Когда то выкладывал сообщение с ссылкой на этот материал, но тамошний сервер не то околел, не то еще чего. На всякий, полежит у меня целиком.
... I'm sorry that there are no diagram here because the files are too big to show ^^;
The Value of One Move and Komi
Hong Soonseon
(Incheon Guweol baduk institute current president,
previously leader of Seoul University baduk club)
If they want to teach well, I think it is very important for baduk teachers to have a clear understanding of the baduk education principles. First of all, if the teacher has a deep knowledge about baduk principles and specificities, he can teach more easily and efficiently. So, I have been conducting a research on baduk basic principles during 3 years teaching students from beginner to almost professional level. As a result, I discovered some basic rules and standards. Among those results, there is one theory in particular I would like to introduce.
While playing, along the game, everybody has already been wondering about the value of each move. Recently, a baduk book has been explaining about the value of one move, but it mostly depends on professional perception or on the author’s subjectivity.
Roughly speaking, in the beginning stage, we can assume the value of 2 moves is more or less 20 points, or one move’s value is 6 to 10 points. But until now, this theory didn’t have exact foundations.
It is all the same about komi. For the famous Kiseong title holder Wu Qingyuan, black has approximately a 10 points advantage. Taiwanese Ing Changqi analyzed the winning ratio with black from 1,971 game results published in Japanese newspaper during 1978. Black had a 54% winning percentage with a 5.5 points komi, and a 53% winning percentage with a 6.5 points komi. With a 8 points komi (or 7.5 points), the winning percentage was 50%. Thus, he advocated for a 8 points komi. This assertion has been criticized because results are established on the base of games actually played with a 5.5 points komi. But still, the Ing’s cup is now played with a 8 points komi.
Another example, about 10 years ago, ChoEliason formula was announced as a mathematic attempt to define the value of komi. But this estimation was objectively not reliable. The formula was based on professional players’ opinion that a 3 stones handicap would be like a 35 points advantage, and 9 stones would be like 140 points. From this, for even games, a basic equation stated that the komi should be 6.5 points. Because it is based on a subjective opinion, this theory is not reliable.
Komi is generally affecting the opening stage, the fights occurring during the game, and the endgame. If the komi gives the advantage to one player, that player can win even if he avoids fighting and passively manages his advantage, and the other player should play very aggressively. So, komi is determinant for players’ state of mind.
1st principle: the value of one move in the opening stage is same as komi.
Usually, the first move is the biggest and the last move filling a neutral point is worth zero point. So, the value of each move is decreasing along the game. If you estimate the value of each move by drawing a line between the value of the first and last move, you can find some differences along the game but generally, this is an acceptable approximation. Assuming that in the beginning stage, each move is worth around 20 points and that each side takes alternatively the biggest remaining point, there is a 10 points difference between the 2 players if there is an arithmetic progression of the value of the moves along the game.
In the beginning stage, each player’s move is worth 20 points. So, 2 consecutive moves from one side are also worth 20 points. Thus, the value of one move is 10 points. Komi has the same value as one move in the beginning stage. Inversely, if the current 6.5 points komi is exact, we can deduce that one move in the beginning stage is worth 6.5 points, and 2 moves are worth 13 points.
Here is an illustration example of this theory. If the game is finished after 200 moves and if the first move is worth 20 points, a linear approximation of each move’s value from the first to the last move is:
20, 19.9, 19.8, 19.7, .... , 0.3, 0.2, 0.1
From this, the advantage for black due to having the initiative in the game is:
(20  19.9) + (19.8  19.7) + .... + (0.4  0.3) + (0.2  0.1)
= 0.1 * 100
= 10
If the game lasts 100 moves, the theoretical value of each move is:
20, 19.8, 19.6, ... 0.6, 0.4, 0.2,
and the difference between black and white is:
(20  19.8) + (19.6  19.4) + ... + (0.8  0.4) + (0.4  0.2)
= 0.2 * 50
= 10
So, the number of moves in the game doesn’t matter if we know the value of the first move. Here is the official explanation:
Let's call a the first move's size (2 moves are worth a points), –d the common difference is and n the number of moves:
d = a/n and
(odd numbered move’s size sum) – (even numbered move’s size sum)
= dn/2
= a/2
This gives the theoretical value of komi.
Next, we will discuss about exceptional situations through illustration examples when, at the endgame, the value of the remaining moves is like that:
Ex. 1. 10, 9, 8, 7, 6, 5, 4, 3, 2, 1
Ex. 2. 10, 7, 6, 4, 3, 1
Ex. 3. 10, 9, 7, 6, 4, 3, 1, 1
In those examples, the difference between having sente and having gote is:
Ex. 1. (10  9) + (8  7) + (6  5) + (4  3) + (2  1) = 5
Ex. 2. (10  7) + (6  4) + (3  1) = 7
Ex. 3. (10  9) + (7  6) + (4  3) + (1  1) = 3
In examples 2 and 3, the move values distribution is not linear. In this case, it shows a difference with the first example where the theoretical value of having sente was 5 points.
