There is a classic principle of "Play where your opponent wants to play." This isn't quite true during a fight where stones can get captured, but there are related principles that I'm just making up now. Tentatively:
"Try to punish your opponent's move. Play to take a key area that they should have attacked." "If your opponent has a weak point, you can jump out further than normal." "Threaten your opponent's peep stone if it is weak rather than connecting solidly." "Fight for the head in the centre between multiple weak groups by leaning on your opponent's weak points and weak groups." "Don't play forcing moves unless you have to. Instead exploit the threat of them to play more quickly." "If your opponent has made an earlier exchange that makes you more solid, refuse to answer defensively when they play another threatening move even if that would have been the normal response before." (I often heard this referred to a fighting spirit - kiai) "The value of stones comes from their support for life and death and their territory (e.g. endgame) follow ups. Even rescuing a large clump of stones can be bad if it is next to your opponent's wall." "If you are alive with no serious aji by the opponent (e.g. opponent only has 5 point threats rather than 40 points threats), then any opponent stones nearby should be studied carefully to see if you can attack/kill even if they have a lot of stones there." "In a cross cut position, you can only realistically save one group at a time and even then both sides will still be able to threaten the opponent's stones. Take the more valuable direction, which is normal the side where you have spent more moves preparing." "If you are weaker locally, you must keep threatening your opponent's weakest points or else all your stones will die." "If you are stronger locally, try to take more big points while attacking." "When attacking a weak group from the centre, if the weak group plays too low and submissively, they may have even less eyespace and connection due to being low and may need to keep paying for it."
From studying with AI, I think when it comes to perfect play, one way of thinking about shape is
In the diagram, W has repeatedly defended the shape at the top with the marked stone but there are still weaknesses. This was in order to make W's move at a forcing on the corner's life and death (though B will still have a lot of liberties and aji).
Both b and c are possible next. b would emphasise W's shape problems on the left and make it more difficult to find the timing to play a. Whereas c would squeeze W's shape on the top, but W doesn't have to respond since the main role of W's group is to reduce B's upper right shimari and make B disconnected in the centre, not to build territory.
I just found it curious that if B plays c, then W at d is the best response. This has a weakness in shape (wedging the one space jump), but in order to aim at that, B must aim at the two W chains around it, or poke it themselves probably with b. But b would be a terrible empty triangle. This is one sign that W at d is a good fighting shape since the opponent doesn't have an efficient follow up with the move they just played.
As for whether this helps, well, the best W response to B at b is also W at d, so perhaps this wasn't the best example.
Overconcentration is key in the opening. Some opening principles: "Keep a territory balance - if you have some control over the centre and don't have weak groups (e.g. several 3rd line stones making 2 space extensions), then take territory more aggressively." "When there are weak groups, you can imagine them running. If they run into your moyo, careful about gambling it away by attacking. Choose attack direction carefully." "Almost no goal that can be described by words can be accomplished with just one move. Your opponent may play a move that seems to defend their group or take an area, but it remains unresolved if you have any strong stones nearby." "If your opponent threatens to make your group a running weak group but you can't find a big move to run, then if your opponent has some weaknesses created by your group running, consider probing those weaknesses first. If they respond submissively, you might add a move to connect to the probe. If they cut, you defend. If they attack, you strike the weak points."
The point of 9 seems to be in order to prepare to add another move immediately. Although d would be normal, the jump seems best when the opponent has a moyo on the left or a potential one, and B has the ladder in the upper right. Basically, B expects W to add another move on the left, a shimari in the upper right, but then B defends at b and it isn't easy for W to make the left into a good moyo when the 3 stone wall in the lower right is weak and can't even get out easily. The move c is standard and good, but still leaves a vulnerable group with a cutting point. Although W should play more solidly with the disadvantage of the second move, in this position, the whole board plan for W seems slightly overconcentrated if they keep investing in the left.
The AI have repopularised the move at , and was always the standard response. However, in moyo games, seems preferred by AI.
Hence, the conversation these moves say to me has now changed somewhat.
