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 Post subject: Sente, gote and endgame plays
Post #1 Posted: Mon Mar 12, 2012 4:23 am 
Gosei
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I have started Get Strong at the Endgame recently, I just got into the first endgame tesujis (prolems 43 and up). And I've found that most of them (well, most of the first 12) are 1-2 points better than the standard response, but the standard response usually is sente and the best (point-wise) response is gote.

My question is (and I know I could find a suitable answer in The Endgame, but I want something I can understand better and I have faith in the L19 community :bow: ): how do I order these moves? I just count double for sente? Count in half for gote? Count the best sequence as a gote play? Count the best sente sequence as sente play? Resign before yose?

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 Post subject: Re: Sente, gote and endgame plays
Post #2 Posted: Mon Mar 12, 2012 4:30 am 
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In general, I think reverse sente is approximately worth double gote (as you're taking a gote sequence that your opponent could get for free, so for the remaining endgame points, if you take reverse sente you get one, and your opponent gets one, whereas if you take gote, your opponent takes the reverse sente as sente and then gets a gote point as well - effectively a free move of that value). Sente moves are just to be played really, ideally before the reverse sente if your opponent becomes their biggest move. Sente moves can be left as ko threats, but not left so late that you don't get them.

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 Post subject: Re: Sente, gote and endgame plays
Post #3 Posted: Mon Mar 12, 2012 4:32 am 
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As far as I know: Count double for sente (because you can play on a equally big point after the sente sequence). Thus count 4 times the points for double-sente.

Then the order is simply pointwise.

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 Post subject: Re: Sente, gote and endgame plays
Post #4 Posted: Mon Mar 12, 2012 4:46 am 
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p2501 wrote:
As far as I know: Count double for sente (because you can play on a equally big point after the sente sequence). Thus count 4 times the points for double-sente.

Then the order is simply pointwise.


As far as I know, the value of sente and double sente is effectively infinite, as they are "free" plays and therefore don't lose initiative. Reverse sente is double gote because even though it's gote for you, it deprives your opponent of a move for your opponent and therefore forces him to lose one more initiative, and has double value for the reason that it removes one move from his next sequence.

Double sente should be played immediately unless there's something big enough that your opponent would ignore your move and allow you to follow up to get something even bigger - which is fairly rare outside of big group loss situations.

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 Post subject: Re: Sente, gote and endgame plays
Post #5 Posted: Mon Mar 12, 2012 6:37 am 
Judan

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Let's suppose that you have a position where you can move to a local count of G with gote or S with sente, and the your opponent can move to a local count of R with gote, and G > S > R (from your perspective).

Just looking at the local position, compare S with (G + R)/2. If the position is gote, then its count is (G + R)/2, and if it is sente its count is S. If S > (G + R)/2 then it is sente, if S < (G + R)/2 it is gote, and if S = (G + R)/2 it is ambiguous (See http://senseis.xmp.net/?AmbiguousPosition .) You can also compare G - S with S - R. If they are equal it is ambiguous, if G - S > S - R it is gote, and if S - R > G - S it is sente.

As for which option to play, you should normally play the sente option if the position is sente and the gote option if it is gote. But that is not always the case. If you can read the whole board out, that's cool. However, here is a rule of thumb. Suppose that the ambient temperature is T, i. e., that the gain from making the largest play elsewhere on the board is T. (Note that if the above position is gote, then the gain from playing it is (G - R)/2, not G - R, since you are starting from a position worth (G + R)/2.) Then we can estimate the gain from playing out the rest of the board as T/2. That gives us the following comparison:

S + T/2 >?< G - T/2 ,

or

T >?< G - S .

If T is larger, then make the sente play. If G - S is larger, then make the gote play.

Note: In the problems in the book where the gote is presented as correct while the other option is sente, the sente play should be a losing sente, i. e., G - S > S - R. :)

Edit: The above discussion is about which option to choose, not about playing a losing sente, which I discuss next.

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Last edited by Bill Spight on Mon Mar 12, 2012 7:25 am, edited 1 time in total.
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 Post subject: Re: Sente, gote and endgame plays
Post #6 Posted: Mon Mar 12, 2012 7:24 am 
Gosei
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Although it looks odd, this makes sense Bill, thanks :) (and now I understand some of all that CGT gibberish I see in Senseis :)). Indeed, the "wrong sente" I have checked are all losing sente, although they are sente plays they lose points against the best answers to the gote play (and it's usually not a gote play, but that they end in gote). Since they are local problems, a losing sente is bad (i.e. we have to assume T=0) and gote is good. Does this makes sense?

