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.
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.
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.
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...
...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.
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