Thoughts upon the rules

Introduction

Before the game : board size, handicap setting, etc.

The game in progress : legal moves

Counting the final position : general principles

Japanese troubles : what is a dead group ?

Chinese puzzles

Ways of counting

Are White's complaints justified ?

Choosing the right komi

Introduction

This document is written for any go player familiar (well, reasonably fluent ) with one of the existing set of rules (for French players, this would probably be the "French rules", which are in fact a translation of the American (AGA) ones), but eager to know more, and bothered by, for instance, the subtleties of the pass penalty when White passes first, or the strange Japanese exceptions (like "bent four in the corner", not to mention "three points without playing"), and similar oddities.

Many technical explanations can be found on the Web, from those I rewrote (in French, following the previous version of François Mizessyn), to what is probably the most complete analysis, by Robert Jasiek (on his home page). Moreover, Robert Pauli recently published an analysis of the many weaknesses of Japanese rules, and a rewriting of them under the name of "Logical Japanese Rules of Go". But I thought a more general approach could be useful, pointing to the central issues, and maybe helping to put at ease the amateur player troubled by those divergences between rules : for the most part, these complications almost never arise in actual play, and, as will be shown below, it is very exceptional that the choice of the rule set affects the result of the game.

Before the game : board size, handicap setting, etc.

Go (or Weichi, if we are to mention its history) has at stake occupation and control of a grid, but this was probably not initially the object of the game, and actually the initial motivation of Weichi is clouded in mystery. It is well-known that the present grid (the 19×19 goban) is not the only one possible, and was for a long time coexistent with the 17×17 one, which may have appeared first, but, besides experimental games on 21×21, and beginner's games on smaller boards, it is only, oddly enough, games on 9×9 which may at times be seriously played by high-level players. It would appear logical if Weichi had always been played starting with an empty board, but this is not the case : in ancient China, games began with the four corner star points already occupied, and there existed even an old Korean variant where 17 stones were first placed on the board before play began (maybe under the influence of chesslike games, but the reasons are unclear) ; of course, these starting positions hindered the development of opening theory ...

It is quite possible that those early customs are an explanation for the marks on hoshis (the "star points"), and for the usage of fixed-style handicap, a practice which some may resent as a gross injustice, invented for making White's task easier, but which has some pedagogical advantages, like leading to standardization of teaching practice. Anyway, we will not debate in this document questions related to handicap, such as calculations of its true value, methods for having fair games and tournaments with it, or discrepancies between area and territory counting in handicap games. On the other hand, one may consider that reaching an agreement about the value of komi is also a necessary part of the preliminaries ; this will be discussed later.

The game in progress : legal moves

If the game had really be created around the idea of control of a grid, the capture rule would probably have appeared as a consequence of the necessity of disposing of the invading opponent's stones. It is quite unlikely that this was the real order of events, and in fact modern rules are probably derived from an early primitive version (like the game of the first capture, aka atari-go), refined by the players' experiences (for instance, it becomes quickly clear that something must be done in case of ko, if one wants the game to end at all ). Interesting analysis along those lines can be found in "Mathematical Go Endgames", or on line, for instance in this article at the "Sensei Library" site.

Whatever the history, the way of playing must have been codified uniformly in quite ancient times, and it is reasonably easy to formalize completely the moves allowed during the game (however, even for this comparatively simple question, interesting oddities can be found, for instance a strange prohibition of immediate recapture in the Tibetan rules. While on the subject of oddities, here is a complete (?) list of everything you will probably not want to do with a goban ; you will find there interesting variants (similar to fairy chess), like phantom go, or random go…

When the capture rule is agreed on, the only serious problem left is "what to do in case of repetition", with two possible options : A) Japanese (and Korean) rules, where only two-moves cycles (kos) are forbidden, and everything else (triple ko, chosei, etc.) draw the game ("mushobu", i.e. "no result" ; it must be replayed) ; B) Chinese rules (and also American, French, etc.), where every global repetition is forbidden (with small variations of negligible consequences, though…) : this is called the "superko" rule.

Usually, suicide moves (moves leaving one's own chain without liberties) are also forbidden, but in the Taylor-Tromp rules (applied in New Zealand ; they are, by the way, a model of a concise and rigorous set of rules), those are legal moves ; this is again of very little practical importance, as of course, after a suicide, it is the other player's turn (one will nevertheless wonder at some quite unrealistic positions (composed by a Dutch problemist) where this rule can be put to good use, changing the status of groups).

Finally, it is necessary to know when the game is over ; except for no-pass go (a strange variant not played anywhere, I believe), this happens after a certain sequence of consecutive "pass moves" (usually two, but questions related to ko threats may require three, or even four). That's it ; the only question left is to determine the winner, which is where the real trouble begins.

Counting the final position : general principles

To understand the following analysis, one must keep in mind, as has already been said, that the "historical" rules (as developed in China, and then in Japan) are only a codification of players' common sense (as a matter of fact, for a very long time, there were no written rules, and, in case of a dispute, one would simply consult the strongest player available). Even starting from the "first-capture game" it will soon appear to astute players that it is a good strategy to build large territories, where one will be able to play stones for a long time without the risk of being captured. .

