from these positions that it will sustainable adjust its network, and also come up with the then "correct" line of play during "play" / "analysis".

To come back to the beginning:
The effect you described with Elf might have a similar reason. In a position that is "unknown" to the network used I assume it definitely possible that a human choice will be assessed better than the AI's favourite, once the stone has been put on the board. If the AI cannot rely on the knowledge in his network, but is dependent on its processing power only, you cannot be sure that the AI's favourite is really the "best" move.

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 Post subject: Re: Principles from Basic Endgame Trees (Daniel Hu)
Post #23 Posted: Fri Apr 23, 2021 1:55 pm 
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Cassandra, I think you might be overfitting too much to your experience on igoh120. It is rare for normal games to have such massively globally entangled tactics at the point where the endgame values of moves becomes small. And it's rare in normal games for the effect of every endgame exchange to be masked behind a long sequence of liberty filling of a huge capturing race before the position visually collapses down to the final result.

KataGo does care about points, about as much as human pros care about points in tournament games. Which is to say - just like human pros, KataGo will give up small amounts to play thickly when ahead, and escalate to risk large losses sometimes when looking for a place to resign. But it won't obviously play locally suboptimal moves, defend where completely pointless, fill in its own territory, etc, like some prior bots. And when ahead, it will still take free points when offered, kill groups if it is not risky, play decent basic endgame, etc - playing accurately enough so that even if not perfectly optimal, against amateur players, it tends to continue to gain many points in the endgame even under the constraint of not doing anything tremendously risky.


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 Post subject: Re: Principles from Basic Endgame Trees (Daniel Hu)
Post #24 Posted: Fri Apr 23, 2021 2:47 pm 
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@Bill or anyone else doing analysis - Iif you're doing any form of analysis, I recommend setting "analysisWideRootNoise" to a value like 0.05, or 0.10. This setting makes KataGo explore much more widely at the root, judging a lot more moves, and makes it a little less likely that a move is missed in the way you described.

You can also easily configure KataGo to search wider in general, not just at the root. For example, "nnPolicyTemperature=1.1" and "cpuctExploration=1.5" and "cpuctExplorationLog=0.6". This will also slightly reduce the risk of missing improved moves within the tree. Or try yet-more extreme numbers, play with it and see.

The reason these settings aren't default, of course, is that they make play worse on average, holding compute fixed. Because for play, you care a little bit more about finding a move that's good enough and being a bit more sure of the move you're planning to play that there is nothing wrong with it when you go deep. Whereas for analysis, depending on the goal of your analysis, maybe you care more about comparing alternatives, or finding the most likely optimal move, rather than a good-enough move, at the cost of searching less deeply and being less sure about any individual move that you might pick.


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 Post subject: Re: Principles from Basic Endgame Trees (Daniel Hu)
Post #25 Posted: Fri Apr 23, 2021 7:26 pm 
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lightvector wrote:
KataGo does care about points, about as much as human pros care about points in tournament games. Which is to say - just like human pros, KataGo will give up small amounts to play thickly when ahead, and escalate to risk large losses sometimes when looking for a place to resign. But it won't obviously play locally suboptimal moves, defend where completely pointless, fill in its own territory, etc, like some prior bots. And when ahead, it will still take free points when offered, kill groups if it is not risky, play decent basic endgame, etc - playing accurately enough so that even if not perfectly optimal, against amateur players, it tends to continue to gain many points in the endgame even under the constraint of not doing anything tremendously risky.

Dear lightvector,

I am afraid that my brief statement was too bold.

In my understanding, the training of AI programs (i.e. the network) follows the main goal of "winning the game" (as this is the final measurand at the end of a selfplay game). In doing so later during "play" / "analyse", the AI will optimise its prospects for winning the game, as you described in detail, and similar to what human professionals are used to do in important games.
(Probably too) Strongly abbreviated, this behaviour would correspond to "Care about win rate". E.g. avoid moves whose potential benefits (in points) are out of proportion to their likely risk (in percent). Do not struggle to maximise the final score, no matter the costs.

However, "winning the game" (seen globally) cannot be the principle object for "modern endgame theory", which is much more likely to stress the maximisation of the final score of the game.
Board positions for deriving and / or application of the "scientific" principles might be as artificial as the starting position of Igo Hatsuyôron 120 is.
Board positions, which were created for demonstrating that programs are not perfect in the endgame, will be artificial as well. "Solving" that endgame "problem" would be required in principle, and so would run into the same basic difficulties as with solving a very special classic middle game problem.