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 Post subject: Re: Sente, gote and endgame plays
Post #7 Posted: Mon Mar 12, 2012 7:43 am 
Judan

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RBerenguel wrote:
Although it looks odd, this makes sense Bill, thanks :) (and now I understand some of all that CGT gibberish I see in Senseis :)). Indeed, the "wrong sente" I have checked are all losing sente, although they are sente plays they lose points against the best answers to the gote play (and it's usually not a gote play, but that they end in gote). Since they are local problems, a losing sente is bad (i.e. we have to assume T=0) and gote is good. Does this makes sense?


First, in an endgame book you should assume that T = 0 only for whole board problems. :) For instance, we might classify a local position as sente, which is how it is normally played, but if T = 0 and G > S, then you play it as a gote.

Second, the book should show a sequence up to the point where the local temperature drops. (Or sometimes where it stays the same, as with ambiguous plays.) However, endgame books have many mistakes where that is not the case. In part that is because they were ghost written by amateurs, in part because the pro author goofed. These errors are biased towards the sequences being too long.

I remember when I found a play that the book said was sente but I calculated it as gote. It should not normally have been answered. I was feeling pleased with myself. Then a couple of weeks later I saw the play in one of Sakata's games (against Fujisawa Hideyuki I think). It was not answered. ;) So the top pros knew it was gote, but the books still said it was sente.

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 Post subject: Re: Sente, gote and endgame plays
Post #8 Posted: Mon Mar 12, 2012 8:34 am 
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Bill Spight wrote:
First, in an endgame book you should assume that T = 0 only for whole board problems. :) For instance, we might classify a local position as sente, which is how it is normally played, but if T = 0 and G > S, then you play it as a gote.


But... if the "correct play" is a gote move (because the equivalent sente is losing sente), doesn't that mean that T=0? Because the correct move is G, because G>S. Hmmm....

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 Post subject: Re: Sente, gote and endgame plays
Post #9 Posted: Mon Mar 12, 2012 8:54 am 
Judan

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RBerenguel wrote:
Bill Spight wrote:
First, in an endgame book you should assume that T = 0 only for whole board problems. :) For instance, we might classify a local position as sente, which is how it is normally played, but if T = 0 and G > S, then you play it as a gote.


But... if the "correct play" is a gote move (because the equivalent sente is losing sente), doesn't that mean that T=0? Because the correct move is G, because G>S. Hmmm....


The sente option is a losing sente when G - S > S - R. That says nothing about T. :)

I should not have made the comment about playing the sente "even if it is a losing sente" because, while true, it is irrelevant. The sente will not be a losing sente. (More later.)

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Last edited by Bill Spight on Mon Mar 12, 2012 10:23 am, edited 1 time in total.
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 Post subject: Re: Sente, gote and endgame plays
Post #10 Posted: Mon Mar 12, 2012 9:26 am 
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This is why there is "research" on endgames :) Waiting for more ;)

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 Post subject: Re: Sente, gote and endgame plays
Post #11 Posted: Mon Mar 12, 2012 10:29 am 
Judan

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Sente option or Gote option?

Suppose that you have a position where you have two options of play, one sente (ending at a count of S) and one gote (ending at a count of G), while your opponent has one gote option (ending at a count of R), with G > S > R (from your point of view). How to play it?

OC, if you can read the board out, go ahead. :) If you can't here are the basic rules of thumb.

First, let's classify the position. Is it sente or gote? If it is sente, then normal play is to choose the sente option, if it is gote, normal play is to choose the gote option. As mentioned earlier, if G - S > S - R the position is gote, and that is normally the option to choose, while if S - R > G - S the position is sente, and that is normally the option to choose.

The great majority of the time, normal play is correct, but there are exceptions. Exceptions depend on the ambient temperature, T, which is how much a play somewhere else on the board gains. An example of a play that gains T is one where if Black takes it the resulting position has a local count of T while if White takes it the result position has a count of -T (from Black's point of view).

Exception 1: In a sente position to play the gote option.

Condition: G - S > T.

Exception 2: In a gote position to play the sente option.

This exception requires a large gote (ending in a local count of A if you play, B if your opponent plays, A > B). We also need to know the size of the sente threat. If you play the sente and your opponent ignores it, and you play again, let the resulting local count be H (for huge ;)).

Case 1. The sente threat is at least as large as the other gote: H - S >= A - B.

