Is ‘fair’ komi necessarily an integer?

Published 20 Jul 2020 by antti (last edited 21 Jul 2020)
tags: tldr

One popular topic among go players is: what is the ‘correct’ value of the komi?

There are many ways to approach this question. For example in tournaments, it is useful to have the 0.5 in 6.5 (or 7.5) to prevent draws; after all, tournaments need a winner, and a game that ends in a tie often has to be replayed. In casual games, on the other hand, the players might actually prefer it if the possibility of a draw exists.

Usually this discussion quickly moves into the abstract: if we had two perfect players, what would be the ‘correct’ value for the komi that leads into a draw?

Recent evidence, especially from the ai, seems to point to that this ‘fair’ komi value is probably 7 points under Chinese rules and 6 or 7 points under Japanese rules.

Why cannot the ‘fair’ komi value involve a half-point? This is because when a non-komi game ends, the final score difference is always an integer. The value for the komi that offsets this integer difference also has to be an integer.

Or so I had thought until today, when I came up with the counterexample below. Japanese rules (1989) are used.

Dia. 1
Black to play

Let the position in Dia. 1 – it is Black’s turn. This position is of course highly artificial and simplified; in mathematical terms, it is enough if we can show that a counterexample can exist.

Let’s see how ‘perfect’ play changes depending on the value of the komi.

Let komi = 6 points.

Dia. 2
6 at a

Assume that Black plays in the simplest possible way in Dia. 2. After black 7, the game is over, and both players have 11 points – the game ends in a draw.

Let komi = 7 points. Now, playing as in Dia. 2 leads to a one-point loss for Black.

Dia. 3
6 at a

Black 1 threatens to cut White’s stones apart. If White connects with 2, then the moves up to 9 follow. Even if white 8 tried to fight the kō at 9, White cannot win it because of Black’s threat at ‘b’.

With komi = 7, this game results in a draw.

Let komi = 6.5 points. Now, White loses by half a point if he plays like in Dia. 3, so instead of 2...

Dia. 4

...White has to cut Black with 2 like in Dia. 4. Black counter-cuts White with 3, and after white 4 and black 5, the left side turns into a triple kō – ‘no result’ under Japanese rules.

To summarise: when the komi is 6 points in this game, Black can choose between a draw and a ‘no result’. When the komi is 7 points, White can choose between a draw and a ‘no result’. If the komi is instead 6.5 points, both players have to opt for the ‘no result’ or else lose the game – is the ‘fair’ komi for Dia. 1, in other words, not exactly 6.5 points?

While the position in Dia. 1 is artificial, it is not impossible that an effectively similar position could occur on a 19×19 board after perfect play – after all, triple (and quadruple) kō fights have also happened in top professional games. Furthermore, Dia. 1 shows just one example of a probably much larger number of whole-board situations where only a decimal komi will cause a ‘fair no result’. The existence of the position in Dia. 1 proves that it is not ‘certain’ that the correct komi under Japanese rules is either 6 or 7 points.

Comments (8)

sig wrote 3 years, 4 months ago:

>‘fair’ komi value is probably 7 points under Chinese rules and 6 or 7 points under Japanese rules

According to it's 9 on 7x7

sig wrote 3 years, 4 months ago:

Also, what about rulesets that prevent "no result" (basically have superko, I guess)

lightvector wrote 3 years, 4 months ago:

>>‘fair’ komi value is probably 7 points under Chinese rules and 6 or 7 points under Japanese rules
> According to it's 9 on 7x7

antti is probably talking about 19x19, not 7x7, but I'm guessing you're aware of that and just also interested in 7x7?

What's fun is: due to a recent new variation discovered by KataGo, it seems quite likely that the optimal komi on 7x7 is 8 under Japanese rules, not 9. It's still 9 under Chinese rules.

lightvector wrote 3 years, 4 months ago:

(of course, all of this is very far from any sort of proof, it's merely the best we can tell for now. 7x7 is way beyond the reach of known rigorous proof methods, the largest fully rigorous solution is for 5x6 Chinese rules, maybe 6x6 would also be possible using a big enough cluster with modern hardware, but it hasn't been done).

yakago wrote 3 years, 4 months ago:

I thought I would make the example a little more beastly and extend it to 19x19

Here is a 19x19 position, where the 'point of contention' at the top, is worth slightly more! (actually a bit too much I think)

If you apply the same logic as in the post, black can opt to 'play safe' and lose by more than 100 points on the board. by connecting on along the bottom. But if the komi is sufficiently in black's favor, it can lead to a victory.

Black can also choose to connect, at which point white can 'play safe' and lose by a few tens of points. Again, with a big enough komi, this could be a certain victory.

But this means that know there exists a decently sized interval (modify the example to change it if necessary), where both sides should opt for playing the triple ko variation.

So not only can we have half a point as 'fair komi', we can in fact have an entire range of values.

If you go the mathematical route, the most well known (robust?) way we have at looking at these kinds of question, is the Combinatorial Game Theory, as Berlekamp, Conway, etc. has established it.
Under those definitions, the standard way is to require a game to end within a finite amount of moves. I.e. infinite cycles such as triple kos are simply not allowed, and such a game would be considered ill-defined. You would not want to deal with 'no result' or even draws for that matter (although I'm sure there exists some extensions).

What people usually do is to turn to rulesets like Tromp-Taylor, where these kind of things do not exist :-)

None the less, japanese rules are a reality, so I applaud the very cool example :-)

antti wrote 3 years, 4 months ago:

sig: The article assumes a 19×19 board, although komi 6/6.5/7 may be correct for 9×9, too.
sig: As far as I’ve looked into it, this counterexample doesn’t seem apply to rulesets with superkō.

yakago: I think your example is a bit beside the point that I am trying to make in my post. I am trying to show that there exists a type of a whole-board position in which a decimal komi turns out to be the only fair komi. My point isn’t about what komi the players should bid on in Dia. 1, but if it can be possible that perfect play leads to a whole-board position that is virtually identical to Dia. 1, in which case the ‘fair komi’ turns out to be a decimal number rather than an integer.

yakago wrote 3 years, 4 months ago:

Yes, your example shows the half point komi nicely.

What I am trying to point out, is that this extension shows, that if perfect plays leads to a position akin to the one I posted, then 'fair komi' would be an entire interval [a,b] of values.
i.e. we could have that {6,7,8,9} are all 'fair komi' values, where the game is forced into 'no result'.

The point being that 'fair komi' would then not be a single value.

antti wrote 3 years, 4 months ago:

I see, now I think I understood your point! So while I (or somebody) might argue that it is ‘unlikely’ for a perfect game to develop into such a position, it cannot be excluded from the possibilities until a more rigorous proof is found.