Parity of Komi (First Move Compensation Points)

Theory

Preliminary note: Here full count komi are considered. To get Chinese half count komi, divide a full count komi by 2. E.g., the full count komi 7.5 equals the half count komi 3.75.

Both Area Scoring and Territory Scoring can let a game's score assume every integer number. (Of course, under Area Scoring the score is not greater than the number of the board's intersections.) So either player wins by 0, 1, 2, ... or X points. However, in playing practice one often observes the following:

Area Scoring: Nearest scores differ by 2 points.
Territory Scoring: Nearest scores differ by 1 point.

This difference is caused by the following parity aspects:

Pi: Parity of the number of intersections of the board: Usually a 19x19 board is used. It has 361 intersections. This number has the parity odd. As long as boards of odd parity size are used, this parity aspect is constent.
Pn: Parity of the number of not scoring intersections: Usually the number of not scoring intersections in the scoring position is 0 or 2. These numbers have the parity even. As long as an even number (0, 2, 4, ...) of not scoring intersections occurs in the scoring position, this parity aspect is even. The number of not scoring intersections is 0 if there are no coexistences in the scoring position. The number of not scoring intersections is 2 if there is eaxctly one coexistence with exactly two "neutral intersections" in the scoring position or if there are exactly two coexistences each with exactly one "neutral intersection" in the scoring position. - If the number of not scoring intersections in the scoring position is odd (1, 3, 5, ...), the parity aspect is odd. E.g., this is the case if there is exactly 1 not scoring intersection in exactly one coexistence in the scoring position. This is scarce. Therefore usually (but not always) the number of not scoring intersections in the scoring position is even.
Pk: Parity of the integer part of komi: The parity of the integer part of komi is odd if the komi are (2n + 1) or (2n + 1.5) for some positive or negative natural number n. The parity of the integer part of komi is even if the komi are 2n or (2n + 0.5) for some positive or negative natural number n.

All parity aspects could also be considered together and together they are either even or odd.

Now we shall consider the effect of the komi parity in the usual case that (Pi + Pn) has the parity odd. Then with so called Standard Area Komi (2n + 1.5), the winner of a game at the scoring position is the same under Area Scoring or under Territory Scoring, unless exceptional circumstances (like asymmetrical coexistences or one-sided "neutral intersections") create additional scoring differences! This has been proven in general mathematically. Besides, under Area Scoring rules, both the komi (2n + 1.5) and the komi (2n + 0.5) always create the same winner, as long as (Pi + Pn) is odd. E.g., if the komi 7.5 lets Black win by 1.5 under Area Scoring rules, then Black wins by 0.5 under Territory Scoring rules and Black wins by 0.5 under Area Scoring rules with the komi 6.5.

Comments

The nearest score difference 2 versus the nearest score difference 1 is an aesthetical aspect. (With Chinese half counting, this becomes 1 versus 0.5.) However, the aspect is not one of practical strategic complexity of the endgame. In both cases, the aim of the game is to win. Strategic skill is the ability to distinguish move-sequences that lead to a win of Black from those move-sequences that lead to a win of White. Before the last few moves of a game, reading ahead move-sequences is always by far beyond human capability. This is independent of nearest possible scores. Only the omniscient player would read so far ahead already during the middle game that he can distinguish move-sequences not only due to the winner but also due to the parity of not scoring intersections in scoring positions.

For Area Scoring, it makes sense to use some Standard Area Komi so that usually the score is the same as under Territory Scoring. If winning percentages suggest a komi increment and if auction komi should not be considered an option, then the komi increment should be 2 for Area Scoring or 1 for Territory Scoring.

However, if one does not care whether in games with tiny winning margins occupation of the "neutral intersections" often lead to different winners under Area Scoring versus Territory Scoring, then one may as well select komi on the grounds of the most profound among the available statistics of professional games.

Examples on an Odd Board (7x5) with an Even Number (0) of Neutral Intersections

Example (1)

. . . # . . .
. # # # # # .
# # # # # # #
O O O O O O O
. O O O O O .

Example (2)

. . . . . . .
. # # # # # .
# # # # # # #
O O O O O O O
. O O O O O .

Discussion for Area Scoring and Territory Scoring

Komi	Score for Example (1) under Area Scoring	Score for Example (2) under Area Scoring	Score for Example (1) under Territory Scoring	Score for Example (2) under Territory Scoring
6	1	1	0	1
6.5	0.5	0.5	-0.5	0.5
7	0	0	-1	0
7.5	-0.5	-0.5	-1.5	-0.5
7.75 (Button Go)	-1.25	-0.25

With a Standard Area Komi like 7.5, the winner (here White) is the same in all cases.

Discussion for Spight Area Button Rules

Spight Area Button Rules are a possible compromise ruleset. They use Area Scoring and exactly one 1/2-point button (think of it as a joker playing card) available as a third move type. It has the strategic effect that "neutral intersections" are worth 0 points.

In example (1), White gets the button. In example (2), Black gets the button.

Spight Area Button Rules ought to use Standard Area Komi plus 1/4. Then the winner is the same for Button Go, Area Scoring, and Territory Scoring. (For Button Go, this has not been proven in general mathematically yet, but the author believes a proof to be possible.)

Other komi are not suitable for Button Go because the winners differ in the case (1) for Area Scoring and Territory Scoring.

Playing inside one's territory as one's last play before the button is taken cedes the button to the opponent. Thus, strategically, removals should occur after taking of the button. The button, if taken after the endgame, separates exciting alternation from the removals encore.