The influence buzzword
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I'd like to play Devil's Advocate for a moment, and suggest that influence, as a term bandied around, has become a piece of jargon used in a facile way.
Of course I'm not saying that it has no content. If I accuse it of buzzword status, it's a more a question of too much content, used in imprecise (if suggestive) ways.
There are a number of possible and interesting approaches, for example by returning to the Japanese technical vocabulary (which would be outside my competence), historical - none of these is decisive.
Go is ultimately a game of territory, meaning that success is in the end judged by the acquisition of secure territory.
One has to point out to beginners that the concept of secure territory early in the game is (i) questionable except for the extremely restricted class of groups against which no ko threat can be made, and (ii) misleading for discussion of strategy.
Then one can say that 'influence' is the portmanteau term that is introduced in relation to (ii) to cover everything left out when one equates go strategy with early acquisition of secure territory. Of course this is unbearably pompous and arid if put that way - but accurate, at least, in my opinion.
A useful syllogism about the game is this: secure territory is quite hard to make directly, the indirect methods of making territory come down to attacking (threatening to capture), so influence comes down to what you need to attack successfully.
This is quite a good start to discussion. The player who takes secure territory is likely to be cast as defender rather than attacker.
This says that positions designed purely to take territory are likely to lack influence. That is one substantive point - for example the 3363 enclosure (by which I mean 3-3 point plus 6-3 point) is made up of two stones on the third line. It can be pressed low by a play at the 5-4 point. This enclosure is sometimes seen in pro play, but not so often.
The other side to the story is that a player who likes to attack often builds formations that don't guarantee any secure territory. A contrasting example to the one just given would be the 4474 enclosure. It is easy to invade; but a play from outside, for example at the 6-5 point, becomes a bad exchange when answered at the 6-4 point. The opponent can only diminish the outside influence of this enclosure by 'forcing' it to become a good territory.
The term 'threat' as applied in military intelligence is used, I believe, to describe what the enemy is capable of doing to you, as distinct from the question of intentions. This is quite useful as thinking about go positions in which your opponent has many outward-facing stones and solid groups. These will become actively threatening to any weak group you happen to create near them.
The word 'power' has been used almost synonymously with 'influence'. One has to understand that the power of influential stones is something that can be cancelled by settling a group in the vicinity.
Influence therefore interacts with weak groups - that is how you can recognise it. Weak groups expand/connect or eventually die.
There is a major distinction between 'hard' influence and other kinds ('ragged' outward-facing stones, 'soft' formations such as frameworks built from loosely connected stones). As defender of a weak group you steer away by choice from the 'hard' places your opponent has built - the extreme case of thickness will look very much like a place designed as a killing field.
On the other hand the attacker normally plays to expand frameworks, by playing from the framework side rather than chasing the target weak group into the framework.
One should qualify that in a couple of ways. The point of expanding frameworks is to make territory in the not-too-distant future. If there are counterindications for solidifying an area (for example open skirts, 'walls' that aren't worthy of the name because of defects) one should reconsider.
More complex patterns (billiards?) are set up by driving weak groups in directions chosen so as to create further influence which is more usefully placed - phase two of an attack may create the territory, and so on.
Some formations are such that they are incomplete (may be invaded, or have portions cut off) - but gain in influence if that happens. Other formations are such that if you leave them too early, you will have to concede influence when your opponent plays to 'bend them back'.
Strong players generally are much more interested in 'bending back' than 'snipping off' as a way of punishing the opponent who jumps around the board.
It is very noticeable if a player tries to take a whole side of the board. This cannot however be done with fewer than five plays (on the 19x19 board). So formations such as the Chinese styles and sanrensei, with stones in two adjacent corners and one in the middle of the side, are definitely frameworks. The role of the stone in the middle is above all to prevent the opponent easily settling a group to break up the side. So one can talk about this stone developing the influence of the corner stones. This takes some understanding in the case of the 4-3 point in the Chinese - which isn't normally thought of as having influence in that direction. (It would be the 3-5 point standing in that corner which possessed that influence.)
Harleqin: I'd like to object here - the 4-3 Point does have more influence on the side where it stands on the 4th line.
Therefore one should be careful about assuming that individual stones have localised influence.
Central frameworks are often built up using the 10-10 point (or a neighbour) at a late stage. It really must be the case that the 10-10 point play has a similar role in preventing the opponent setting stones easily in the centre. But I'm sure many amateur players believe that it has a powerful intrinsic influence. Well, I think it does too, in cases where it is in harmony with the rest of the framework; but I'm not a subscriber to the Barbara Woodhouse 'no bad frameworks' theory. I'd better stop here, I can see ...