Double Ko Seki is a Terminal Anti-Seki
When we discussed anti-seki earlier above (see Odd Cases 🤔 in the Japanese Rules - #3 by yebellz and Odd Cases 🤔 in the Japanese Rules - #4 by yebellz), we considered some cases that were not “terminal” positions, i.e., they were unsettled in the sense that at least one of the players (if not both) should have played another move to maximize their score.
It might seem that all potential anti-seki positions would be of this unsettled/non-terminal nature. However, the double ko seki is an example of a terminal anti-seki position. Specifically, the single black stone and single white stone that are both in atari will be considered dead, but not removed (according to article 8) since they do not reside inside territory.
Neither player can force capture of the other. If Black plays at A, White can play at B to avoid capture, but then the cycle cannot locally continue due to ko. Likewise, if White plays at B, Black can play at A to avoid capture.
Unless some other ko arises on the board, this position will persist until the end of the game. After passing, during the confirmation phase, both players can demonstrate that the single stones in atari are dead, since they can be taken, and although it would allow the player to again play another stone, that new stone would not be uncapturable:
Article 7. Life and death
1. Stones are said to be “alive” if they cannot be captured by the opponent, or if capturing them would enable a new stone to be played that the opponent could not capture. Stones which are not alive are said to be “dead.”
The confirmation phase would actually get stuck in an infinite loop:
- Black captures a ko stone by playing at A.
- White captures a ko stone by playing at B.
- Black passes in order to retake the ko at B.
- White passes in order to retake the ko at A.
- Black retakes the ko at B.
- White retakes the ko at A.
- Black passes in order to retake the ko at A.
- White passes in order to retake the ko at B.
… return to step 1.
After step 8, we are back at the original position before step 1, where they can continue repeating the sequence. By symmetry, a similar sequence would occur with White starting. This confirms that neither player can capture the bulk of the other player’s stones. However, the single stones being traded in ko captures are dead, since although their capture permits another stone to eventually be played in it’s place, it would ultimately be captured as well.
Note that the passes during the confirmation sequence above are necessary in order to allow ko recapture and continue the sequence according to this rule:
Article 7. Life and death
…
2. In the confirmation of life and death after the game stops in Article 9, recapturing in the same ko is prohibited. A player whose stone has been captured in a ko may, however, capture in that ko again after passing once for that particular ko capture.
One might find it peculiar that there could be an infinite sequence during the life and death confirmation. The “no result” rule (which I will talk much more about in another post) would seem to be relevant.
Article 12. No result
When the same whole-board position is repeated during a game, if the players agree, the game ends without result.
However, I believe this “no result rule” is not triggered since the confirmation phase is not the “game” (alternating play phase). After all, the confirmation phase is only a consideration of hypothetical play, for the purposes of demonstrating that the stones cannot be captured, since even playing out the cycle indefinitely still never results in the stones being captured. This type of reasoning perhaps presents a major difficulty for algorithmic implementations of the Japanese rules.
Double ko seki is a source of infinite ko threats and further complications
I mentioned earlier that the double ko seki would persist to the end of the game, unless another ko situation arises. Basically, the double ko seki is two active kos, where a player could always take one to force the other player to take the other (assuming the life of that group is big enough to compel a response). Thus, the double ko seki can act as a source of infinite ko threats should another ko arise.
If another simple ko were to arise, then the game very well could end up as a “no result” by “triple ko” according to rule 12 (more about this in another post). However, it could also create another completely different kind of beast, Moonshine Life, which is not necessarily a “no result” situation (I will also post more about this in another post).