Conditions:

1) S - R > T

2) A - B > G - S + T

Case 2. The sente threat is smaller than the other gote: A - B > H - S

Conditions:

1) H - G > T

2) H - R > A - B + T

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 Post subject: Re: Sente, gote and endgame plays
Post #12 Posted: Tue Oct 11, 2016 10:36 pm 
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Bill Spight wrote:
Let's suppose that you have a position where you can move to a local count of G with gote or S with sente, and the your opponent can move to a local count of R with gote, and G > S > R (from your perspective).

Just looking at the local position, compare S with (G + R)/2. If the position is gote, then its count is (G + R)/2, and if it is sente its count is S. If S > (G + R)/2 then it is sente, if S < (G + R)/2 it is gote, and if S = (G + R)/2 it is ambiguous


While I admire the elegance of these conditions because of their independence of move values, the gote / sente unequations are wrong. The conditions 1 below are correct, according to the proofs below.


Consider a local endgame and a player, here Black, having a gote sequence resulting in the count G or a sente sequence resulting in the count S and the opponent having a gote (or reverse sente) sequence resulting in the count R.


Conditions 1:
The local endgame is
a local gote if S > (G + R) / 2,
Black's local sente if S < (G + R) / 2,
ambiguous if S = (G + R) / 2.


The initial position's tentative 'gote' move value is MA = (G - R) / 2.

With GB and GW being the counts of G's followers, we have
S = GW, // From the initial position, Black moves to G, then White moves to GW.
the count G = (GB + GW) / 2,
the move value MG = (GB - GW) / 2.


The following conditions define local gote or sente according to de-/increasing move values.

Conditions 2:
The local endgame of the initial position is
a local gote if MA > MG, // decreasing tentative 'gote' move values
Black's local sente if MA < MG, // increasing tentative 'gote' move values
ambiguous if MA = MG.


Proof of equivalence of the "ambiguous" conditions 1 and 2:

MA = MG <=> (G - R) / 2 = (GB - GW) / 2 <=> G - R = GB - GW <=>(*1) G - R = (2G - GW) - GW <=> G - R = 2G - 2GW <=> -G - R = -2GW <=> G + R = 2GW <=> (G + R) / 2 = GW <=> (G + R) / 2 = S.

Proof of equivalence of the "gote" conditions 1 and 2:

MA > MG <=> (G - R) / 2 > (GB - GW) / 2 <=> G - R > GB - GW <=>(*1) G - R > (2G - GW) - GW <=> G - R > 2G - 2GW <=> -G - R > -2GW <=>(*2) G + R < 2GW <=> (G + R) / 2 < GW <=> (G + R) / 2 < S.

Proof of equivalence of the "sente" conditions 1 and 2:

MA < MG <=> (G - R) / 2 < (GB - GW) / 2 <=> G - R < GB - GW <=>(*1) G - R < (2G - GW) - GW <=> G - R < 2G - 2GW <=> -G - R < -2GW <=>(*2) G + R > 2GW <=> (G + R) / 2 > GW <=> (G + R) / 2 > S.

(*1) This transformation is possible because of G = (GB + GW) / 2 <=> 2G = GB + GW <=> 2G - GW = GB.

(*2) Multiplication by -1 inverts the unequality sign, as can be seen in this example transformation: 2 < 3 <=> -2 > -3.


Note:

If the local endgame is sente, we replace the tentative 'gote' count (G + R) / 2 and tentative 'gote' move value (G - R) / 2 by the sente count S and sente move value S - R.

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 Post subject: Re: Sente, gote and endgame plays
Post #13 Posted: Tue Oct 11, 2016 11:40 pm 
Judan

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RobertJasiek wrote:
Bill Spight wrote:
Let's suppose that you have a position where you can move to a local count of G with gote or S with sente, and the your opponent can move to a local count of R with gote, and G > S > R (from your perspective).

Just looking at the local position, compare S with (G + R)/2. If the position is gote, then its count is (G + R)/2, and if it is sente its count is S. If S > (G + R)/2 then it is sente, if S < (G + R)/2 it is gote, and if S = (G + R)/2 it is ambiguous


While I admire the elegance of these conditions because of their independence of move values, the gote / sente unequations are wrong.


The conditions are not, repeat, not independent of move values. The sente condition depends upon the miai value of the threat, which must be greater than S - R.

Quote:
Consider a local endgame and a player, here Black, having a gote sequence resulting in the count G or a sente sequence resulting in the count S and the opponent having a gote (or reverse sente) sequence resulting in the count R.