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 Post subject: Re: Principles from Basic Endgame Trees (Daniel Hu)
Post #26 Posted: Sat Apr 24, 2021 3:20 am 
Judan

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Back to the paper, chapter 1:

Chilling applies to all trees interatively by imposing the tax on the currently moving player.


Chapter 2:

Instead of the exercise, I would like to see two theorems and their proofs. Later, I will publish a proof related to the second part of the exercise for simple gotes without follow-ups (or real numbers, if you prefer) and decreasing-or-constant move values. For that, induction is not needed. Took me 20 minutes to prove. (Several others failed to prove this a couple of years ago at the DGoB forum. Taking the larger cake or calling it obvious do not count as a proof.) I suppose, proving for arbitrary gote trees must consume more time.

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 Post subject: Re: Principles from Basic Endgame Trees (Daniel Hu)
Post #27 Posted: Sat Apr 24, 2021 3:44 am 
Judan

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Section 3.1:

n seems to be the number of local endgames but what is 'a'? When considering the three games abd, bac, bca, how can one start with a but the others start with b when there is only one (n=1) local endgame?

What boundary and why at a = c?

I cannot verify the optimim scores and their conditions as long as it is unclear how many and what local endgames we have.

Principles 2+3 are informal and I suspect counter-examples exist but it depends on clarification as before.

Principle 4 and its argument are plainly wrong.

Consider as counter-example the local gote endgame (in move value | follow-up move value annotation) 100|1 and another local gote endgame 4|2.

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 Post subject: Re: Principles from Basic Endgame Trees (Daniel Hu)
Post #28 Posted: Sat Apr 24, 2021 3:59 am 
Judan

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Section 3.2:

"Decreasing difference" I and some number theorists studying such things call it "alternating sum" because the sign of each summand alternates.

Proposition 3:

Basic and important. I have proved such but would expect your proof if you state it as proposition. Right, telescoping terms are useful here but this phrase is new to me. Have you invented it or is it used regularly?

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Post #29 Posted: Sat Apr 24, 2021 4:12 am 
Judan

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Proposition 4:

Basic and useful. The proof is straightforward. I have been using such, too. I suppose |A+| means number of numbers in A+.

I prefer shorter annotation:

∆A := ∆(A)

A|b := A ∪ { b }

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Post #30 Posted: Sat Apr 24, 2021 7:13 am 
Judan

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Lemma 5:

How is the proof trivial? Either you or I make a sign mistake. Here is my attempt of a proof:

(Each u is in superscipt. Sorry, without mouse, marking individual letters is too difficult on iPadOS.)

∆A - ∆A|b
= ∆A+ + (-1)u∆A- - ∆A+ - (-1)u(b - ∆A-)
= (-1)u∆A- - (-1)ub + (-1)u∆A-
= (-1)u2∆A- - (-1)ub

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 Post subject: Re: Principles from Basic Endgame Trees (Daniel Hu)
Post #31 Posted: Mon May 24, 2021 8:15 am 
Judan

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Lemma 6:

The proof is correct.

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 Post subject: Re: Principles from Basic Endgame Trees (Daniel Hu)
Post #32 Posted: Mon May 24, 2021 10:27 pm 
Judan

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3.3 General case of n / introduction:

Assuming b, c, d unequal to all a_i move values may be necessary to proving but means that cases of equality demand further propositions.

What is "a difference game argument (strategy stealing)" for proving that choosing the largest depth 1 move value or play b? If something is trivial, write it down, please!

The paper claims

s_Bi+1 - s_Bi = a_2i+1 - a_2i+2 - ∆(a_2i+3,...,a_n,c) + ∆(a_2i+1,,...,a_n,c).

Let me try by applying the definition given in the paper:

s_Bi+1 - s_Bi = (a_1 - a_2 +... + a_2i+1 - b + ∆(a_2i+2,...,a_n, c) - (a_1 - a_2 +... - a_2i + b - ∆(a_2i+1,...,a_n,c)) =

a_1 - a_2 +... + a_2i+1 - b + a_2i+2 - a_2i+3 +... -+ a_n +- c - (a_1 - a_2 +... - a_2i + b - a_2i+1 + a_2i+2 -... +- a_n -+ c) =

a_1 - a_2 +... + a_2i+1 - b + a_2i+2 - a_2i+3 +... -+ a_n +- c
- a_1 + a_2 -... + a_2i - b + a_2i+1 - a_2i+2 +... -+ a_n +- c =

2a_2i+1 - 2b +- a_n-1 -+ 2a_n +- 2c.