In baduk there are subtle irregularities in the distribution due to life and death issues. But even though some irregularities may occur, there are generally more than 200 moves because of the 19*19 size board and it can be thought as a standard convergence.
Application of a joseki
In baduk, when both players are playing the best moves during a fight, especially in the corner, the result of this handtohand fight can be called joseki. In some joseki, black plays one or two more moves than white, but in all other standard cases, we can estimate the situation more precisely. For example, let’s have a look at dia. 1. In this joseki, black played one move more than white. If we consider this shape, black has an about 3 points advantage in territory and his influence is also better than white’s. If black’s influence advantage is worth 4 points or more, we can say this joseki is not bad for black.
Dia. 1
However, how would it be if White didn’t play elsewhere but defended like in dia. 2?
If we compare both side’s thickness and territory, the situation looks balanced. From white’s point of view, he can choose to play elsewhere or to defend. For Black, if getting thickness doesn’t give him a more than 4 points positional advantage, it is hard to choose this joseki.
Dia. 2
In the same idea, when one side plays 2 moves more than the other side in a joseki, we can say he gets a good result if his positional advantage is 13 points or more.
In dia. 3, white played 2 moves more than black. In this shape, like in dia. 4, black played elsewhere and the sequence from 8 to 17 followed. If white’s development in A, B or C direction is worth more than 13 points, we can estimate that locally, white is not bad.
Dia. 3
Dia. 4
Elsewhere,  , Elsewhere
2nd principle: the total value of several stones is greater than the sum of each stone’s values.
How about placing stones like in a handicap game? In dia. 5’s case, according to the first principle, 1 stone should be worth 6.5 points, 2 stones 13 points, and a 4 stones handicap should be a 26 points advantage.
Dia. 5
In Dia. 5’s situation, ‘sanrensei’is a very wellknown position. If the 3 stones are valued as described in the 1st principle and with the recent 6.5 points komi, those 3 stones are worth 19.5 points. In that case, should we just see those stones exactly as a 19.5 points advantage? The conclusion is ‘certainly not’. In fact, for masters, the value is higher. Asking 31 young Korean professional players the equivalence in points for a sanreisei, 30 of them said had this opinion.
Chart1. Sanrensei's territorial value
Territorial value
20~30 points
30 points
40 points
50 points
Asked professionals opinions
4 people
13 people
9 people
4 people
For each player, the value of a Senrensei is different because they use their own criteria but for sure the value is more than 19.5 points. Why this? In handicap games, white moves are influenced by black’s handicap stones. In other words, from the beginning until the end of the game, it is hard for white to play moves with same value as black's. Or, speaking again about the 13 points advantage, white cannot play moves whose value is same as black’s. The bigger the handicap, the more difficult it is for white to play. For example, on some internet sites, there are life and death games like in dia. 6 and dia. 7.: one player has to give a 17 stones handicap and he wins the game if he can live anywhere on the board.
If black’s handicap is more than k stones and both players are of the same strength, we can deduce that white cannot make any living group. Let’s call this the ‘maximum possible handicap’. If the maximum possible handicap is 18 (k=18), the power of one move is (361 – handicap) / handicap. In this case, the value of one move is almost 20 points. If one move is worth 20 points, it doesn’t match with the 1st principle saying that one move is worth 6.5 points.
Dia. 6
Dia. 7
For example, if we represent on a graph the efficiency of one move as a function of the number of moves, it looks like that. In this graph, the value of one move is the total value of the position divided by the number of stones. In this situation, the move between k1 and k can be seen as a brilliant move but we need more research and experiments about this idea.
Application in even games
Similarly, in an even game, when black plays a sequence with more moves than white and white gets a balanced result looking like a joseki, it can be called a success for white. As for black, he should make good use of the power of his already placed stones to keep his advantage. As an illustration, in the previous figure (Dia. 1 joseki), the result might be unsatisfactory for black because both players obtain similar shapes although black started with more stones in place. For this reason, this joseki is in accordance with principle 1 but not principle 2. Therefore, the value of shapes resulting from fights might be different depending on the surrounding situation.