It is reasonable to think that the purpose of the game then would move towards one of the "modern" rules we know : the Chinese way of counting, built on the notion of spaces occupied or controlled ("area counting"), or the Japanese one, using only the controlled spaces ("territory counting"). As soon as this is done, the rules must be made to work so that territories (in the "naïve" meaning of the word) are scored in such a way as to bring the same number of points that one would expect at first sight.

Everywhere, the logic of the game has driven players to discover the following basic principle : when a black territory (say) is built, its value must decrease (or stay constant) if Black plays inside, and increase (or stay constant) if White plays inside. As any argument following the lines of "I play a stone, you need 4 to capture me, so I win 3 points" must be eliminated to respect this fundamental intuition, one is led to the easy Chinese solution of counting the capturing moves as points for the one playing them, or, for the reasons discussed below, to the Japanese idea of dead stones, and the well-known rule of counting dead stones as prisoners after the end of the game.

An important consequence is that a move in one's opponent's territory is now a bad move (if it is not, this shows the territory was not really his yet) ; this is true in Chinese and in Japanese versions, even if is is not so obvious in the Chinese one. In fact, it is even less obvious that playing in one's own territory is a bad move under Chinese rules. But a little thought will show that this is of even less value (if such a thing is possible ) than playing a dame, as you win one point for your stone, but lose one for the space you filled.

It is nevertheless true, in area counting, that playing in one's own territory is not penalised, if it is done late enough (after the last dames are played) ; it is a common mistake to resent this as an opportunity given to overcautious players, no longer punished for their extra defensive moves (finding where the mistake lies is an exercise left for the reader ; the solution will be found here) ; this resentment may have been one of the reasons behind the birth of territory counting, in accordance with the feelings and practice of Japanese professionals, who instead try to play (before counting) the fewest possible useless moves.

Japanese troubles : what is a dead group ?

Now the real problems appear: some decision process must be created to determine which stones are dead. Of course, such a procedure must not require either player to lose points (otherwise, the previous difficulties will reappear) ; in rules where the proof of the death is done by actual play (this was for instance the case in my first writing of the (old) french rule, in 1972 ; one will find it in "Le jeu à 9 pierres"), the position must be counted as it was before the dispute (this leaves various problems unsolved, of which I was blissfully unaware in those times )

As it happens, Japanese sensibilities (and maybe also their conception of the law) resulted in the treatment of this question neither by a practical procedure nor by a abstract general ruling, but in a case-by-case fashion. In fact, a rigorous definition of life and death is extremely difficult to formulate (see on this subject the rule "J89" and its revision (1997) formalized by Robert Jasiek). Moreover, all decisions already made becoming precedents, common sense arguments give no longer necessarily the results professionals have already agreed upon. Three examples will show how this is not at all a trivial matter.

The most famous of those precedents is probably the one known as "Bent four in the corner" (Figure 1). Black has no move left. White can wait till the end of the game, then sacrifices 4 stones, plays back (at 1-2) to initiate a ko in his favour (i.e Black must find the first threat), and White wins the ko, since the game has ended and all the ko threats have disappeared.

Obviously, this "demonstration" suffers from two weaknesses. First, it can be said that White should pay to eliminate Black threats. This argument is easily refutable : in area counting, it does not cost anything (he plays inside his own territory, but that doesn't diminish it); in Japanese-style rules, those are "virtual" moves, only having to be shown as a proof that the killing can be done. But the other weakness is a real one : there may exist non-removable ko threats (in fig. 1a, Black may sacrifice 4 stones, and get 2 ko threats).

In cases like that, Japanese counting (which rules nevertheless the group as dead) seems essentially "unfair". Moreover, its rigorous formulation is not an easy task : 1) what happens if the ko threat has some relation to the fight in the corner? 2) what is exactly the definition of a " bent four in the corner" ? Japanese rules used (before J89) to give a list of the three possible shapes, which, of course, created painful discussions when there was found (during a professional tournament in the 1960's !) a new situation of arguably similar kind

 Figure 1 : « Bent four in the corner »

 Figure 1a :  Same one, with a non-removable ko threat : by Japanese rules, White would win by 7 points ; in area counting , he may : a) not play, and lose by 9 points b) start the ko, not answer the threat, and lose by 8 points c) answer the threat, let Black recapture and win the ko (as White has no threats); White loses again by 9 points...(the reader should check those calculations ; he will note that in the initial position, Black and White played the same number of moves )

A careful analysis of the fight of figure 2 (left to the reader ) shows that White cannot move (if he does, he loses both his groups), and that Black can capture any group she likes, but that this capture lets the other one live. Arguing that "the two white groups are dead" (meaning capturable against any defense, even if White starts) is hardly contestable, yet in this case, Japanese jurisprudence (and, of course, the logical application of area counting) is "Black must play (and capture the upper group, if she has any sense)" ; this seems a lot more sensible .