I cannot confirm the paper's claim and, at least not in its way, subsequent conclusion s_Bi+1 - s_Bi >= 0.

Therefore, I am not motivated to
- try exercise 7 (What is its answer?),
- verify s_Wi+1 - s_Wi <= 0 (If it is "amazingly simple", please write down its proof!),
- verify the alleged conclusion "Hence it is correct strategy for Black after 2i moves to delay playing move b if and only if [he can gain] from doing so and White cannot stop him. That is [...,] there is [...] s_Bi+j > s_Bi but s_Wk > s_Bi for all i <= k < i+j. The correct timing to play at b is just before the increasing s_Bi and the decreasing s_Wi cross. [...A] greedy algorithm (looking only one move ahead) works."

EDIT

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 Post subject: Re: Principles from Basic Endgame Trees (Daniel Hu)
Post #33 Posted: Mon May 24, 2021 11:06 pm 
Judan

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You study one local gote with both players' simple follow-ups in an environment of simple gotes without follow-ups. I studied this four years ago but still need to publish it. My study allows equalities. I found major cases depending on how the temperature compares to the two follow-up move values. For low temperature, play locally. Medium or high temperatures are rather demanding and require different alternating sums of the intermediate move values of the environment.

Later in your paper, you also speak of sente. I have still to read the remaining paper. Since it declares to study local gote endgames only, I need to see later what you mean by "sente".

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Post #34 Posted: Mon May 24, 2021 11:23 pm 
Judan

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Principle 5 says: "Don't play on b unless your opponent can get more from playing b on their next turn. Otherwise play at b."

What is "gaining more on the local reply than the first local play gains"?

In a local gote, the reply gains less. In a local sente, the first local play and its reply gain the same: the follow-up move value. If, however, the first local play (such as a self-atari) gains less than the local reply, then the first local play is a mistake!

So what shall principle 5 tell us?!

My suspicion is that you do NOT mean gain(s). Maybe you mean move values and that the opponent's reply has a larger move value than the player's first local play? That would be a local sente. However, your paper has declared to only study local gotes (decreasing move values).

So what shall principle 5 tell us?!

Is the subsequent text on 1 1/3 pages supposed to be its proof? A proof of what meaning of principle 5?


EDIT:

Oh, maybe principle 5 has a typo: "unless" -> "if". If so, it would read:

"Don't play on b if your opponent can get more from playing b on their next turn. Otherwise play at b."

Now, THIS is self-explaining:) Do you mean this? (We presume no kos now or later. In a ko fight, it can sometimes be correct to incur a local loss.)


EDIT 2:

The second sentence "Otherwise play at b." demands context. It is only a truth for low temperature. For high temperature, playing locally at b can be wrong. E.g., if the temperature is 100 but the local move values are small.

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 Post subject: Re: Principles from Basic Endgame Trees (Daniel Hu)
Post #35 Posted: Tue May 25, 2021 12:23 am 
Judan

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RobertJasiek wrote:
s_Bi+1 - s_Bi = (a_1 - a_2 +... + a_2i+1 - b + ∆(a_2i+2,...,a_n, c) - (a_1 - a_2 +... - a_2i + b - ∆(a_2i+1,...,a_n,c)) =

a_1 - a_2 +... + a_2i+1 - b + a_2i+2 - a_2i+3 +... -+ a_n +- c - (a_1 - a_2 +... - a_2i + b - a_2i+1 + a_2i+2 -... +- a_n -+ c) =


Presumably I have made a mistake here. In 3.3, c is not specified. However, now, I guess that c is assumed to be like in lemma 6. That is, it need not occur after a_n. The paper should clarify the presuppositions for 3.3, in particular for c. So we have:

s_Bi+1 - s_Bi = (a_1 - a_2 +... + a_2i+1 - b + ∆(a_2i+2,...,a_n, c) - (a_1 - a_2 +... - a_2i + b - ∆(a_2i+1,...,a_n,c)) =

a_2i+1 - b + ∆(a_2i+2,...,a_n, c) - b + ∆(a_2i+1,...,a_n,c) =

-2b + a_2i+1 + a_2i+2 - ∆(a_2i+3,...,a_n, c) + ∆(a_2i+1,...,a_n,c).

Instead, the paper writes

a_2i+1 - a_2i+2 - ∆(a_2i+3,...,a_n, c) + ∆(a_2i+1,...,a_n,c).