Crude but I don't care about my lack of liberties since you can't exploit it without invading the lower side after which your left side moyo will evaporate. I have no choice since there are too many cutting points otherwise. gimme corner now not later You are letting me connect all my stones. This is what I call a key centre point, where one move defends all my weaknesses as well as getting out into open space, rather than a flimsy move with lots of cutting points. I have some choices but this is simplest and good. Your shape is very solid that's true but is the left side star point well placed? You don't need to make a moyo with such an alive group when you don't even have blocked off. If I play a few more jumps I can still threaten your lower left chain.
I am taking the left and your stone is going to suffer at the same time. Haha fool, your move makes no territory, I will make territory with my weak stone now. Calm as Takemiya. I've scooped out a lot of the corner now and didn't even need to connect So have I and how are you getting into the left side? Why do I need to, my moyo is big too. Key centre point Ooh, I still have several cutting points and you only have one. I'll invade the upper left later. I agree the centre is too important for me to invade the lower left that you built. Thank you.
Author:
dhu163 [ Sat Nov 13, 2021 12:05 pm ]
Post subject:
Re: local fighting shape principles
An attempt to glean some principles from a PETC game.
More fuzzy thoughts influenced by wondering how influence functions and endgame theory can be applied in the opening. This is most stark when an unusual move occurs such as a 5-3.
Here the upper left is standard for an invasion especially as that will naturally reduce black's lower left too (a living group in the upper left with the relatively unalive 5-3 stone is nice for W).
B's approach in the upper right is designed to make the upper side big, to be consistent with the 5-3.
How should W invade the upper left. The AI prefers the 3-3 over the 3-4 most of the time. In this case, the 3-3 arguably seems to directly threaten big follow ups at b and c, so it can actually help to reduce the upper side. Even in the standard 3-3 invasion joseki, a cutting point is left in B's shape at d, so that can help invade/reduce the upper side.
The AI's choices here are interesting to me. I correctly predicted (since the upper side is relatively important and W knight's move on the upper left helps set up an attack on B's wall, so you don't want to strengthen the O17 stone. seems to say that I want the corner now even if you get strength in the centre. The fact that it wasn't at instead (which leads to settled groups) perhaps means that with the wall in the upper left, B had more confidence fighting in the centre rather than the endgame on the side. This helps explains though it was still a surprise to me as well as the follow up a then b!!
If W defends solidly and so does B, how to play the unusual area at the top? The AI line tenukis for the next 20 moves or so for the 3-3s and then W wedges on the left side, completely ignoring the upper right, probably because the upper left and upper right corners are relatively solid so fighting is less interesting and yet if W plays, W would prefer to invade the upper side (starting a fight) since the upper side is wider than normal. If B plays, the 3-4 in the upper right is very natural, but B doesn't have time to do that, and such a move is most attractive if it makes W's upper right weak, the B upper side into territory, or overconcentrates W on the right. It does all these things but in the particular surroundings perhaps only to 80% not 100% efficiency.
If B defends the upper side solidly, then W prefers to opt for a less attacking move with the 3-4 in the upper left (which doesn't try to attack the 5-3 stone as much), which gets out better for left side potential and still leaves a big kick in the upper left instead of trying to invade the upper side.
In the upper right, likewise neither side much wants to continue there. W's corner is normally very big, but if W takes the corner, even if B just jumps (a for b), this exchange seems questionable for W (the moyo is big).
If W invades or tries to reduce there isn't a great shape just yet (B is pretty solid above). Later I would still prefer to curl around at c (to be consistent) maintaining pressure on B's "thickness" rather than messing around in B's strong area with d or e.
However, B doesn't have a great shape to continue themselves. Playing 3-4 in the upper right is big, but W naturally curls around and reduces the moyo from the centre. Jumping at B immediately is slightly overconcentrated.
Even though g is always instinct for B to play next when a 3-3 hasn't been invaded this late on, the approach at f is consistent with a moyo (threaten another moyo on the other side of the board and keep white playing on there own side of the board).
Author:
dhu163 [ Sat Nov 20, 2021 1:03 pm ]
Post subject:
Re: local fighting shape principles
When stones can be captured, the opponent's best move tends not be your best move but they tend to be adjacent since they short each other's liberties.
[go]$$ Black to play $$ --------------------------------------- $$ | . . X . a b . . . . . $$ | . . X . . . . X X . . $$ | . . O O O O O O . . . $$ | . . . , . . . . . , . $$ | . . . . . . . . . . .[/go]
In this connection problem, why is A the right move not B? In particular, why play closer to the stronger group than the weaker group?