Proposition 1. The komi should be slightly favorable for black
Since the past, komi has been continuously increased. Nowadays, it is worth 6.5 points in Korea and 7.5 points in China. Ing rules set up a 8 points komi. In korea, when the komi was worth 5.5 points from 1989 to 1996, based on the 4 representative players Cho Hoonhyun, Seo Bongsoo, Yoo Changhyuk, Lee Changho on 509 games, the winning percentage was 52.5% for black against 47.5% for white. And, based on 8 players  Cho Hoonhyun, Seo Bongsoo, Yoo Changhyuk, Lee Changho, Yang Jaeho, Choi Gyubyeong, Choi Myeonghoon, Ma Xiaochun  the winning percentage was about 53.8% for black and 46.2% for white on 749 games.
Chart 2. Past Korean 4 top players wins and loss
White
Lee
Changho
Cho
Hoonhyun
Yoo
Changhyuk
Seo
Bongsoo
Black
Lee Changho

63W 31L
25W 17L
17 W9L
Cho Hoonhyun
47W 52L

23W 16L
33W 13L
Yoo Changhyuk
10W 26L
15W 23L

10W 2L
Seo Bongsoo
6W 18L
17W 30L
1W 5L

Next, we see the winning percentages after the value of komi has been changed in 3 countries.
Chart 3. Komi and winning ratio: Korea, 1999  2003
Black wins
White wins
5.5 points komi
305
255
6.5 points komi
332
336
Chart 4. Komi and winning ratio: Japan, 2002  2004
Black wins
White wins
5.5 points komi
430
368
6.5 points komi
382
400
Chart 5. Komi and winning ratio: China, 2000  2003
Black wins
White wins
5.5 points komi
212
196
7.5 points komi
54
71
If we consider only the statistics, the komi of 6.5 points in Korea seems to be the most appropriate. However, the psychological primary factor can influence the results. If the players have preconceived ideas about the komi, the statistics also may be affected. Anyway, according to the first principle, the komi's value is same as one move in the opening stage. Because this will never change, it means if the komi is changed, then both sides can have an advantage.
But we have to mention the existence of mimic baduk. Mimic baduk is, in an even game, when white plays all his moves symmetrically with black's moves. Then, if we look at the value of the moves, until the symmetricality is broken, the value of white's move is almost same as that of black's moves. In this situation we can assume that, like in the first principle, the value of a move is not decreasing following an arithmetic progression. When the symmetricality is broken, the balance is destroyed. This depends on the method black uses to prevent white from playing mimic baduk any longer. For a long time, two methods have been used for this. Playing a ladder like in Dia. 8, or playing on the tengen like in Dia. 9.
Dia. 8
Dia. 9
Actually, in the past, because there was a smaller komi or not at all in tournaments, mimic baduk was often played during the Wu Qingyuan period in Japan. Wu Qingyuan (alias Go Seigen) played mimic baduk for the first time in 1929 against Kitanu Minoru, the strongest opponent in his life. He started with black on the tengen, then played symmetrically until move 63. Kitani was upset and complained to the referee. Later, Wu Qingyuan said like this: 'in that time, there was no komi, so I guessed if I played mimic baduk continuously until the end, black would win by 2 or 3 points. Kitani left the room several times and was displeased. I think maybe he didn't understand the strategy behind mimic baduk. As I thought, in the beginning, things were going well for me. I played mimic baduk until move 63 and when I stopped, my situation was favorable. But I was careless and Kitani played a brilliant move, so I lost that game.' (Ref. Tygem news  www.tygem.com)
Still recently, the game showed in Dia. 10 was played by Chinese professionals (18th Chinese Myeongin title best 8 players, 2005/06/20). In Dia. 10, mimic go occured with ladders but when white played 86, the question for everyone was to know wether or not black could afford the komi. Actually, Yu Bin 9Dan said 'this kind of moves is severe and after white stopped mimic baduk, it seems black was leading. (Ref. Cyberoro  www.cyberoro.com). In that case, using ladder is not a good method. So, how about the other method? Generally, we occupy the board in this order: corner, side, then center. So, moves played in the center have a lower value. In this idea, when white plays mimic baduk and the komi is a burden for black, isn't it difficult for black to break mimic baduk? If mimic baduk winning percentage increases, the charm of baduk is also reduced. So, I think the value of the komi should be slightly favorable for black.
Dia. 10
References
Kim Jeongwoo, 2002, Value of the endgame, Oromedia.
Kim Jinho, June 1995, Research about the Advantage of Playing First in Go, Monthly baduk magazine.
Moon Yongjik, 1998, Discovery of baduk, Booki.
Moon Yongjik, 2005, Discovery of baduk2, Booki.
Lee Seonggu, October 1992, “Is 5.5 points a fair komi?”, Monthly baduk magazine.
Sakada Eio, 1965, Endgame miracle, Seoul: Yook Minsa.
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