Line 3 v line 4 in the opening

[Bill Spight]

About the average points per move in a final position, you cannot judge the points per move of early play. At the end of a fairly typical game there will by around 240 stones on the board and around 120 points of territory, for an average of about 1/2 point per stone. But early in the game stones gain around 14 points of territory.

The stones in the middle of a center facing wall on the 4th line early in the game basically gain points in only one direction. That would be around 6 points. However, overconcentration and the difficulty of making territory in the center bring it down. My estimate is a bit over 3 points.

The territory made by stones in the middle of a side facing wall on the 3d line is easier to estimate. Once the wall is safe, each stone makes 2 points of territory.

Later in the game the value of center facing walls is typically less, while the value of side facing walls remains the same.

[Aidoneus]

About the average points per move in a final position, you cannot judge the points per move of early play...Later in the game the value of center facing walls is typically less, while the value of side facing walls remains the same.

I think that this is precisely the problem with the contrived theoretical examples. They ignore the time factor. And this goes back to my question about limiting the discussion to a corner sequence between territory/influence. Please correct me if I am completely mistaken but I thought that an important reason for delaying 3-3 invasions against hoshi points is that the territory gained is less than the influence granted during the early game? Though I have no idea when such a balance might swing the other way. (Of course, joseki are not pure 3rd versus 4th line, which complicates the analysis.)

[skydyr]

Please correct me if I am completely mistaken but I thought that an important reason for delaying 3-3 invasions against hoshi points is that the territory gained is less than the influence granted during the early game? Though I have no idea when such a balance might swing the other way. (Of course, joseki are not pure 3rd versus 4th line, which complicates the analysis.)

This is correct, and figuring out the correct timing for an invasion, so that you don't help your opponent more than you help yourself, but before they remove the option without giving away too much by losing sente, is tricky.

Generally, you have multiple ways of dealing with a 4-4 stone, of which invading the 3-3 point is only one of them. Ideally, you keep all of your options open, since as the game progresses, you may prefer one or the other. Once you pick one, the others are gone. If you are going to invade, ideally you do so one move before your opponent closes it off to consolidate his moyo. Another good time is when you have a good followup to reduce the influence of the wall he creates, like a good capping move or something, that you would would want to play anyways. If the reduction isn't something you'd consider playing without the influence the invasion would give, don't play the invasion.