Kee Rules of Go
KEE RULES OF GO
Written by Wilton Kee on 18-May-2008. Dedicated to Wing.
Homepage: http://www.geocities.com/kee_rules/kee_rules_of_go.html
I. JUSTIFICATION OF KEE RULES
To prevent cycles, one might argue that traditional go rules already serve the purpose. Simple ko rule prohibits cycle within 1 turn (with each player playing once) and superko rule prohibits longer cycles. Why do we need something different? The rationale of each component of Kee Rules is illustrated as follows.
A. Why is Primary Prohibition superko (rather than simple ko) in nature?
Ans: Cycles like 3-ko, 4-ko and chosei can be longer than 1 turn. Superko can provide a sensible resolution to these long cycles.
B. Why is Primary Prohibition positional in nature?
Ans: A good rule shall penalize the one who produces the repeated board position. Consider a 3-player game with "Board A -Player 1-> Board B -Player 2-> Board A -Player 3-> Board A -Player 1-> Board B". It is more reasonable to prohibit "Board B -Player 2-> Board A" (if positional) than to prohibit the second "Board A -Player 1-> Board B" (if situational).
C. Why is Primary Prohibition lifted after N-1 consecutive pass plays?
Ans: In classical ko, after a player has placed a stone to produce a board position, the first local repetitive response by opponents would be prohibited so as to allow the same player to further locally place a second stone freely, with hope of the possibility of such repetition being permanently extinguished by the second stone.
"Pass plays by all opponents" can be viewed as successful prohibition of their local repetitive response. To be consistent with classical ko, the game shall start freely (i.e. without Primary Prohibition) since then.
As an example, consider this simple ko on a fully occupied board. If Primary Prohibition is not lifted, no opportunity can be given to White when Black has nothing to do after winning the ko.
Also consider this "sending two returning one" situation where Black keeps playing as a cycle after White has already passed (in White 2). If Primary Prohibition is not lifted, wrong timing of cycle (White 4 instead of Black 5) would be prohibited.
D. Why is Primary Prohibition not lifted until at least N-1 consecutive pass plays?
Ans: From the perspective of each player, only "own play" and "combination of plays of opponents" can be seen. Consider a 3-player game with "Board A -Player 1-> Board B -Player 2-> Board B -Player 3-> Board A -Player 1-> Board B".
If Primary Prohibition is lifted by the pass play of Player 2 so that only "Board A -Player 1-> Board B" is prohibited, it is unfair to Player 1 as he/she can only see "Board A -Player 1-> Board B -Opponents-> Board A".
"Board B -Player 3-> Board A" shall have been prohibited like simple ko because there are actually still at least two players fighting using Primary Prohibition.
E. Why does Secondary Prohibition apply on the board which has ever been consecutively passed by N-1 players?
Ans: It is because repetition cannot be extinguished even after a free local second stone has been allowed.
If there is no such Secondary Prohibition, the player would at best encounter the same pass plays by all opponents again. Afterwards it is just the same free local second stone on the same board position by the same player.
Without Secondary Prohibition, the game would be destined to certain cycle with some board positions from which repetition can actually be extinguished if given a free local second stone.
F. Why does the Playing Phase end upon 2N-1 consecutive pass plays?
Players shall be free from Primary Prohibition after N-1 consecutive pass plays. If all players still pass after N-1 consecutive pass plays (i.e. under an environment free from Primary prohibition), then Playing Phase shall end as no different plays would be expected afterwards if there is no change in both board position and prohibition status.
G. What is the purpose of the Affirming Phase and why does Reversed Secondary Prohibition apply in the Affirming Phase?
Ans: The purpose of the Affirming Phase is to release Secondary Prohibition and see whether any player can suggest dissatisfaction about the final board position by using his/her stone play if there is no Secondary Prohibition.
The principle is that the game shall not end with a board position from which certain player can make a change by his/her stone play assuming the game starts there (i.e. no cyclic prohibitions exist).