The best answer I can give is that however you try to connect, white will be able to play adjacent to that stone in order to obtain more forcing moves (i.e. ataris) in the future in order to disconnect. The key here is to play so that white's ataris are themselves in self-atari which is why playing closer to the strong group helps.
Black is locally connected on the left and on the right, but the key is to ensure that white's move at the boundary line doesn't impact both connections simultaneously (i.e. by capturing A or B).
Author:
John Fairbairn [ Sat Nov 20, 2021 1:51 pm ]
Post subject:
Re: local fighting shape principles
Since it would come under this heading, I would be interested to hear your take on the 'five alive' principle. I first heard about this from Korea. In the case of five alive, I think Bruce Wilcox also highlighted it, as he too seemed to have had a fascination with local fighting shape principles.
The five alive concept is that, in a local fight, a group with five liberties is temporarily safe enough to leave, but with four liberties or fewer you need to take urgent action.
It seems to work out well in practice, but have you got any insight into why five is the magic number and not six, or seven?
PS I've never come across five alive in Japanese (except that some suji combinations seem, fortuitously perhaps, to coincide with it). I have not seen it myself in Chinese either, but I've been told there's an equivalent, or something similar. Anyone?
Another 'proverb' that seems to be exclusive to Korea is: Each stone in a moyo is worth 6 points. That, however, may be an amateur proverb. Five alive I've seen explained by a pro - or at least so my memory tells me.
Author:
dhu163 [ Sat Nov 20, 2021 5:01 pm ]
Post subject:
Re: local fighting shape principles
I'm not that familiar with that proverb and SL didn't seem to have much explanation. All I can offer is analysis of several examples without any generalisation for now.
If you only have one or two liberties, you should probably consider saccing the group. Three or more and your shape is often a bit too heavy to sacrifice but likely also less easy to kill without aji.
[go]$$c $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . O O . . . . . . . . . . . O O . . | $$ | . . X O . . . . . . . . . . . O X . . | $$ | . . X O . . . . . , . . . . . O X . . | $$ | . . X O . . . . . . . . . . . O X . . | $$ | . . X O . . . . . . . . . . . O X . . | $$ | . . . . . . . . . . . . . . . O X . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . O . | $$ | . . j . . . . . . . . . . . . . . . . | $$ | . h b k . . . . . . . . . . . X O . . | $$ | . a e f O . . . . . . . . . . X O . . | $$ | g X X X O . . . . . . . . . . X O . . | $$ | . O O O X . . . . , . . . . . O X X . | $$ | . . O X X . c . . . . . . . . O X . . | $$ | . . O O X . . . . . . . . . . O X . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]
In the upper side, black is cramped against the side. On the left there are 5 liberties but letting white double hane is normally too much. On the right, black has 6 liberties and is ok leaving the group for now.
In the lower left, black has 4 libs with both groups and white has 5 in the corner and 4 in the centre. The corner is normally safe enough (many liberties even if not locally alive).
White at a or b practically captures black's stones and on the other side c is severe. Black's stones are stuck on the edge of the board. What can white expect from one attacking move? An attacking move is normally a loose surround and not connected to any white chain in order to be speedy. Hence normally it has 4 liberties. What are those 4 liberties controlling? Let's analyse a white attack at b.
3 of the liberties are directly facing black's weak group (h, e, k). e is a mutual liberty, k means that black can't push through at f, and shorting a liberty of h makes it more difficult for black to escape in that direction. Controlling j may be helpful later (say for defending the b stone), but isn't yet working to short black's escape route.
So black has 4 liberties now. After one move by white, essentially black loses 3 liberties (a, e, f) as being useful). The last liberty is at g which goes towards the edge of the board and dies. Hence 4 liberties needs a defence. QED .
Black still has a and g as partially useful liberties and should try to escape through a. However, playing a directly is a silly empty triangle. So h is the move (since black can use the control they have over a to make it more difficult for white to play there (they lose liberties). However, locally, white blocking can capture black.
