Values of moves

[Bill Spight]

Bill & Robert have taken mathematics, applied it to the endgame of Go and have created what we may call "end game theory based on mathematical approach".

The only thing in this theory (about non-kos) that is original with me is the colored masts of thermographs and the concept of ambiguous positions and plays. I discovered the method of assuming that a position or play is gote until you can show that it isn't, but I'm sure that I'm not the only one. I also discovered that double sente does not make any sense for evaluation. But I'm not the first to do that, either.

It not only includes mathematical devices such as value trees and statements in formal logic, it also defines new terms as "count", "value" and "gain". Their aim and interest is precise calculation and unambiguous description.

Not for the masses. I may make up problems that depend upon very small differences, but have always said that precise calculation in rarely needed in practice. As for those terms, value and gain are simply English. Count comes from Berlekamp. When I was starting my endgame study as a 4 kyu, I simply called it territory, like the books did. But I discovered on rec.games.go over 20 years ago, the instant you call it territory, somebody jumps up and says, "That's not territory!" or "You can't have ½ a point of territory!" So I now say count in self defense.

I see three hurdles when trying to cross the chasm: the math, the new concepts, some of which verbally overlap with old ones, and the high degree of precision which is held by the current way of articulating things. Of these, I believe the concepts should remain: the definitions of count, value, gain ... counterintuitive as they may be.

If those terms seem counterintuitive to you, all I can say is Mea culpa.

As Robert points out, precisely the confusion they create when holding onto tradition is the progress we're looking for. The math may have to go, or should not be intertwined with the text so as to represent the mathlish hurdle. The biggest problem may lie in letting go some degree of the precision reached by Bill & Robert in favor of practicality.

I don't know how to do this.

The main problem, IMX, is not the math, it's the concepts. Old ideas die hard. I never got my paper contra double sente published, and I never will. It's not like the best players don't understand. While kibitzing a game I made a joke to Jujo about a player ignoring a double sente and he looked at me like I was crazy. He did not even think about double sente. (OC, he does not read books aimed at amateurs.) The example from the Nogawa and Shimamura book, which came out in the mid-20th century, of a whole board with several "double sente" positions, was played correctly, with no mention of double sente in the text. But the latest (I think) edition of the Nihon Kiin's Small Yose Dictionary, has a similar example where whoever plays first gets all the double sente.

O Meien's book has turned the corner, and he has prestige. He does not criticize double sente, but he makes no use of it. And he focuses on gain (the value of one move). As I have pointed out, his definition of local sente is not quite right, but it's close, and his examples are correct. (Maybe one is ½ pt. off, though. )

Edit: One new idea is calling positions, as well as plays, sente or gote. People seem to resist that.

[Pio2001]

A lot of this discussion does not pick me up where I am. Where I am is not knowing when to play a sente, and not having a rough knowledge of its value or urgency compared to playing a big gote.

Hi Daal,
Here is how I see things at my modest level : we must evaluate if the first move is "sente enough", or "not enough sente" to be played first.
If the opponent ignores the sente move and takes the gote instead, we must be able to punish him. Therefore the gain of the sente threat must be higher than the gain of the gote move. If not, it means that the first move is not sente enough.

Example : we have two endgames to play.
Endgame A is sente for us. If we play it, we gain 5 points. If the opponent plays it, he gains 5 points.
Endgame B is double gote. If we play it, we gain 50 points, if the opponent plays it he gains 50 points.

Question : should we play A or B ?
My answer would be : how much sente is A ?

Let's say that if the opponent doesn't answer A, the follow-up gains 100 points gote.
Then we must play A first. The opponent must answer, or he'll loose 100 points. Then we can play B too.

Now, imagine that if the opponent doesn't answer A, the follow-up gains 20 points gote.
Then we must play B first. We gain 50 points. The opponent plays A and gains 5 points. We have gained 45 points.
If we had started with A, the opponent would have answered B, gaining 50 points gote, and we would have punished by executing the sente threat... worth 20 points only. We have lost 30 points !

Question for the experts :
If we calculate the moves values, we find
For A : (5-(-5)) = 10 points (for a sente endgame, the difference is taken as it is)
For B : (50 - (-50))/2 = 50 points (for a double gote endgame, the difference is divided by 2)

In the case where the move A is sente enough, I come to the conclusion that I should play the move with the smallest value first. Is it correct ?

In other cases, for example:

If you have some number, N, of a position such that the total score, S, of all of them together is the same, regardless of who plays first (they are miai), then the average value of each of them is S/N. That works for gote positions.

are beyond my ability to rephrase into something I can understand.