To effectively test whether possible stone play was once blocked by Secondary Prohibition in the Playing Phase, no stone play other than those really blocked in the Playing Phase would be allowed in the Affirming Phase.
H. Why does the game end with Cyclic Scoring if the last play was a stone play in the Affirming Phase?
Ans: The stone play suggests that the game ends in the Playing Phase just because of Secondary Prohibition. The game is therefore cyclic in nature and the final board position is not single but multiple. The previous board position which was identical to the final board position and produced by a crystallized stone play was the starting point of such cycle.
To avoid putting all weights to the single final board position which is just one within a cycle of many board positions, it is reasonable to argue that any intersections which were ever changed by stone plays during the cycle were neutral and should be excluded from scoring for or against one player. Only the intersections which are permanently with stone of the same color or permanently surrounded by stones of the same color should be counted in scoring.
II. STRENGTHS OF KEE RULES
1. Game ends in finite plays on all types of finite boards.
2. Game ends by consecutive pass plays of all players.
3. Game ends with a definite score.
4. Applicable to any number of players (even 3 or more).
5. Compatible no matter whether suicide is allowed.
6. Compatible with both area and territory scoring systems.
III. CONTENTS OF KEE RULES
1. Playing Phase
The game starts with an initial board position as the Playing Phase.
2. Alternate Plays
Players play alternately either by stone play or pass play.
3. Crystallization
If a stone play is immediately followed by N-1 consecutive pass plays (where N denotes the number of players), such stone play is crystallized.
4. Primary Prohibition
In the Playing Phase, a stone play may not produce a board position which is identical to any previous board position (including the initial board position), unless there have been a crystallized stone play on or after such previous board position.
5. Secondary Prohibition
In the Playing Phase, a stone play of a player may not produce a board position which is identical to any previous board position produced by a crystallized stone play of such player.
6. Affirming Phase
The game changes from the Playing Phase to the Affirming Phase immediately after 2N-1 consecutive pass plays.
7. Reversed Secondary Prohibition
In the Affirming Phase, a stone play of a player may not produce a board position which is not identical to any previous board position produced by a crystallized stone play of such player.
8. Termination
The game ends immediately after a stone play or N consecutive pass plays if all such plays are within the Affirming Phase.
9. Cyclic Scoring
If the last play was a stone play, the resulting board position must, by definition, be identical to the previous board position produced by one and only one crystallized stone play in the Playing Phase. The game would be scored using the final board position, but any intersections ever changing from the previous board position produced by such crystallized stone play to the final board position would be uncounted.
10. Single Board Scoring
If the last play was not a stone play, the game would be scored solely using the final board position.
IV. TYPES OF CYCLES
Type 1
Examples: Boards with 3-ko, 4-ko or chosei
Type 1 can be solved using traditional superko.
Type 2
Examples: 2x3 board, some but not all players disturbing on a fully occupied board (e.g. sending two returning one)
"Free local second stone" can be utilized if given to type 2, i.e. can be solved using secondary prohibition.
Type 3
Examples: 2x1 board, 2x2 board, all players (no matter whether 2-player or 3-player game) disturbing on a fully occupied board, 1x8 board (see below)
"Free local second stone" cannot be utilized if given to type 3, the game should be concluded under "Cyclic Scoring" mechanism.
V. TYPES OF CYCLIC PROHIBITIONS
A. Simple ko
Too weak to disallow all kinds of long cycles (all long cycles concluded as "no result").
B. Superko
Too strong to turn types 2 and 3 to anomaly. "No result" would be eliminated.
C. Spight Rules
Spight Rules give scores to types 2 and 3 using the single last board position upon game termination despite the fact in some situation that such single board position was just one within a cycle of many board positions.
D. Kee Rules
Kee Rules give scores to all types. Type 2 is given score by "Secondary Prohibition" and type 3 is given score by "Cyclic Scoring" which considers the oscillation during the cycle.