[go]$$Wc $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . O O . . . . . . . . . . . O O . . | $$ | . . X O . . . . . . . . . . . O X . . | $$ | . . X O . . . . . , . . . . . O X . . | $$ | . . X O . . . . . . . . . . . O X . . | $$ | . . X O . . . . . . . . . . . O X . . | $$ | . . . . . . . . . . . . . . . O X . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . 6 . . . . . . . . . . . . . . O . | $$ | . 3 4 5 . . . . . . . . . j c b . . . | $$ | 9 2 1 8 0 . . . . . . . . e d X O . . | $$ | . 7 . a O . . . . . . . . h f X O . . | $$ | . X X X O . . . . . . . . . g X O . . | $$ | . O O O X . . . . , . . k . . O X X . | $$ | . . O X X . . . . . . . . l . O X . . | $$ | . . O O X . . . . . . . . . . O X . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]
In this sequence black cuts through but white a wins the race as black has already lost one liberty at so that after white plays a, black only has two liberties and can't cut. If white had a few less liberties this sequence might work but not in this diagram.
In the lower right, b is normally a vital point. I suppose if we are talking about cross cut fights, we normally expect that all of the stones have already suffered a hane at the head of n stones. Hopefully they haven't been haned on both sides as then the liberties are really very low. Curiously again the lower right black corner has 5 liberties and actually very many liberties in the corner though not alive.
Consider black's outside three stones. They have 4 liberties but being in the centre they have much more space and ability to live than the examples, so the vital points are about capturing the stones. Instead they are about the ability for the opponent to block off key points in sente.
Black's outside three stones face a situation where if white plays b, even though black can still counter-hane with c, locally black can't deal with (i.e. capture or squeeze) the cutting stone at d if white cuts later. This is because you must start with e but then f is atari and black has to play g forcing white h. This is split shape for black (a broken ladder), so black owes a move but how to defend? Playing h is the shape point but that still hasn't defended the cut tightly which means that white can threaten the cut by threatening the liberties of black's stones adjacent to the cut such as the clamp at j or the push at g.
Basically h is good and proper shape but if forced to play there, that is slack since it doesn't put direct pressure on either white weak group and hence can't make much territory or eyeshape locally and needs to keep running out for several moves.
If black had just one more liberty then a loose net would work against a cutting point created by a hane because white escaping wouldn't be atari against black's key cutting group.
In comparison black's ideal centre shape is probably to have two moves stones at b and one at k or l.
Author:
pwaldron [ Sat Nov 20, 2021 10:38 pm ]
Post subject:
Re: local fighting shape principles
John Fairbairn wrote:
It seems to work out well in practice, but have you got any insight into why five is the magic number and not six, or seven?
I remember wondering the same thing some years ago. My thoughts started from Bruce Wilcox, who commented that having five liberties defeats 99% of tesujis. A more restricted set of options than general safety of a group.
I didn't reach a firm conclusion, but in studying I realized what makes the tesuji a tesuji is that it often means a player can't increase their group's liberties even with the next move. In that case, consider what happens to a Black group with four liberties. With a single move White reduces it to three liberties. Black's next turn has no effect and White quickly reduces Black down to two liberties. Scope for play becomes limited then.
Starting with five liberties means Black will usually have three liberties and enough scope to deal with an attack.
Author:
John Fairbairn [ Sun Nov 21, 2021 2:44 am ]
Post subject:
Re: local fighting shape principles
Somewhat at random, but my own memories are:
The Korean examples were couched in terms of haengma (surprise, surprise).
One was something like the lower left where Black can jump to triangle without fearing A because his three-stone group has 5 liberties, but he can't do likewise in the example with B (4 liberties). There was also something to do with the kosumi, where C is pointless because it leaves Black, after D, with 5 liberties. White instead should be aiming at D himself.
In Japanese, this principle is probably subsumed under various other headings. There is, for instance, the case of the two proverbs: "Cut at the waist [or better: cut across] a knight's move" and "Don't cut at the waist of a knight's move." In the former, more normal case, the admonition is really to do with not pushing into a knight's move (that just confirms the opponents' five+ liberties). Cutting across in tesuji fashion, as Phil says, cuts down the liberties. The contradictory proverb is a bit of a joke, of course, but there's a serious element to it, as Kitani notoriously hated knight's moves, for various reasons, including very bad aji (and so the "don't cut" in that case means preserve your aji for later).
Japanese discussions, without mentioning any numbers, tend to focus on examples like E often being better than F, again a combination of liberties and aji.