Whether the gote sequence is a gote or the sente sequence is a sente depends upon the the intermediate positions having a higher local temperature than the original position. Move values are relevant. I am sorry if I did not make that clear in the earlier text.

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 Post subject: Re: Sente, gote and endgame plays
Post #14 Posted: Wed Oct 12, 2016 12:09 am 
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Bill Spight wrote:
The conditions are not, repeat, not independent of move values.


The conditions themselves depend on G, S and R, which are counts. So evaluating the conditions can be done independently of move values; we do not need to calculate move values to decide whether we have a local gote or local sente. At least, that is how I interpret and want to apply the conditions.

Of course, the conditions are also related to move values and, in this sense, they are not independent of them.

If for the moment we restrict ourselves to Black playing to G, you seem to have stated that the local endgame is
a local gote if S < (G + R)/2,
Black's local sente if S > (G + R)/2.
Contrarily, I say
a local gote if S > (G + R) / 2,
Black's local sente if S < (G + R) / 2.

Please confirm which unequations are right.

I applied your conditions to an example, which obviously is a local sente and got your condition's statement to have a local gote. So I tried to prove. At first I made a mistake in the proofs, having forgotten my school maths that multiplication by a negative number inverts the unequation sign. When correcting the proof, suddenly the example made sense when identified as a local sente.

Click Here To Show Diagram Code
[go]$$B Black's local sente
$$ -----------
$$ | X X X O .
$$ | . O O O .
$$ | . O . . .
$$ | X O O O O
$$ | X X X X X
$$ | . . . . .[/go]


G = -3
S = GW = -6
R = -7

S < (G + R) / 2 <=> -6 < (-3 + (-7)) / 2 <=> -6 < -5. We have Black's local sente according to my conditions.

Quote:
The sente condition depends upon the value of the threat, which must be greater than S - R.


I guess this is so but we do not need to calculate the value of the threat. Simply applying the conditions, it is sufficient to calculate the counts G, S and R. Of course, the value of the threat is hidden in these values and in particular the count G = (GB + GW) / 2. We do not, however, need to explicitly calculate the value of the threat to decide whether we have a local gote or local sente.

Quote:
Whether the gote sequence is a gote or the sente sequence is a sente depends upon the the intermediate positions having a higher local temperature than the original position. Move values are relevant. I am sorry if I did not make that clear in the earlier text.


Nevertheless, according to my current understanding, we need not calculate these move values to decide if we have a local gote or local sente...! We may calculate them and apply conditions 2, but the short method for distinguishing local gote / sente is using conditions 1, that is, just calculating the counts.

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 Post subject: Re: Sente, gote and endgame plays
Post #15 Posted: Wed Oct 12, 2016 6:28 am 
Judan

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RobertJasiek wrote:
Bill Spight wrote:
The conditions are not, repeat, not independent of move values.


The conditions themselves depend on G, S and R, which are counts.


They also depend upon the threat carried by the sente, which is not shown. My bad, I guess. I simply posited that S was the result of sente, without spelling the threat out.

Quote:
So evaluating the conditions can be done independently of move values; we do not need to calculate move values to decide whether we have a local gote or local sente. At least, that is how I interpret and want to apply the conditions.


I did not make what I meant clear, then.

Quote:
I applied your conditions to an example, which obviously is a local sente and got your condition's statement to have a local gote. So I tried to prove. At first I made a mistake in the proofs, having forgotten my school maths that multiplication by a negative number inverts the unequation sign. When correcting the proof, suddenly the example made sense when identified as a local sente.

Click Here To Show Diagram Code
[go]$$B Black's local sente
$$ -----------
$$ | X X X O .
$$ | . O O O .
$$ | . O . . .
$$ | X O O O O
$$ | X X X X X
$$ | . . . . .[/go]


That is not an example of what I was talking about, because there is only a sente option for Black, but no gote option. There is no G, only S.

Here is what I was talking about.

Code:

                   P
               /  / \
              G  /   R
                Q
               / \
             BIG  S


Black has two options in P, one to G in gote and one to S in sente (through Q). White has one option, to R. BIG is big enough to make the play to Q sente.

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 Post subject: Re: Sente, gote and endgame plays
Post #16 Posted: Wed Oct 12, 2016 7:17 am 
Tengen

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Bill,

in simple examples, such as mine, we must also be able to distinguish local gote from sente. In fact, in a different thread, you showed a similar example with White's local sente and four stones to be connected and we used the example to distinguish local gote from sente by tentative 'gote' counts and move values and conditions 2.