VI. EXAMPLES
A. Simple ko on fully occupied board
1. Simple ko in 2-player game
All black stones would be captured if Black loses the ko but white stones would not be captured even if White loses the ko. Even without hope to capture the white stones, Black places to capture the ko.
With Primary Prohibition in place, White cannot immediately snap back. Since the board is fully occupied and there is no other empty intersection for White to place stone, White 2 can only pass. After such pass, Black is subject to Secondary Prohibition upon the resulting board position of Black 1.
Black has nothing else to do and Black 3 passes as well.
Under traditional superko rule, White can never snap back because otherwise the board position would be repeated. However under Kee Rules, (unlike White 2) can snap back because with the pass of White 2 such Primary Prohibition has been erased.
With Secondary Prohibition in place, Black 5 can only pass.
This time all white stones would be captured if White loses the ko but black stones would not be captured even if Black loses the ko. With help of the third player, we are going to see whether stronger Black can capture all white stones.
With Primary Prohibition in place, White cannot immediately snap back. Since the board is fully occupied and there is no other empty intersection for White to place stone, White 2 can only pass. Please note that after such pass, Black is still not subject to Secondary Prohibition upon the resulting board position of Black 1 because Red has not yet passed.
Now the third player, say Red, captures the Black stone to help White (probably due to consideration out of this area).
Black cannot immediately snap back with Primary Prohibition. Black 4, White 5 and Red 6 then pass. After Black and Whites's passes, Red is subject to Secondary Prohibition upon the resulting board position of Red 3.
Under traditional superko rule, Black can never snap back because otherwise the board position would be repeated. However under Kee Rules, (unlike Black 4) can snap back because with the passes of Black 4 and White 5 such Primary Prohibition has been erased.
With Secondary Prohibition in place, if White 8 is a pass, Red 9 can only pass.
So should snap back. This is also not prohibited because of the passes of Black 4 and White 5. Red 9 also passes. Please note that after such pass, White is still not subject to Secondary Prohibition upon the resulting board position of White 8 because Black has not yet passed.
With Primary Prohibition in place, Black 10 can only pass. After such two passes, White is subject to Secondary Prohibition upon the resulting board position of White 8. White and Red have nothing else to do and White 11 and Red 12 pass as well.
Black 13 can now snap back.
With Primary Prohibition and Secondary Prohibition in place, White 14 can only pass.
With Secondary Prohibition in place, Red 15 can only pass.
Black wins the game by Black 16.
B. Sending two returning one on fully occupied board
1. Sending two first
is prohibited under both traditional positional superko rule and Kee Rules (with Primary Prohibition). Notice that the effect under situational superko rule can be different but we are not going to discuss into details here.
With board fully occupied and Primary Prohibition in place, White 2 can only pass. After such pass, Black is subject to Secondary Prohibition upon the resulting board position of Black 1.
Black places again to prolong the game by sending two. Is Black still without hope to win the game?
White cannot return one under traditional superko rule (both positional and situational). However under Kee Rules, can return one because with the pass of White 2 such Primary Prohibition has been erased.
As discussed above, with Secondary Prohibition in place, is prohibited. Therefore, Black 5 can only pass.
With Primary Prohibition in place, Black cannot immediately snap back. Since the board is fully occupied and there is no other empty intersection for Black to place stone, Black 3 can only pass. After such pass, White is subject to Secondary Prohibition upon the resulting board position of White 2.
White has nothing else to do and White 4 passes as well.
Under traditional superko rule, Black can never snap back because otherwise the board position would be repeated. However under Kee Rules, (unlike Black 3) can snap back because with the pass of Black 3 such Primary Prohibition has been erased.
As discussed above, with Secondary Prohibition in place, is prohibited. Therefore, White 6 can only pass. After such pass, Black is subject to Secondary Prohibition upon the resulting board position of Black 5.
Black has nothing else to do and Black 7 passes as well.