Also in Japanese, you will see reference to Honinbo Shuei's famously strong L shapes that face the centre. A white stone at G would make such an archetypal shape, but even as shown with a black stone at G, White still has 5 liberties (and Black has made bad shape in terms of the example with C/D).
Yoda Norimoto has tried to explain all this in his own way by introducing the concept of sujiba, which he teaches in his school. He has also written a book about it. Yoda is not the world's best writer, however, and his book does not seem to have garnered much interest. I found it close to incomprehensible.
Thinking in terms of liberties can be interesting in various contexts. With josekis, for example, there are a couple of mind tricks you can use to get a penetrating sidelight on them. One is what I call the Catherine Wheel (the firework) effect, where many josekis seem to spin round in a spiral. It's fascinating to work out why. Another is to look at the liberties after every move. Can be very illuminating.
In practice, most of us only look at liberties when on the scaffold, such as when we realise a group is dead because of, say, the Rooster on One Leg tesuji. This was beautifully encapsulated in an English go proverb from the first London Go Centre days. I associate it, perhaps wrongly, with Brian Castledine: "Give me liberties or give me death."
In the case of China, although I have seen nothing directly relevant in Chinese, I can remember a group of us talking about this with Victor Chow at a London Open (40+ years ago). He (a very strong Chinese amateur) insisted that this sort of thing was taught as basics in all Chinese go schools, but in the hurly burly of tournament play and grabbing a bite to eat, we never got round to actual examples. But Victor was then living in South Africa (or move there soon after), and so Clive Hunt said he would pursue it with him there. I have a memory that Clive did eventually publish something, but he moved to Australia and so didn't complete the project. Maybe someone has a better memory or a contact. Or, better yet, knowledge of what is taught in Chinese go schools.
Author:
kvasir [ Tue Nov 23, 2021 12:54 am ]
Post subject:
Re: local fighting shape principles
I am not sure it is a west-east divide but we often tend to see moves in terms of their effect on the other color's stones rather than what they do for the same color. For example how we talk about preventing the opponent from playing hane at the head of two stones, rather than that it is good to extend to make three-in-a-row. It is logical to try to view the game in terms of actual sequences, which does match how we often talk about these things, but many situations call for us to see further (and more accurately) than most of us are able to do. Though I don't use the 5-liberty rule, I have found that thinking about making three-in-a-row shapes really helps when the position calls for thick, strong shapes that contest control in the center. For examples in some running fights and when making contact plays on the edge of two moyos. It can be a slow move in contact fights when it is possible to hane or play a keima but often it is solid and allows a more forceful hane at the next juncture.
Similar thing applies to the bend as opposed to the keima. Also the choice between a more solid extension/bend or a faster hane/keima can often be informed by considering if the opponent will reply immediately. The more solid way is rarely too slow when the opponent replies anyway but the faster way can be thin. At least thinking this way has helped me achieve more consistent results.
Author:
dhu163 [ Tue Nov 23, 2021 5:40 am ]
Post subject:
Re: local fighting shape principles
kvasir: agreed.
There are also many examples of this from corner joseki fighting for centre control. In Asia it is mainly described as "gote no sente" versus "sente no gote" though that is mainly about fighting shape not joseki.
[go]$$c Solid at a or push with b? $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . O X . . . . . . c . c . X O . . . | $$ | . . O X . . . . . , . . . . X O . . . | $$ | . O X . . . . . . . . . . X O . . . . | $$ | . O X . c . . . . . . . . b a . . . . | $$ | . O X . . . . . . . . . . . . . . . . | $$ | . . O X . . . . . . . . . . . . . . . | $$ | . . a b . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . b a . , . . . | $$ | . . . . . . . . . . . . X O . . . . . | $$ | . . . . . . . . . . . . X O . . . . . | $$ | . . . . . c . . . . . . c X O O . . . | $$ | . . . . . O X X b . . . . . X X O . . | $$ | . . . . X X O O a . . . . . . . . O . | $$ | . . X X X O . . . , . . . . . X . . . | $$ | . . O O d O . . . . . . . . . . X . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]
In all these examples except the lower left, a tends to be better than b. This is because one move defends all your weaknesses, and still you have a serious threat against your opponent's shape at c. In the lower left, d remains a problem even after a, so that is probably why b is often better than a (to get so many liberties with the a,b group that the d cut isn't a serious threat, or if black cuts at a to fight in the centre and defend d naturally or sacrifice accordingly).