Yes, the original sentence is hard to undestand. I don't know what is "a number of a position" (bad english ?), but I suppose that it means "N independant local sequences". However, in this case I don't know what would be the "score" of a sequence. Maybe its "value" ? But in this case, is it its "swing" value or "miai" value ?
So it should rather mean the "count" of N different local positions. But in this case, I don't understand the part "regardless of who plays first". The count of a position is the count of the position. No one is supposed to "play".
So maybe this is the count of the followers. But in this case are we talking about N positions with 2xN followers ?

Well, I don't understand anything either... But I don't think that it is because of a lack of "mathlish" ability on our part. Either the sentence needs to be reworded, or just lack the formal definitions of "score" and "value".

[Bill Spight]

As long as everything is simple gote without follow-ups, understanding it is easy and sorting move values is easy. Gote with one follow-up move is maybe half as easy.

Already a simple sente is much less easy, unless you ignore most of its behaviour and pick only one aspect.

Local endgames with several follow-ups or combinations of several local endgames (of which at least some are not simple gotes) are much more difficult than simple gotes. . . .

The jump from understanding simple gotes to understanding other selections is steep.

Unfortunately so. But the resistance to ideas that help us to understand the more difficult positions comes well before we get that far. {sigh}

BTW, when I talk about resistance I am not talking about our current audience. But to take a recent example, the reason only the first part of my article in Myosu was published, with the rest supposed to come later, was the resistance from one of the editors, a very strong amateur. Apparently my explanation of the correct play offended his sense of sente. {shrug}

[Knotwilg]

If those terms seem counterintuitive to you, all I can say is Mea culpa.

They are counterintuitive, not because you explained badly or found the wrong word but, intuitively

- count means: (verb) to count something and ending up with a whole number; (noun) the count, which can be a final score, a number of moves, or perhaps the value of something
- value means: what something is worth (to me); intuitively, a move should be equally valuable for two players; it isn't but that's not your fault
- gain means: how much something (a move) gains (to me); that assumes a starting position, a zero towards which a move adds something (dare I say a value)

Let me give some examples of intuitive concepts:

- a position: we all understand what that is
- a move: same
- Lizzie's probabilities: the chance of me winning, if I play a certain move; incidentally, the chance of the opponent winning is 100% minus my chance. This is intuitive.
- Lizzie's plies: the number of games she has evaluated

Being non-intuitive is not a privilege for endgame theory or mathlish:

- miai counting: this is a salad of a Japanese go term and a basic English verb in the math domain; my reference for miai is "(making) equivalent options for a certain objective"; this doesn't help me at all in "miai counting"
- deiri counting: same but worse/better, I don't even have a reference for "deiri"
- sente/thickness: remember the days ...

Even when translating sente as "having the initiative" or "keeping the initiative", it's problematic at the intuitive level. It's either my turn or it's your turn. How is "having the initiative" different from "it being my turn" and how on earth can I keep the initiative with alternating moves - intuitively speaking.

(rant off)

[Bill Spight]

Hi, Pio. If you don't mind, I'll make a comment or two.

Example : we have two endgames to play.
Endgame A is sente for us. If we play it, we gain 5 points. If the opponent plays it, he gains 5 points.
Endgame B is double gote. If we play it, we gain 50 points, if the opponent plays it he gains 50 points.

This indicates that you do not really understand sente. Not your fault, OC, given what's out there. You may not understand gain, either. If so, nobody taught you, and you are making a reasonable guess.

In other cases, for example:

If you have some number, N, of a position such that the total score, S, of all of them together is the same, regardless of who plays first (they are miai), then the average value of each of them is S/N. That works for gote positions.

are beyond my ability to rephrase into something I can understand.

Yes, the original sentence is hard to undestand. I don't know what is "a number of a position" (bad english ?), but I suppose that it means "N independant local sequences".

Actually, it is talking about positions. See my example of two of one position in this note: viewtopic.php?p=236716#p236716

However, in this case I don't know what would be the "score" of a sequence. Maybe its "value" ? But in this case, is it its "swing" value or "miai" value ?

The two positions together have a score, because they are miai.

So it should rather mean the "count" of N different local positions. But in this case, I don't understand the part "regardless of who plays first". The count of a position is the count of the position. No one is supposed to "play".
So maybe this is the count of the followers. But in this case are we talking about N positions with 2xN followers ?

I hope the example has made all of this clear.

Edit: There are positions, or combinations of positions, that have a score, but the rules will not assign them a score unless they have been played out. Sometimes you will see such positions at the end of ancient games, before they had written rules.

[RobertJasiek]

Pio2001, when to play a sente depends on what is in the environment. Your question may presume the early endgame with a "rich" environment and its temperature T (largest move value). Let us simplify assuming the local sente has a move value M and, after the first move of the sente sequence, a follow-up move value F.

As long as the temperature is high with F < T, play in the environment. Matters become more interesting when the temperature becomes low with F > T. We want to delay playing the sente as long as possible to 1) retain ko threats and 2) give the opponent the chance of making the mistake of playing reverse sente prematurely. Simply speaking, for that purpose, we compare M and T. Bill has become more precise but we might be defensive anyway and play the sente a bit earlier than at the last moment suggested by the simplifying theory, which presumes a possibly slightly unrealistic ideal environment.