With Secondary Prohibition still in place during the Playing Phase, White 8 can only pass and the Playing Phase ends with 3 consecutive passes.
In the Affirming Phase, Black 9 passes. is not prohibited because it was prohibited under Secondary Prohibition in the Playing Phase (i.e. Reversed Secondary Prohibition). This leads the game to Cyclic Scoring (from
to
) and both intersections have ever changed and thus are just neutral points. The game (2x1 board) would therefore be scored as a draw.
2. 2x2 board
Black puts one stone back. The outcome is the same no matter is on the upper or lower corner. Readers may verify by themselves.
White 6 passes. After such pass, Black is subject to Secondary Prohibition upon the resulting board position of Black 5.
Black is not satisfied with the current board position and places stone on to capture the two white stones.
The outcome is the same no matter is on the upper or lower corner. Readers may verify by themselves.
It is unwise for Black 9 to place stone. Please try to verify this.
Assumes Black 9 passes. After such pass, White is subject to Secondary Prohibition upon the resulting board position of White 8.
White captures the two black stones again. This capture is allowed because with pass of White 6 such Primary Prohibition have been erased.
Black is prohibited to place stone on the upper corner with Secondary Prohibition. Black therefore places stone on the lower corner instead.
White 12 passes. After such pass, Black is subject to Secondary Prohibition upon the resulting board position of Black 11.
Black is not satisfied with the current board position and places stone on Black 13 to capture the two white stones. This capture is allowed because with pass of Black 9 such Primary Prohibition have been erased.
White is prohibited to place stone on the upper corner with Secondary Prohibition. White therefore places stone on the lower corner instead.
It is unwise for Black 15 to place stone. Please try to verify this.
Assumes Black 15 passes. After such pass, White is subject to Secondary Prohibition upon the resulting board position of White 14.
White captures the two black stones again. This capture is allowed because with pass of White 12 such Primary Prohibition have been erased.
Black is prohibited to place stone on both the upper and lower corner with Secondary Prohibition in place. Black 17 can therefore only pass. White 18 is satisfied with the current board position and passes as well.
With Secondary Prohibition still in place during the Playing Phase, Black is prohibited to place stone on both the upper and lower corner. Black 19 can only pass and the Playing Phase ends with 3 consecutive passes.
In the Affirming Phase, White 20 passes because White was prohibited under Reversed Secondary Prohibition to place stone on any intersection (because the resulting board position was not prohibited by Secondary Prohibition in the Playing Phase).
Black 21 is not prohibited because it was prohibited under Secondary Prohibition in the Playing Phase (i.e. Reversed Secondary Prohibition). This leads the game to Cyclic Scoring (from Black 5 to Black 21) and all 4 intersections have ever changed and thus are just neutral points. The game (2x2 board) would therefore be scored as a draw.
It should be noted that if Black 21 was placed on the lower-left, it still leads to Cyclic Scoring from Black 11 (instead of Black 5) to Black 21 and the score would be the same.
3. 2x3 board
White 8 is fine to respond by a pass. After such pass, Black is subject to Secondary Prohibition upon the resulting board position of Black 7.
Black 9 is not satisfied with the current board position and places stone on .
may be on the opposite corner but it does not matter.
White 10 is again fine to respond by a pass. After such pass, Black is subject to Secondary Prohibition upon the resulting board position of Black 9.
Black is not satisfied with the current board position and places stone on Black 11.
This is the only choice for White 12.
It is unwise for Black 13 to place stone. Please try to verify this.
Assumes Black 13 passes. After such pass, White is subject to Secondary Prohibition upon the resulting board position of White 12.
The outcome is the same no matter White 14 is on the upper or lower corner. Readers may verify by themselves.
It is unwise again for Black 15 to place stone. Please try to verify this.
Assumes Black 15 passes. After such pass, White is subject to Secondary Prohibition upon the resulting board position of White 14.
White captures the three black stones again. This capture is allowed because with the pass of White 8 such Primary Prohibition have been erased.