Author:
dhu163 [ Tue Aug 02, 2022 9:41 am ]
Post subject:
Re: local fighting shape principles
I realise that if the principles in weak point theory are right, the most difficult extension is probably what happens in a fight. It can probably already mostly be explained, but there may be subtleties that are useful to dig up.
Let our model be that we have a weak group A of value a + ab' (potential given reasonable 1-1 exchanges between A and B) ac' that is under attack. However, A can cut the opponent into two groups B,C by cutting at D. What is the value of such a move and when is it good?
Perhaps D only just has enough aji to live by playing at the weak points of say B in particular (say B is weaker than C). Say the swing from living at D is d.
Compare this to if the opponent had the first move. Perhaps they could attack at A1 (a weak point of A) that gains ab' already (which might be more than a) in a way that B,C are much stronger groups as A is about to die (or otherwise be severely attacked) so that it isn't worth it anymore to live in at D. So the opponent gets
b+c+d+a''-b''-c''
where a'' is the potential of further attacks on A, which could be small or big depending on how big A is compared to the surroundings. (if A is large then the attacks are certainly forcing but only if A isn't really dead yet. If A is small, then attacks are smaller but the opponent may be able to count nearly the whole of A as captured). b'' and c'' are the value of A's remaining threats on B and C (if A can be saved first). Normally b'', c'' will be small if A is weak. So perhaps a''~ab'+ac', and this is ~ b+c+d+ab'+ac'
What if A lives in at D? Then the equation above is similar but lets write it as
b+c+a'''-b'''-c'''
The main thing is that a''' may be much bigger than a'' as D needed to use many forcing moves to live, giving B,C extra moves. However, b''',c''' are also much bigger as B,C are split and need to live independently.
Most likely ab' becomes dominated by B, so perhaps a'''~ ab' + lambda*(a+ac'). Mostly likely the life and death of A depends very much on C, so c''' also ~ (1-lambda) * (c+ac') where 1-lambda can be thought of as aliveness of A. Note that I have squeezed the value of the initiative also into lambda. Next perhaps b'''=0. This approximates to ~ b + lambda(c+a+2ac')+ab'.
The swing is simply D if the groups are all alive to a similar extent. -1<Lambda<1 usually and it will depend on the particular fight. In general, the swing is
d - lambda(a)+(1-lambda)(c)+(1-2*lambda)(ac').
If lambda is high, the cut can be called a sacrifice of A (intentional or otherwise). If low, probably A lived at D in sente (or close to) and it can be called an efficient defense of A.
If A instead develops directly without playing D what happens? Assume the opponent goes back to defend D by defending B. Say this makes the opponent instantly alive at B,C with their control of D. Then they have
b+c+d-a-ac' - d' -T
Where a+a' is the value of a and its potential, and d' is the value of A's forcing moves on B,C to threaten D, which could include potential. Perhaps an additional "+a''''" should be added to indicate that A remains not yet alive. I have treated ab' as neutral since only one move has been played (though strictly it could vary too). We can assume d' is small. T is the temperature as A gets the initiative.
The swing is
a+ab'+2ac'+d' + T
It isn't unusual for lambda =0. In that case, the balance between the two strategies occurs when
d+c ~ a+ab'+ac' + T
That is, compare the territory d and weakness of c with the strength of a and its development.
In my opening example, we actually have lambda ~ -1, so the balance is closer to comparing
d+2c+2ac'~ ab' + T
which turns out to be heavier on the left as the corner is so big, even though T is very large. Note however that ab' is unusually big as it comprises much of the corner value of d also.
Note I have assumed that D doesn't connected to much potential. But potential can occur when cutting into open space.
20220907
The above is the right idea I think but remains quite some distance from proving say that 3-4 and 4-4 are worth equal in points.
Important to work out how strong a group needs to be in order to beat local cutting points, or tenuki even if opp plays first. Basically to control 3-3, either need stone there or surround it with stronger groups. Even if played there, opp still threatens to surround with say 3 moves to kill. Hence, you need at least 1 connected move locally to live connecting to side, and even that won't block their endgame. Alternatively, can use 2 moves around that in tiger mouth as long as each has more liberties than the connection point (which should have only 1 without support). But if play like this, opp has ways to aim with peep at weak point and then although you have larger group, opp might still have forcing moves if they are strong enough.