Still somebody must have made a transformation mistake:)

we do not further transform the alternating sums but apply lemma 6 directly to them. Ok, but first let us clarify which transformation is right!


Last edited by RobertJasiek on Tue May 25, 2021 4:43 am, edited 2 times in total.
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 Post subject: Re: Principles from Basic Endgame Trees (Daniel Hu)
Post #36 Posted: Tue May 25, 2021 3:21 am 
Judan

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On pages 5+, I expected a proof of something before but find a new study. I see. However, I am unsure what is being studied there.

The first sentence there suggests a study of generalisation to games of larger depth. What shall have larger depth? Only b? Also any a_i? In the subsequent study, I do not recognise generalisation to games of larger depth. Now I wonder whether the first sentence there stands on its own and is entirely unrelated to the subsequent study on pages 5+, or whether that study is meant for generalisation to games of larger depth. Please clarify!

In the book Combinatorial Game Theory, Aaron N. Siegel, chapter VII and other CGT literature, one finds the topics of stop, score, rich environment, orthodoxy, sentestrat and (without rich environment) orthodox accounting. They study generalisation to games of larger depth but the error of an approximation in a thin environment can be T/2 (half the temperature).

You presume an irregular environment and want optimal scores and correct first moments of playing locally. For that, generalisation to games of larger depth still appears to be too hard. For an environment of simple gotes without follow-ups (you would say depth 1) and one local gote endgame of depth 2 with the initial move value b and simple follow-ups of both players (whose move values you call c and d), a complete analysis is possible. As I have said, I have done it. My guess is that, on pages 5+, you also try to do it, although still without equalities.

Daniel, please confirm that, on pages 5+ except for page 5 sentence 1, you study an environment of simple gotes without follow-ups (you would say depth 1) and one local gote endgame of depth 2 - or tell us what you mean to study there!

As long as this is unclear and I am not convinced of all now applied earlier results, I do not study every detail on pages 5+. Nevertheless, I have noticed some things during my earlier readings as follows.

What justifies that the following conditions determine optimal play?

s_Wi-1 > s_Bi > s_Wi

s_Bi+1 > s_Wi > s_Bi

My own study has been different: I have studied the cases Black starts or White starts separately. Instead, you seem to combine and relate both cases so I want to see your justification why you may do so and why these inequations express the right last / first moments.

You are on track when mostly the intermediate parts A_| of alternating sums remain.

In your case analysis (cases 1 to 4), you compare a_2i+1 to c and d. You do so because you concentrate your study on the verge between the last moment of playing in the environment and the first moment of playing locally. My own study instead compares the (dynamically changing) temperature T (which you might call a_1) to c and d (for which I use different letters;) ). Basically, my cases are the same but I also allow equality:

Case 1: T >= c, d AND NOT T = c = d
Case 2: c > T > d
Case 3: d > T > c
Case 4: c, d >= T (including T = c = d here is easier)

Furthermore, you are also on track that some values are multiplied by 2 but others aren't.

In my study, the notation greatly differs from yours but the necessity to fight against signs is familiar:) Therefore, I cannot say yet whether (except for equality, which you postpone) your results agree to mine for an environment of simple gotes without follow-ups and one local gote endgame of depth 2.

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Post #37 Posted: Tue May 25, 2021 4:42 am 
Judan

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RobertJasiek wrote:
s_Bi+1 - s_Bi = (a_1 - a_2 +... + a_2i+1 - b + ∆(a_2i+2,...,a_n, c) - (a_1 - a_2 +... - a_2i + b - ∆(a_2i+1,...,a_n,c)) =

a_2i+1 - b + ∆(a_2i+2,...,a_n, c) - b + ∆(a_2i+1,...,a_n,c) =

-2b + a_2i+1 + a_2i+2 - ∆(a_2i+3,...,a_n, c) + ∆(a_2i+1,...,a_n,c).
[...]
Still somebody must have made a transformation mistake:)


Me, of course:) Indices must be substituted i -> i+1 properly and carefully! Since the inserted index term is multiplied by 2, signs keep their parities. Let me try 3.3 / introduction again:

s_Bi+1 - s_Bi =

(a_1 - a_2 +... - a_2(i+1) + b - ∆(a_2(i+1)+1,...,a_n, c) - (a_1 - a_2 +... - a_2i + b - ∆(a_2i+1,...,a_n,c)) =

(a_1 - a_2 +... - a_2i+2 + b - ∆(a_2i+3,...,a_n, c) - (a_1 - a_2 +... - a_2i + b - ∆(a_2i+1,...,a_n,c)) =

a_2i+1 - a_2i+2 - ∆(a_2i+3,...,a_n, c) + ∆(a_2i+1,...,a_n,c)),

as the paper states.