[Bill Spight]

Bill has become more precise but we might be defensive anyway and play the sente a bit earlier than at the last moment suggested by the simplifying theory, which presumes a possibly slightly unrealistic ideal environment.

I remember when, as a 4 kyu, after I had learned how to evaluate sente, I noticed pro endgames where, all of a sudden, the players started taking their sente "early", when there were larger plays on the board. OC, the players had realized that they did not need to save their sente for use as ko threats.

[Knotwilg]

OK, I have reread this whole thread and I've understood the fairly simple concepts of "value of a position" and "value of a move". This is my attempt at rewriting the ideas laid out by Bill & Robert, in a structure that is consumable by John.

1. Introduction

The aim and claim of modern endgame theory:
Playing moves in order of their value (as in value of the move) will give the best possible result for both.

The value of a movecan be easily derived from the "value of a position".

The value of a position is the crucial concept.
- It is recursive, meaning the value of a position is obtained from the value of the resulting positions, the one where Black plays first and the one where White plays first.
- It is probabilistic, meaning it is not known who will play first.

We always look at the position from Black's perspective.
We will first simplify the discussion in two ways:

1) there is no recursion.
2) chances for White or Black to start are equal, i.e. 50-50, i.e. 1/2 each

We will then add recursion and unequal chances.

2. The simplest case: no recursion, equal probability

Definions, in text:

The value of a position is the average (1/2) of the territory Black has if Black plays first and the territory Black has if White plays first.
The value of Black's move is the difference between the territory Black has if Black plays first and the value of the position.

We can rewrite this in math, using the following abbreviations
P = the value of a position
Tb = the territory Black has if Black plays first
Tw = the territory Black has if White plays first
Mb = the value of Black's move
Mw = the value of White's move

By definition, rewriting above text:
P = 1/2 (Tb+Tw)
Mb = Tb - P
likewise
Mw = Tw - P

As Mitsun has shown you can also write
Mb = Tb - 1/2 (Tb+Tw) = 1/2 (Tb - Tw)
Mw = Tw - 1/2 (Tb+Tw) = 1/2 (Tw - Tb)

So Mb = -Mw and we can speak of "the" value of a move M = Mb, from Black's perspective. The value of White's move is the negative, from Black's perspective and equal from White's perspective, so we don't have to keep track of either perspective.

-
3. Adding recursion

Adding recursion just replaces the values obtained by counting the territory, with the values of the next position, which you can compute from the next position ... and so on, until you can count the territories. This is where the game tree comes into the discussion.

In text:

The value of a position is the average (1/2) of the value of the resulting position if Black plays first and the value of the resulting position if White plays first.
The value of a move is the difference between the value of the resulting position if Black plays first and the value of the position.

In math:
P = 1/2 (Pb+Pw)
M = Pb - P

Where
Pb = the value of the resulting position if Black plays first
Pw = the value of the resulting position if White plays first

(as you can see, the math makes it shorter)

4. Adding unequal probability

Things become more complicated if the probability that Black will play there is not equal to the probability White will play there. Here the concept of sente comes into the discussion.

If a move is sente for Black, then the probability Black will play there is 1, not 1/2 and White's probability is 0, not 1/2. So you can just play it, let White answer it, and then average the resulting position, like above.

Double sente would mean that the probability is 1 for Black and 1 for White. This is impossible in probability theory. This is why Bill says "double sente does not exist" and "sente gains nothing". He does not mean "you shouldn't play sente", on the contrary.

Now, rewriting "sente for Black" as "the probability Black will play there is 1" is of course avoiding the question why this would be so. We know that the heuristic is that a follow-up for Black would be bigger than anything else on the board. This includes the follow-up of White's tenuki, should she not answer Black's move.

This is, I believe, where the "follower" term comes into the discussion, but this is also where I will stop for now, giving Bill & Robert the chance to point out misrepresented ideas of theirs, and John and others to say that this still makes them lose the will to live.

Dieter

PS: Side note about fractions

Fractions necessarily arise from averaging. You will see values like 2,5 from dividing by 2. You will see values like 1,25 from an extra level in the game tree. And you will see a 3 in the denominator due to kos.

As Robert says, rounding can be done if there's no impact on the order of playing moves according to their value. In late endgame, this is almost impossible. As such, it is probably undesirable at all. Rounding creates the illusion that there is no game tree with probabilities, only whole values which can directly be assigned to moves. This is an illusion.

[Bill Spight]

1. Introduction

The aim and claim of modern endgame theory:
Playing moves in order of their value (as in value of the move) will give the best possible result for both.

Except when it doesn't.

