There used to be a lot of discussion in L19 about the true meaning of thickness. There is an old heuristic of counting 3 points for every height of a thick wall (note that the thickness of the wall doesn't directly contribute to this equation, and instead is normally discounted for the value of an attack by the opponent, assuming the wall can be defended).

I will add a theorist's perspective. If a group G is completely alive in empty space, they regardless of how the opponent approaches, unless they occupy the empty positions neighbouring G, then G has at least 50% chance of occupying any (dame) point adjacent to G. There is also good chance of making territory locally too. Even if the opponent is strong, they still can't afford to play too loosely near to wall, or else they can be captured.

If we estimate, a la Spight influence functions, the influence of strength decays by a factor of two with every move played, then under stone counting, the wall has a value of 1 + 1/2 + 1/4 + ... = 2 for each height. I can't really justify that the possibility of making territory doesn't change this value, but atm, I can't even tell you if it should increase or decrease this value.

If the opponent has a strong group nearby, then they compete for value, perhaps by cancelling out value somewhere along the way so it becomes 1 + 1/2 + 1/4 + 1/4 = 2 (sente-gote tends to double the probability a boundary play occurs). This is no change at all. (note we haven't counted the value of the opponent's group)

If the opponent doesn't have a strong group nearby, then pushing the value of the wall up to 3 per height seems a good estimate. There is probably an argument for why it must be less than 4.

Now compared this to if G wasn't completely alive or could be cut. For every move that the opponent plays, it is more likely to be sente. If the group is killed, then though it still has lingering influence with threats to save it, this becomes a drastic reduction. For example, perhaps we should instead count influence as 0 + 0 + 1/4 if the capturer expects to answer threats two moves away from saving the group. Then for every move that the opponent plays nearby, if we assume the owner of G may ignore the threats (ignoring the sente reduction), we can add a value of (2-1/4)/(2^n) if the threat is n moves from capturing G to the opponent's moves.

Accounting for the sente reduction (the amount depends on the local temperature), then we should add less value. However, be careful as cuts can be double attacks if the player has another group H nearby that depends on G for support, increasing the value.

Overall, in summary, a good rule of thumb seems to be that having thickness can reduce the value even of strong moves by the opponent nearby by up to a factor of 2x. (and weak dead moves by as much as the global temperature). There were many weak assumptions in this derivation, but I think this is a good summary regardless. Playing near thickness is like playing in the centre. It is less valuable than the corner, but far from worthless.

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Give me triangles strong enough and I can measure the universe.

When Venus transits, we can align our clocks to one event. By measuring the angle to flat Earth at two places far apart on Earth, we can compute the distance to Venus and the Sun.

See https://www.lifein19x19.com/viewtopic.php?f=17&t=18734 for the good definitions!

Maybe influence functions go back all the way to Zobrist?

The three points per length of wall should be compared to counter examples. Such rules seem to fail because they are too mechanical and there isn't a criteria when to not "count" thickness. Personally I like the QARTS system because it works with the players understanding (or lack of) and not against it. What I mean is that QARTS subtracts 20 for each weak group that isn't able to make eyes and 10 for weak groups that can make a single eye, the problem for the player is then to find the eyes and group safety is what most of us should be thinking about before counting thickness. It is also straight forward to extend to cases where you make judgment to subtract anywhere between 0-20 points for a weak group.

I'm not sure if "the evaluation theory of thickness" is supposed to work in the opening, middlegame or the endgame? It can't be the case that thickness has the same value at every stage of the game. Probably there should on average be a gradual decline in the value of thickness from the opening to the endgame, and in the middle game it is probably more concreate in that you try to effect the thickness in the middle game. Every game is different though and there is the kind of whole board thickness that only really becomes useful in the endgame.

My own experience with counting three points for walls in the late middlegame / early endgame is that it doesn't work: 1. because it overvalues the thickness when there aren't weaknesses to exploit; 2. it is a biased estimate and doesn't get you the right answer when you are close to solving the endgame; 3. it doesn't seem to guide where to play. Maybe others have a more positive experience (and I probably put it in a more negative way than needed)?

Far too simplistic.



Always, of course!



Groups representing thickness can survive or die so the connection and life aspects of thickness decrease or increase. For surviving thickness, its aspect of new territory potential can increase if a) the connection and life aspects of thickness increase or b) the opponent's stones in the environment become weaker to increase the new territory potential of thickness. Otherwise, for surviving thickness, its aspect of new territory potential decreases to eventually zero at the game end while it should be realised.

The values of thickness must be reevaluated after each move.

My model allows all that.

I'm encouraged that at long last at least some people are making the distinction between thickness and influence. But there's still a long way to go if we want to get into synch with the Japanese pro usage of the words.

I think in particular that the first step is to make a case that a numerical evaluation of thickness can (or even should) be made. To say it is necessary for computer algorithms to work is no sort of case for humans.

In my now rather vast compendium of index of Go Wisdom concepts, thickness is one of the biggies. For example, in Kamakura there are about 40 instances for just 10 games, and that is without counting closely related topics such as thinness, walls, and influence. In not one of those instances, as far as I can recall (and likewise in all the other GW books), is there even a hint of a pro attaching a number to thickness.

It is true that a couple of Japanese pros have written books in which they appear to put a value on potential territory associated with thickness, but (a) the value is on the territory not the thickness and (b) they don't appear to use such numbers in their own games/commentaries. I infer these books are just sops to lazy amateurs, and maybe even ghost-written by amateurs. The much quoted 3 points per stone heuristic is something I associate with Bill Spight, though I think he told me once that he got it from someone else - certainly not a pro. The related heuristic of 6 points per stone in a moyo is something I heard from Korean amateurs. I've never seen it linked with a pro.

So, apart hearing why thickness should be counted, it appears we need also an explanation why amateurs cleave so much to counting thickness (and other things) whereas pros don't.

In real, pro-commentary life, the way thickness is talked about is rather about the way it adumbrates the game. It provides context. It determines strategies (including strategic mistakes). It's a gross form of signposting. It tells you what you can or should do next, or shouldn't do. And, along those lines, the one phrase that comes up most often in pro talk about thickness is "keep away from thickness - including your own". That's seems a lot more valuable than numbers of spurious accuracy.

Reply to John: https://www.lifein19x19.com/viewtopic.p ... 32#p273032