Now (see the paper for the detailed case analysis c > a_2i+2 or c < a_2i_2) lemma 6 implies

-∆(a_2i+3,...,a_n, c) + ∆(a_2i+1,...,a_n,c)) >= a_2i+1 - a_2i+2 >= 0 so

s_Bi+1 - s_Bi = (a_2i+1 - a_2i+2) + (-∆(a_2i+3,...,a_n, c) + ∆(a_2i+1,...,a_n,c)) >= 0 + 0 = 0.

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Post #38 Posted: Tue May 25, 2021 6:53 am 
Judan

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Conjecture s_Wi+1 - s_Wi <= 0. Proof:

s_Wi+1 - s_Wi =

(a_1 - a_2 +...+ a_2(i+1)+1 - b + ∆(a_2(i+1)+2,...,a_n,d)) - (a_1 - a_2 +...+ a_2i+1 - b + ∆(a_2i+2,...,a_n,d)) =

(a_1 - a_2 +...+ a_2i+3 - b + ∆(a_2i+4,...,a_n,d)) - (a_1 - a_2 +...+ a_2i+1 - b + ∆(a_2i+2,...,a_n,d)) =

(a_1 - a_2 +...+ a_2i+3 + ∆(a_2i+4,...,a_n,d)) - (a_1 - a_2 +...+ a_2i+1 + ∆(a_2i+2,...,a_n,d)) =

-a_2i+2 + a_2i+3 + ∆(a_2i+4,...,a_n,d) - ∆(a_2i+2,...,a_n,d) =

// We have d < b < a_2i+3.

-a_2i+2 + a_2i+3 + ∆(a_2i+4,...,a_n,d) - a_2i+2 + a_2i+3 - ∆(a_2i+4,...,a_n,d) =

-2a_2i+2 + 2a_2i+3 <= 0.

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Post #39 Posted: Tue May 25, 2021 7:36 am 
Judan

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The changes in scores are also described by the following more general proposition of combinatorial game theory for enriched scores of a local position P comprising one or several local endgames, which can be iterative, in a rich environment. We presume no kos now or later. B means Black starts, W means White starts, the index is the temperature.

For two real numbers T > S > 0, we have
T - S ≥ B_S(P) - B_T(P) ≥ 0,
T - S ≥ W_T(P) - W_S(P) ≥ 0.
In particular, B_T decreases in T, W_T increases in T and each is continuous in T.

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 Post subject: Re: Principles from Basic Endgame Trees (Daniel Hu)
Post #40 Posted: Tue May 25, 2021 10:26 am 
Judan

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Exercise 7:


"if and only if": I do not buy this. E.g., equality can also mean a_1 = a_2 =... = a_n = b = c = d including an ambiguous local endgame or some subset of such equalities.


"If max (c, a_2i+2) >= a_2i+1 => s_Bi+1 = s_Bi": Proof:


Case max (c, a_2i+2) = c:

Study s_Bi+1 = s_Bi <=> b - a_2i+1 + c = a_2i+1 - b + c
because these might differ but played in either stated order.

By definition of b and c, we have b >= c.
By definition of a_2i+1 and a_2i+2, we have a_2i+1 >= a_2i+2.
By the initial and case assumptions, we have c >= a_2i+1.
Hence b >= c >= a_2i+1 >= a_2i+2.
In the second played order, we also have a_2i+1 >= b.
Hence b = c = a_2i+1.
Hence b - a_2i+1 + c = a_2i+1 - b + c.


Case max (c, a_2i+2) = a_2i+2:

Study s_Bi+1 = s_Bi <=> b - a_2i+1 + a_2i+2 = a_2i+1 - b + a_2i+2
because these might differ but played in either stated order.

By definition of a_2i+1 and a_2i+2, we have a_2i+1 >= a_2i+2.
By the initial and case assumptions, we have a_2i+2 >= a_2i+1.
Hence a_2i+2 = a_2i+1.
In the second played order, we also have a_2i+1 >= b >= a_2i+2.
Hence a_2i+1 = b = a_2i+2.
Hence b - a_2i+1 + a_2i+2 = a_2i+1 - b + a_2i+2.

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