In such simple examples, G is the intermediate follower in the sequence from the initial position to the tentative 'sente' follower S. As a subproblem, we can look at G as a starting position, and G is a local gote endgame with the count G = (GB + GW) / 2.

We can apply conditions 1 or 2 to such a simple example to distinguish local gote from sente. The only trick here is to have a two-phase analysis: I) speak of tentative local gote or sente, II) determine the real local gote or sente status.

***

Now, IIUYC, you imagine different kinds of examples, in which a player, say Black, can start one gote sequence to the gote follower G or can start a different, sente sequence via some other intermediate position, say Q, to the sente follower S.

In such a different kind of example, I suppose we can use conditions 2. Again, we have G = (GB + GW) / 2. We do, however, not have S = GW. Therefore, for such a different kind of example, we may not apply conditions 1. Currently, I do not know yet if conditions 2 help us. I think we need more: MQ, which shall be the move value at the intermediate position Q, and is, IIUYC, the value of the threat, right?

Can you please show some such example and the necessary calculations to distinguish local gote from sente incl. the calculations involving the value of threat?

***

Then, for positions without kos, which method is generally applicable to all kinds of positions to distinguish local gote from sente?

***

My apology for apparently misrepresenting you. Has it been about you presuming the different kind of examples?

***

I am still at a loss trying to understand a) how certain conditions depend upon the threat carried by the sente and b) how your conditions "a local gote if S < (G + R)/2, Black's local sente if S > (G + R)/2" might be involved or derived.

***

EDIT: replace 'P' by 'Q' to fit your added tree.

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Post #17 Posted: Wed Oct 12, 2016 10:15 am 
Judan

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RobertJasiek wrote:
Now, IIUYC, you imagine different kinds of examples, in which a player, say Black, can start one gote sequence to the gote follower G or can start a different, sente sequence via some other intermediate position, say Q, to the sente follower S.


Right. One player has two live options, one gote and one sente, while the opponent has only one, which is gote.

Quote:
Can you please show some such example and the necessary calculations to distinguish local gote from sente incl. the calculations involving the value of threat?


I am pressed for time, these days, so I'll beg off on locating or constructing such a position. But they do show up in real games. :)

Quote:
My apology for apparently misrepresenting you.


My apology for not being clear. :)

I will discuss this kind of position. Remind me in a few days if I have not. :)

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Post #18 Posted: Wed Oct 12, 2016 12:01 pm 
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Maybe tomorrow I will start discussion of your (so far) abstract example (a monkey with gote or sente options is not clean enough, I suppose) and see how far I come. Then, in a couple of days, you might fill the rest or correct my mistakes:)

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Post #19 Posted: Thu Oct 13, 2016 12:16 am 
Judan

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Code:

                   P
               /  / \
              G  /   R
                Q
               / \
             BIG  S


Black has two options in P, one to G in gote and one to S in sente (through Q). White has one option, to R. BIG is big enough to make the play to Q sente. G > S > R (for Black).

Since G > S, if this is the last play on the board, Black to play should play to G. But is P characteristically sente or gote? We can answer that question by the method of multiples. (See http://senseis.xmp.net/?MethodOfMultiples )

Let there be 3 copies of P, with Black to play. Should Black play it as gote or sente? Except the last time, OC.

1) Gote. Black plays to G, then White plays to R, then Black plays to G. Result: 2G + R.

2) Sente. Black plays to Q, then White replies to S; Black plays to S with sente again; then Black plays to G. Result: 2S + G.

Black should play P as gote if 2G + R > 2S + G ; i.e., if (G + R)/2 > S . If S > (G + R)/2 then Black should play P as sente, except for the last one.

Let there be 5 copies of P, with Black to play. What are the results if Black plays it as gote versus the results if Black plays it as sente?

1) Gote. Result: 3G + 2R

2) Sente. Result: 4S + G

Black should play P as gote if 3G + 2R > 4S + G ; i.e., if (G + R)/2 > S . If S > (G + R)/2 then Black should play P as sente, except for the last time.

We get the same comparison with 7 copies, with 9 copies, etc.

So we classify P as gote when (G + R)/2 > S and as sente when S > (G + R)/2; when (G + R)/2 = S, P is ambiguous. (See http://senseis.xmp.net/?Ambiguous )

To put it another way, when the mean gote value of P is greater than the mean sente value of P, P is gote, and vice versa. When the two values are equal, P is ambiguous.

