Please help with this exercise III

[Bill Spight]

Hi Bill,

Thanks for being patient with these 30k questions.

$$ A
$$ +---------------------+
$$ | . O . O . . X . X . |
$$ | O O O O O O X X X X |
$$ +---------------------+

Click Here To Show Diagram Code

[go]$$ A
$$ +---------------------+
$$ | . O . O . . X . X . |
$$ | O O O O O O X X X X |
$$ +---------------------+[/go]

Black has 2 pts. White has 2½ pts.

The whole number 2 is clear (for both B and W).
How to derive the ½ for white?

The easy way is through the method of multiples (i.e., the mean value theorem).

$$B 2(A) Score = -1
$$ +---------------------+
$$ | . O . O . 1 X . X . |
$$ | O O O O O O X X X X |
$$ | O O O O O O X X X X |
$$ | . O . O . 2 X . X . |
$$ +---------------------+

Click Here To Show Diagram Code

[go]$$B 2(A) Score = -1
$$ +---------------------+
$$ | . O . O . 1 X . X . |
$$ | O O O O O O X X X X |
$$ | O O O O O O X X X X |
$$ | . O . O . 2 X . X . |
$$ +---------------------+[/go]

This board contains 2 copies of A. The score is -1. That means that A is worth ½ pt. on average. The same is true for 2N copies, no matter how large N is.

and are miai. It does not matter who plays first, or which copy they play in, the score will be the same.

I can see one can argue there's a 50% chance W will get the extra 1 point.
But how to derive the (magical?) 50% odds?

There is more than one way to derive the probability. But IMO the clearest way to think about this is in terms of fuzzy logic, that is, in terms of possibilities. Unless we know better, the possibility that Black will play first in A is equal to the possibility that White will play first.

In any event, the mean value justifies the probabilities, not the other way around.

[jlt]

@Jika: 7+1=8

[Bill Spight]

Just to confuse the discussion further (from a 50 k's perspective):

No matter who fills the dame the result is the same. :

)

Maybe the situation in the first diagram is that B has taken the point where W could make an eye, so who gets the left over dame point does not matter, so it does not matter who's move it is?

In the endgame the board divides up into independent regions of play. Unless we can read the play out, we typically do not know who will play first in any given region. If the region is gote, we may assume that the chances are 50-50 that either player will play first. Since a gote dame gains nothing, it does not matter who fills it.

[EdLee]

Hi Bill, Trying to work out a simpler version of post#15:

$$ x
$$ +---------------------+
$$ | . O . O . O X . X . |
$$ | O O O O X X X X X X |
$$ +---------------------+

Click Here To Show Diagram Code

[go]$$ x
$$ +---------------------+
$$ | . O . O . O X . X . |
$$ | O O O O X X X X X X |
$$ +---------------------+[/go]

$$W (a)
$$ +---------------------+
$$ | . O . O 1 O X . X . |
$$ | O O O O X X X X X X |
$$ +---------------------+

Click Here To Show Diagram Code

[go]$$W (a)
$$ +---------------------+
$$ | . O . O 1 O X . X . |
$$ | O O O O X X X X X X |
$$ +---------------------+[/go]

(a) Territorial value = ( 's local points) - ( 's local points) = 2 - 2 = 0

$$B (b)
$$ +---------------------+
$$ | . O . O 1 O X . X . |
$$ | O O O O X X X X X X |
$$ +---------------------+

Click Here To Show Diagram Code

[go]$$B (b)
$$ +---------------------+
$$ | . O . O 1 O X . X . |
$$ | O O O O X X X X X X |
$$ +---------------------+[/go]

(b) Territorial value = ( 's local points) - ( 's local points) = 4 - 2 = 2

swing value = (b) - (a) = 2 - 0 = 2 (?)

Since the local position is a gote, it is the average of the value after Black plays first (2) and the value after White plays first (0)

The territorial value of the original local position (x) is ((2 + 0)/2) = 1 ?

[Bill Spight]

Hi, Ed.

Correct.

Correct.

[EdLee]

Hi Bill,

Thanks. So, when we talk about the territorial value of a given local position, what does it mean ?