The value of a movecan be easily derived from the "value of a position".

Except when it's not so easy.

[Knotwilg]

Here is the simplest sente or reverse-sente situation I can think of. Let's see if we can agree on how this is evaluated. No mathlish involved (I hope).

• W to play (reverse-sente) can create a final position worth 0 points. It is then B turn to play, somewhere else on the board.

• B to play (sente) can create a final position worth 2 points after W replies. It is then B turn to play again, somewhere else on the board.

• (The B move, which we are calling sente, leaves a follow-up move worth 100 additional points. This is so much larger than anything else on the board that W will certainly respond locally to prevent it. This justifies treating the B move as sente.)

Questions:
1) What is the value (count) of the initial position (I believe the answer is 2)
2) What is the value of the W reverse-sente move? (I suspect the answer is 2)
3) What is the value of the B sente move? (I know Bill's answer is 0)

Bill or Robert, are these indeed the values you would calculate?
John, would O Meien also calculate these same values?

Rereading this post after having made my above analysis, I come to the same conclusion as Mitsun.
The 100 value is to ensure sente, i.e., that Black will play there with probability 1.

Pb = 2
Pw = 0
P = 1*Pb + 0*Pw = 2
Mb = Pb - P = 0
Mw = Pw - P = -2 (so 2 from White's perspective)

I can only see the 100 value play a role in determining which of the available "local sente" moves is globally sente. Is that what you mean with "sente move value = 100"?

Now I am confused (again)

[Bill Spight]

Sente or gote position?

Enjoy!

[Knotwilg]

I understand this but this is about "local sente", where the value of the follow-up move is bigger than the value of the move itself. (where I don't know anymore what the "value of the move" is, since in global terms it is equal to the value of the resulting position fro a sente play)

Global sente means, the value of the follow-up move is bigger than anything else on the board.

Local sente (for which you provide a good algorithm) and global sente are very different things. They look so different that I would hesitate to use the same noun.

[Bill Spight]

I understand this but this is about "local sente", where the value of the follow-up move is bigger than the value of the move itself. (where I don't know anymore what the "value of the move" is, since in global terms it is equal to the value of the resulting position fro a sente play)

Global sente means, the value of the follow-up move is bigger than anything else on the board.

Local sente (for which you provide a good algorithm) and global sente are very different things. They look so different that I would hesitate to use the same noun.

Global gote is also local gote. If you have a global sente that is also a local gote, its reply is worth less than it is. So it would also be worth playing if it were the same size but not global sente. It doesn't matter what you call it.

As far as the value of a move goes, think in terms of gain.

Edit: Also, for sente it helps to think about how much the reply gains, which is not necessarily what the follow-up gains if your opponent plays elsewhere.

[Knotwilg]

Global gote is also local gote. If you have a global sente that is also a local gote, its reply is worth less than it is. So it would also be worth playing if it were the same size but not global sente. It doesn't matter what you call it.

As far as the value of a move goes, think in terms of gain.

Edit: Also, for sente it helps to think about how much the reply gains, which is not necessarily what the follow-up gains if your opponent plays elsewhere.

Sorry Bill, I don't understand anything of what you say here. I'm really willing to study the subject now, since I want to bridge the gap with those who don't want that, but do want to get insight in the value of modern endgame theory.

But your reply to my attempt at summary was:
- some joking remarks "except when ..." which mystified, not clarified
- a helpful diagram on a topic that seemed complementary to my initial summary, in response to my attempt at interpreting mitsun's earlier try
- and now this, which I don't understand

It may be me, but unless proven otherwise, I would conjecture that you are not in touch with the vast majority here.

Global gote is also local gote.

Maybe understand this. Missing definitions. Missing thought.

If you have a global sente that is also a local gote, its reply is worth less than it is.

Nope, not getting this.

So it would also be worth playing if it were the same size but not global sente.

Nope. Not getting it.

It doesn't matter what you call it.

Mystifying. You and Robert seem careful about definitions, but now the terms don't matter.

As far as the value of a move goes, think in terms of gain.

I don't know how to do that. I am as far as understanding "value of a position" and "value of a move" but don't know yet what "gain" means.

Edit: Also, for sente it helps to think about how much the reply gains, which is not necessarily what the follow-up gains if your opponent plays elsewhere.

I'm trying to define and to reason, not to think about something that may help me. I though we were scanning the potential of precise calculations?

Maybe you're tired of making your point. Please refer to earlier threads. I can scan those.

[RobertJasiek]

Playing moves in order of their value (as in value of the move) will give the best possible result for both.

Insert "Usually,"

Mb = Tb - P
likewise
Mw = Tw - P

The convention for defining the gain of white's move is:
Mw = P - Tw

P is the usually larger value than Tw.

Also correct the consequences of this, please.

Otherwise, I do not like all of your approach, but, well, it is your choice of reinterpretation:)