OC, even if we classify P as sente, there are times when it is correct to play it as gote. More on that later. :)

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This post by Bill Spight was liked by: dfan
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 Post subject: Re: Sente, gote and endgame plays
Post #20 Posted: Thu Oct 13, 2016 4:24 am 
Tengen

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Code:
                   P
               /  / \
              G  /   R
                Q
               / \
             BIG  S


We view values from Black's perspective, know / presume that moving to G is a gote sequence and moving to S is a sente sequence and presume G > S > R.

We presume G, S > R so that Black has an interest in preventing White from starting a sequence to R at all. We presume G > S so that Black has an interest in considering choosing the option G at all and, if P is the last local endgame on the board, Black having the turn should start the gote sequence to G.


In the initial position P, Black can start

- the gote sequence with the move value (G - R)/2 or
- the sente sequence with the reverse-sente-preventing move value S - R.

Unless the move values are equal and P is ambiguous, Black chooses the option having the larger move value so he starts

- the gote sequence if (G - R)/2 > S - R <=> (G + R)/2 > S or
- the sente sequence if (G - R)/2 < S - R <=> (G + R)/2 < S.

This principle choice of Black is, IMO, good enough for a proof of conditions characterising P as a locale gote or local sente.


As it turns out, the conditions have inverted unequality signs when compared to my other type of positions more or less coincidentally because G and Q have different meanings. However, the formula structure is the same, which gives rise to hope that there might be a universal method applicable to several types of initial positions.


The environment affects whether a local gote or sente is a global gote or sente. If the enviroment is empty, Black plays P as a global gote. In an "ordinary" environmment (How exactly must it be?), Black plays P as a global gote / sente if it is a local gote / sente, respectively. If P occurs an odd number of N times and is a local sente, the first N-1 times P is played as Black's sente but the last time it might have to be played as a global gote, in such a case called tedomari. The proof above merely characterises P as a locale gote or sente but says nothing yet about correct global play considering the environment and a possible tedomari.


Bill conveys a sketch of an alternative proof using the method of multiples. It seems that a full proof would rely on correct play for the limit of N->oo copies of P. By dividing the count of the ensemble by N, we get the expected average count in P. Iterating N by 1, 2 or a different number on all integers, only even or only odd integers does not affect the construction of such a proof; for a given P, we would choose the most fitting iteration step and subset of all integers. Nevertheless, I remain to be convinced why any proof using the method of multiples is well-defined.

The proof using the method of multiples suggests that the last copy of P should be played as a gote. In reality, we have only one copy of P. We do not want to say that it must always be played as a gote because of being the last copy of P. In fact, we prefer to use the conditions for when, usually, it should be played as a sente. In other words, the proof using the method of multiples also presumes that the environment affects whether a local gote or sente is a global gote or sente.

Therefore, I do not find the proof using the method of multiples any more convincing that the simple proof for one copy of P comparing the move values for the gote sequence versus sente sequence. The method of multiples is an overkill for distinguishing local gote from local sente.


We do not need to calculate the value of the threat, which I think would be the move value of Black's move from Q to BIG, right? All we need initially is a calculation that BIG is indeed big (how big must BIG be?) and of the counts of the leaves shown.


Still missing are examples of actual positions for Bill's type of positions. Anyone? Examples without follow-ups at the leaves are preferred.


We know how to calculate counts if G, BIG, S or R are terminal or have follow-ups. Meanwhile we have methods for distinguishing local gote from local sente in these types of positions and can imply their symmetric cases with White having the options G and S:

Code:
                   P
                  / \
                 /   R
                Q
               / \
             BIG  S


Code:
                   P
               /  / \
              G  /   R
                Q
               / \
             BIG  S


Which other types involving tentative sente options can exist? If a player has several gote options, we simply choose the dominating option. Same for several sente options of his. Hence, for Black's part of the trees, there are only the two types. White uses a gote sequence to reach R. For each of the two trees shown, there can be two additional types: a) White has White's sente option W only, b) White has White's sente option W and a reverse sente option R. Am I right about the types and the completeness of their classication? Which conditions arise for these other types and how do we derive them?

When we will have identified and calculated all possible types including tentative sente options, let me ask again: For arbitrary such positions P without ko, how to spell out a general method of distinguishing local gote, local sente and the ambiguous case?


How much in this thread is current research and what did already exist, possibly in more general contexts, among CGT researchers or by you, Bill?

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