[EdLee]

Let's try post#15, variation:

$$ x
$$ | O O O ? ? ? ?
$$ | . O . O ? ? ?
$$ | O O O O ? ? ?
$$ | . X X ? ? ? ?
$$ | O O X X X X X
$$ | . O X . X . X
$$ +--------------

Click Here To Show Diagram Code

[go]$$ x
$$ | O O O ? ? ? ?
$$ | . O . O ? ? ?
$$ | O O O O ? ? ?
$$ | . X X ? ? ? ?
$$ | O O X X X X X
$$ | . O X . X . X
$$ +--------------[/go]

$$W (a)
$$ | O O O ? ? ? ?
$$ | . O . O ? ? ?
$$ | O O O O ? ? ?
$$ | 1 X X ? ? ? ?
$$ | O O X X X X X
$$ | . O X . X . X
$$ +----------------

Click Here To Show Diagram Code

[go]$$W (a)
$$ | O O O ? ? ? ?
$$ | . O . O ? ? ?
$$ | O O O O ? ? ?
$$ | 1 X X ? ? ? ?
$$ | O O X X X X X
$$ | . O X . X . X
$$ +----------------[/go]

(a) Territorial value = ( local points) - ( local points) = 2 - 3 = -1

$$B (b)
$$ | O O O ? ? ? ?
$$ | . O . O ? ? ?
$$ | O O O O ? ? ?
$$ | 1 X X ? ? ? ?
$$ | O O X X X X X
$$ | . O X . X . X
$$ +----------------

Click Here To Show Diagram Code

[go]$$B (b)
$$ | O O O ? ? ? ?
$$ | . O . O ? ? ?
$$ | O O O O ? ? ?
$$ | 1 X X ? ? ? ?
$$ | O O X X X X X
$$ | . O X . X . X
$$ +----------------[/go]

(b) Territorial value = ( local points) - ( local points) = 9 - 2 = 7

swing value = (b) - (a) = 7 - (-1) = 8

The territorial value of the original local position (x) is ( (7 + (-1))/2 ) = 3

[EdLee]

Later, we assume the big groups are OK:

$$ x
$$ | ? ? O ? ? ?
$$ | ? ? O ? ? ?
$$ | O O O , ? ?
$$ | . X X X X ?
$$ | O O X ? ? ?
$$ | . O X ? ? ?
$$ +------------

Click Here To Show Diagram Code

[go]$$ x
$$ | ? ? O ? ? ?
$$ | ? ? O ? ? ?
$$ | O O O , ? ?
$$ | . X X X X ?
$$ | O O X ? ? ?
$$ | . O X ? ? ?
$$ +------------[/go]

$$W (a)
$$ | ? ? O ? ? ?
$$ | ? ? O ? ? ?
$$ | O O O , ? ?
$$ | 1 X X X X ?
$$ | O O X ? ? ?
$$ | . O X ? ? ?
$$ +------------

Click Here To Show Diagram Code

[go]$$W (a)
$$ | ? ? O ? ? ?
$$ | ? ? O ? ? ?
$$ | O O O , ? ?
$$ | 1 X X X X ?
$$ | O O X ? ? ?
$$ | . O X ? ? ?
$$ +------------[/go]

(a) Territorial value = ( local points) - ( local points) = 0 - 1 = -1

$$B (b)
$$ | ? ? O ? ? ?
$$ | ? ? O ? ? ?
$$ | O O O , ? ?
$$ | 1 X X X X ?
$$ | O O X ? ? ?
$$ | . O X ? ? ?
$$ +------------

Click Here To Show Diagram Code

[go]$$B (b)
$$ | ? ? O ? ? ?
$$ | ? ? O ? ? ?
$$ | O O O , ? ?
$$ | 1 X X X X ?
$$ | O O X ? ? ?
$$ | . O X ? ? ?
$$ +------------[/go]

(b) Territorial value = ( local points) - ( local points) = 7 - 0 = 7

swing value = (b) - (a) = 7 - (-1) = 8

The territorial value of (x) is ( (7 + (-1))/2 ) = 3

[dfan]

I thought the solution would be 7 (the max points for B):
white taking A creates 1 point (A1).
black taking A kills the three stones (3x2=6) plus takes A1 (6+1=7).

All good so far!

If Black moves there he is up by 7 points (locally). If White moves there Black is down by 1 point (locally). So the difference in score depending on who moves there is 8 points.

[Bill Spight]

Hi Bill,

Thanks. So, when we talk about the territorial value of a given local position, what does it mean ?

Informally, it is an estimate of its eventual local territorial score.

More formally:

1) If the position has a territorial score, that is the territorial value.

2) Otherwise, if the position is a non-ko position, it has a mean territorial value, defined as the average result in the alternating play when Black plays first and when White plays first in 2ⁿ copies of the position, either exactly or in the limit as n approaches infinity.

3) If the position is a ko (or superko) position, its value depends upon theory, and there is no single theory for ko values. Professor Berlekamp, Martin Mueller, Kim Yonghoan, Nakamura Teigo, Bill Fraser, and I have extended thermography to evaluate kos, superkos, and multiple kos and superkos.

