This continues tangentially from the ‘ranking system’s crisis’ post.
The handicap stone system is one of the most lauded features of go. Most purely skill-based board games such as chess and shōgi have to depend on seemingly arbitrary methods to even out a game between two players of different skill; for example in the case of chess, by removing a particular piece or pieces from the stronger player at the start of the game. Go, by comparison, has a natural-feeling solution to handicaps: players are ranked on a scale where the difference of one rank translates to a handicap difference of one stone, and then the weaker player simply gets to play that number of stones at the start of the game.
While this handicap system arguably ‘works’, many go players know that it is far from perfect. An easy example is the setting of one handicap stone. As black gets to play one move at the start anyway, traditionally in the case of one handicap stone the weaker player has black and white’s komi is removed. This brings the value of the one-stone handicap setting to komi, 6.5 points under Japanese rules.
In reality, however, the value of one move is exactly twice the value of komi. Consider an even game with a 6.5-point komi at the start: to reverse black’s and white’s position, black needs to pass her move while white needs to give black 13 points. This way the starting position is reversed: now white plays first and black has the 6.5-point komi. Therefore, in the traditional one-stone handicap setting, black actually only gets half the value that she should.
When moving from a one-stone handicap setting to a two-stone handicap setting, the situation is as if as we started with a one-stone handicap game and white then passed once: black now clearly gets one stone’s worth more. We can see that the handicap system is not translative: if player A beats player B exactly 50% of the time on a one-stone handicap setting as white, and player B beats player C exactly 50% of the time on a one-stone setting as white, it does not generally follow that A should beat C 50% of the time when giving two handicap stones. The total handicap distance for A→B→C sums up to 13 points, i.e., what is usually ‘one stone’; and the handicap distance for A→C sums up to 1.5 stones, or 19.5 points. In the latter setting, player C has a free extra advantage of 6.5 points.
Now that we have KataGo, which can fairly accurately tell by how many points a player is winning or losing in a particular position, we actually have a kind of means to tell what the different handicap stone settings are ‘actually’ worth. The below table should be taken with a grain of salt, and its content might change slightly with future updates of KataGo – but I am confident that it gives a better approximation than the traditional handicap system or most alternatives devised by humans.
|n||Stone value||KataGo points value||Difference to previous value|
Edit 8 May: updated the KataGo points values according to the komi values that KataGo thinks give an even game under the handicap setting in question. The text below has been edited accordingly.
Some go players have for years argued that the value of added handicap stones is not constant; that for example that the jump from 8 stones to 9 stones is bigger than the jump from 5 stones to 6 stones, as there is less space left for white to manoeuvre in. While this may feel so, at least KataGo objects: the difference between 8 and 9 stones is apparently 14.5 points, which is in line with the other incremental differences (which range from 12.5 points to 16.5 points).
The table has a few other interesting anomalies: the jumps from 3 stones to 4 stones and from
6 stones to 7 stones 7 stones to 8 stones are bigger than the rest, at 16.5 points and 16 points respectively. The former is easy to explain with tedomari: in a three-stone handicap game white obviously should play in the last empty corner, as the other moves on the board have a smaller priority; but in a four-stone handicap game, white seems to have a multitude of similarly good choices, meaning that there is nothing particularly urgent to do. As for why the difference between 6 and 7 stones should be bigger, however, I can only guess. Now that the second ‘jump’ stabilised to the 7–8 stone difference, it makes more sense: I assume that the star point stones along the sides are worth more than the tengen stone in the 7-stone setting. This theory is backed up by the fact that the change from 4 stones to 5 stones is only 12.5 points, while the difference from 5 stones to 6 stones is again larger, at 15.5 points.
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