I think that when defending the 2nd line cuts, only need to have at least 3 libs (as cut only has 2), and when defending centre, similarly only need to have at least 4 libs. But it depends on manoeuvring space. If you really want to prevent follow ups then 5 libs makes sense as a minimum, but subtract an equivalent number of liberties if there are cutting points.
So why is 3-4 = 4-4 for points? Presumably there is a constant tradeoff locally if you can win against 3-3. 4-4 doesn't exactly win against 3-3 but maintains pressure inside corner with forcing moves of 2-3 etc and follow ups.
Similarly why does each extension of wall gain 3 points? Well it had better gain as much as the 3rd line territory (which gains 2 points) that it is conceding. But perhaps it is 2+1 because there remain follow ups inside the 3rd line area.
Author:
dhu163 [ Wed Sep 07, 2022 1:58 pm ]
Post subject:
Re: local fighting shape principles
Here is some progress on why is 5 liberties critical
[go]$$c Solid at a or push with b? $$ --------------------------------------- $$ | . . . . . . . $$ | . . d . . . . $$ | . . 2 c 1 . . $$ | . a b 4 5 . . $$ | . . 6 3 . . . $$ | . . 8 7 . . . $$ | . f e 9 . . . $$ | . 0 . . . . . $$ | . . . . . . . $$ | . . . , . . .[/go]
In this AI joseki, this is the mainline after a 5-3, better for W by 1pt.
We can count that W played enough pushes to get 5 liberties! That is, assuming the forcing moves e for f and c for b, the group ends up with 5 liberties. Why?
In this case, there is a cut at B14 where B would have 2 liberties. Hence, for W to tenuki W needs 3 liberties. However, with the support of , B can play d which exposes a in sente. After this, W will have 4 liberties and with the cut follow up at B17 in sente (threatening B18), W only has 3. But this is enough.
Hence, B only has a gote follow up of around 10 points rather than any sente aji. Probably the reason this shape is good for W is because W controls B18, and also W didn't have to crawl many times for B to not have much of a follow up on either side.
Normally they say a every stone of a wall is worth 3 points, but you should probably subtract 1 stone for an important cutting point. Also, probably it is actually only worth 2 points and the other points come from follow ups on the group inside. But here there isn't much of a follow up as W is too alive. Few of B's follow ups even make one eye yet in gote, unlike if the 5-3 was at 4-4 and could challenge the 2-2 point. Like this, W has maybe 10points from 5 stones, as normal (x2) when crawling on the 3rd line. B probably has more like (5-1)*2=8 + 1 from having the initiative to use the potential. This comes to 10-9=1pt in W's favour?
Compared to the 3-3 invading the 4-4, the 3-3 is much less alive and must be treated more lightly, so it pushes less times and doesn't play such a crude move as . But in the main line, it seems that the 3-3 group still aims for 5 liberties, but only after adding an extra move to defend at 3-2.
More thoughts along these lines. Against 3-3 invasion of B's 4-4, B can double hane and then as 3-3 only has one lib left, and B has 3 (from 2 moves in a row). There wasn't much point for 3-3 to fill in own libs, but for B it is important that when shorting W's libs, B also gets several. normally only played when B needs to make life or the wall is noticeably more useful than the ponnuki. (e.g. if B is thick on the side of the wall).
Otherwise, B expects to get 3 of the 4 moves need to surround 3-3. Since this is fighting close to W, B only has 2 libs with sealing move. So W's goal is to get 3 libs (including forcing pushes) before relaxing. So that means B can expect to extend once in sente if B wants, as well as initial block. In any case, this reverts to a 3-4 like shape. At the same time, in the main line where W plays keima to get out, this keima also needs to maintain enough libs, or else B can lean on it to get a move in sente and reduce the libs of 3-3. For the time being it has 4 libs, which can go down to the critical 2 rapidly with B's forcing moves, at which point W has to retreat inside or exchange. W lives cleanly enough like this because B has to sacrifice at the cutting point but B has profit on outside. We say W owes half a move locally, so B can also tenuki rather than committing to developing either side (which makes it heavier which prevents B sacrificing it for the corner later).