[Bill Spight]

I managed to solve 3 exercises on this basis, but now I'm stuck with another one (sigh):

Bildschirmfoto vom 2019-07-26 06-48-22.png

I thought the solution would be 7 (the max points for B):

There is a game theoretic value of a position which depends upon who plays first, namely, the result of perfect play, In this position, when Black plays first that value is 7. When White plays first that value is -1 (i.e., 1 pt. for White). In go we do not evaluate positions that way, because typically we do not know who will play first in any given local position.

white taking A creates 1 point (A1).
black taking A kills the three stones (3x2=6) plus takes A1 (6+1=7).
They can't have the same territorial value both at the same time, so 7-1 does not work either?!

You are on the right track. Since it is gote, and Black gets 7 pts. in one play and White gets 1 pt. in one play. the value of the position is (7-1)/2 = 3 pts. (for Black).

Or maybe it is 3?!

That is the value of the local position. Well done!

So, why 8?

For historical reasons, go players have traditionally used swing values to compare plays (not positions). The swing value for a gote play is the difference in the position after Black makes a play and in the position after White makes a play. (Note that we are actually talking about two different plays, one by Black and one by White. ) In this case the swing value is 7 - (-1) = 8.

Some people get confused because of the way swing values are often taught. Informally, people say that the value of a play in the corner is 8 pts,, when actually it is the value of two different plays taken together, one play by White and one by Black. Each play actually gains 4 pts., and together they gain 4 + 4 = 8 pts. Swing values are fine for comparisons, but people make mistakes when they use them to figure gains and losses.

[Kirby]

Swing values are fine for comparisons, but people make mistakes when they use them to figure gains and losses.

Indeed. In the last game I played in the US Open (I haven't gotten to it yet in my journal thread), I played a peculiar sequence, which lost me several points. I know that the moves were bad, but more than that, I have been contemplating for the last few days about the *reason* I played that sequence: it was weird, and seems obviously bad in review. So what state was I in during the game, which made me think to play those moves?

The answer I've come up with, albeit maybe incomplete, is that I don't feel comfortable having a small lead in a tournament game. I only get comfortable when I feel that I have a large lead. Otherwise, my poor endgame could screw up the result of the game.

While this may be true, I believe that I overcompensate for that feeling, because I overestimate what I can lose in the endgame, due to swing values. When people talk about a 15 or 20 point swing value, to me, I feel that missing a single one of these could change a 10 point lead to a loss. And so, I don't feel comfortable with such a small lead. But in reality, when people are talking about a 15 point swing value, the actual gain is probably around half of that. Add that to the fact that I'll probably play *some* move that's kind of decent, so while I'll lose points by missing these big endgame areas, the loss isn't as extreme as it seems when I think about the swing value. If the swing value I miss is 15 points, and I play one with swing of 10 points instead, that's a loss of 5 swing value points, but with both players, I guess it'd just be a few points?

To be clear, I don't advocate ignoring the endgame, and these few points could very well cost the game (I lost a game by 0.5 last week). But thinking in terms of the swing value, I think I've added more fear than necessary, and it sometimes leads me to feel inclined to make weird moves (i.e. invade in weird ways when I don't need to, adding more risk to the game, and potentially costing me points).

[Jika]

@jlt: Well, this actually occurred to me.

But my confused question remains (as I said with "7-1 is not possible", because Bill's and Ed's discussion seemed to be about subtracted values) how can both values count at the same time?
So, the territorial value is the possible points B and W could make each, added??

The other exercises there are much, much, much more simple.

Like, oh, I see, black can close a gap, so the territory is closed, and look, in this territory there are two empty intersections plus two white stones on the other 2 intersections, oh, I'll say "6" - right!"
321go explain things patiently with exercises that get more difficult.
Here, I feel like I've skipped a class or something.

PS: How do you quote (multiple) posts, or parts of posts, in one reply? Copy-paste-quoteTag, or something else?

[Joaz Banbeck]

...
PS: How do you quote (multiple) posts, or parts of posts, in one reply? Copy-paste-quoteTag, or something else?

For a partial post, click the quote button, and then delete as necessary. Some of us use ellipses to indicate deletions.

[Kirby]

Swing value can be confusing, but I think you are on the right track.

It already seems clear to you that black can get 7 points by playing first (3 captures + 1 more point territory). That’s how much black gets, from black’s perspective.

But there’s additional value in playing there: black prevented white from making points. If white had no chance to make points there anyway, blacks would gain the 7 points, and nothing else. But the fact is, if white plays there first, white gets 1 point.

So black didn’t only gain the 7 points - he gained the fact that he denied white 1 point. That’s why you add them together to get 8.

The more points both sides have POTENTIAL to make, the more value there is to play there.

Example: let’s say there are two endgame moves where black can make 7 points by playing first. But in the first one, white gets a point by playing first, as in this example. But in the second one, let’s say that white could t get any points by playing there first anyway. In both cases, black gets the same 7 points by playing first. But it’s better to play the former one, because I’m doing so, black not only gains 7 points- he prevents white from the possibility of getting that extra point.