Why is it so big for B to respond and take this outside? Because if W plays first, W is able to crawl an extra time, so that the keima has no aji anymore unless B takes gote to prepare. Basically, B still has a 2nd line move in sente potentially, but it can be met by a 2nd line move of similar value unless it comes close to killing W. Also, W can cut B from both sides, so that B is floating without corner 2-3 aji either. So it is like a (reverse) sente. But how big is it?
After an approach to a 3-4, the first player won't play as tightly to cover because such a stone against their opponent's area is too weak. For example, attach on top only has 3 libs, and the cutting point is very valuable. When approach hanes, their cutting point has 4 libs, but you only have 2 left, one of which is the valuable cut. Even after pulling back, approach extending inwards to corner connects your cutting point to strong group. Even though you have local miai, the cut remains, so you have to pay something. Approach already gets 5 libs, more than the 2 of the cutting point (even though a 3rd line cut, it still can't get more liberties directly, mostly only more variety in forcing moves). Basically the attach on top doesn't get enough clear profit, spending a move in the centre potential in gote (in the sense that approach can extend on side, and you still owe side extension). Approach can just respond by pushing in centre always more threatening to you than them, as using you cut may surround your whole group. Normally the approach is happy to make life like this.
Otherwise if kick and jump, corner uses a forcing move that is at least connected to gain most of corner (already had some stake there), at cost of letting opp get thicker in centre. This is normally not an area that corner wants to fight over, but approach will try to use this to attack later. after 3 space, there is still aji to connect up, but can't easily kill approach either. Probably need 2 moves normally to kill opp but after they extend up, probably need an extra move or 2 to surround (but of course more valuable as each such move has value of their own apart from killing extra stones). At what point is a threat sente. Probably only if they can cleanly cut from eyespace. Extending up gives more options and likely just focuses on playing high with many options. Loose control of whole area, can give opp side territory in exchange for solidly connecting up.
Since the kick takes the corner, and centre is less valuable, probably the approach stones are slightly lighter than they look, but they aren't exactly easy to kill as influence stones have many options. They have kosumi or hane and attach to get at least one forcing move in the centre. So we should think of them as having some sente aji value locally (better for killing in some variations if opp gets cut locally) which can help own life, but not connected to much territory value yet. So approach doesn't want corner to get too thick nearby or else their moves can't be used and just have weaknesses.
Where each side's potential extends to is basically which fights they can win if opp plays there.
The value of thickness? depends on how easy it is to reduce. For example, if the opp has a weak point whose best direction by some way (say a 3-4) is to reduce moyo, then your moyo isn't great direction when they play there first. If it is on the 2nd best direction (e.g. kobayashi), it can be much more powerful if thick enough to get compensation. AI seems to think that both kobayashi and chinese openings are suboptimal but both appeared in alphago self-play.
If a bog standard 2nd line move is worth 6 points and you expect to get two in sente, then you play a move worth 12 pts.
Consider 3-4 attachment to 4-4. better access to corner say 3-2 and 2-2 at least are worth a swing of around 6+5 (from a sente move but it uses some aji), which is certainly a big move. Even better if can continue attacking. But opp couldn't get whole corner with one move. But if they spent one move, they would also render your group half dead, and may be able to fight more strongly. For example, taking corner more aggressively with double hane and letting go of 4-4's local potential for the 4-4's counterattack value. basically they play a gote 7pt move at the boundary of eyespace for both sides. And if you don't want to run, you leave a weak point behind on the 2nd line (worth 9pts ish), in exchange for getting out better with atari on their 4-4 which is only worth pretending opp capturing anything too easily. if not dead, only around 1pt sente. but gets some forcing moves. basically expect to lose say 9pts ish as well as opp's attack value if get pincered if you still play this way.
, and 
was always the standard response. However, in moyo games, 
seems preferred by AI.
I have no choice since there are too many cutting points otherwise.
gimme corner now not later
You are letting me connect all my stones.
I have some choices but this is simplest and good.
Your shape is very solid that's true but is the left side star point well placed? You don't need to make a moyo with such an alive group when you don't even have 
Thank you.
.
so that after white plays a, black only has two liberties and can't cut. If white had a few less liberties this sequence might work but not in this diagram.