.  Figure 2 : "Two dead, only one capturable"

On the other hand, both groups of figure 3 are capturable, but whoever captures first leaves the possibility of "under the stones" play. If Black begins, White scores 3 points (neglecting the ko, which could be a problem in itself) ; if White begins and then finds the tesuji of sacrificing three stones,White scores 2 points (and if he misses the tesuji, he scores only one point !). In short, each player would rather have the other one start the fight (as in a seki ), but of course White doesn't get anything if nobody plays here, which is even worse for him. It is a cute exercise in counting to show that in area counting (and by finding the tesuji), White gets in fact the 3 points that (for completely different reasons) Japanese ruling gives him.

 Figure3 : « 3 points without playing »

The reader not yet completely disgusted will find on Robert Jasiek's home page a complete list of those strange positions (as far as one knows  ; Andre Moussa and I discovered in the early 1980's complicated semeais not covered by the rules in those days ; you will find them on my bestiary), and an explanation of their rulings in the formal recent versions (J89 et J97). Other similar analysis (and also litigious positions in Ing rules) can be found at Matti Siivola's site.

Chinese puzzles

At this point, the reader should begin to agree that area counting displays some superiority in litigious positions. To be completely honest myself, I must admit that it suffers also from some imperfections. The main problem is that two "philosophical" visions are always possible : a "theoretical" one, saying that as soon as something is clear in the intuition of every player, like the fact that a white group without eyes and completely surrounded by a live black group is never able to subsist on the board, any rule must respect this knowledge (this is the Japanese point of view) and a "pragmatic" one (more "Chinese") saying (in its most radical expression) that if White can by any means prevent his capture, he has well earned his victory. Here is a spectacular example of what I mean : the "moonshine life". The NW corner group is not capturable, as White has an infinity of non-removable ko threats on the right.

Almost every player feels that this is like cheating, and would rather ask the rule to prevent White from succeeding with this. In fact one discovers, when trying to play the sequence of threats, that the superko rule solves the problem : if White attempts to save his NW group, he loses everything (try it). But it is a close call

In the same vein, some positions has been constructed (to the amazement of almost all strong players) where a normally live group (in seki) can be captured, as in the following situation …

White, with a clever use of the superko rule, can kill the black NW group ! It is not very hard, but the idea is quite hidden ; I will let you try to discover by yourself the devilish trick of the White ko threat in the middle of the sequence, in spite of the absence of any ko, to force Black to pass twice and finally lose his group, because of the prohibited repetition. If you didn't find it, you are in good company ; click here to see the solution.

Lastly, T. Mark Hall (BGA 4d), a few years ago (in a serious game), came across a perfectly plausible position (the "molasses ko") where the literal application of the superko rule leads to a completely absurd and monstrous conclusion. I recently discovered that with a slightly modified formulation of superko rules (due to Bill Spight), these difficulties disappear, but there is no guarantee that other troubles will not appear someday .

All this being said, the above positions are nevertheless exceptional, and they don't seem to give many serious arguments against area counting. So why does one encounter so much reluctance to count the Chinese way ?

Ways of counting

The real problem is that the majority of senior players were taught the Japanese way, and that the Chinese method is unfamiliar to them. When discussing counting, it is important to distinguish between scoring (i.e. number of points in territories, value of prisoners, etc.) and the actual method of counting (with rearrangement of boundaries, placing the prisoners in opponent's territory, etc.) ; the Chinese procedure is much more spectacular, and ends by counting stones by lots of 10 . By the way, if one has at disposal 180 stones of each color (and bowls for measuring the number of stones so as to be sure none are lost), the Ing way of counting is the fastest and the surest of all (essentially, this consists simply of trying to fill one's territory with the stones still in the bowl ; the winner is the player who succeeds).

So, one wants area scoring (if only to simplify the referee's task ), but wants also to use the Japanese way of counting, in order not to disturb the (bad ?) habits of the players (one must admit it is easier, during the game, to count the prisoners and not the stones on the board, even if it comes to the same result in the end). To respect those constraints, it appears there must be a correction for the last move : let's imagine a completely filled 19×19 goban, except for a black territory of 8 points, and a white territory of 2 points. There are no prisoners, and neither player has passed (except for the last two moves). Japanese scoring  : 6 points minus komi for Black (so one half point with the usual komi). 351 stones were played (361-10) ; since Black began, there are (on the board) 176 black stones and 175 white ones ; area counting is therefore 7 points minus komi for Black  and Black has played the last stone. Therefore, in this case, White must be penalised by 1 point to get the result of area scoring, while counting the Japanese way. It is easy to convince oneself that prisoners would not change anything in the above analysis.

Are White's complaints justified ?

At this point, one used to Japanese scoring can think that all of this is quite unfair for White, who will lose half of the games he would have won by territory counting (by half a point). In fact, nothing like that happens, but one needs a little bit more reflection (and calculation) to believe it. It may not be completely mandatory to open the window where the complete analysis of this question is done (if nothing appears, just click here), but this will, I hope, put algebraically minded readers at ease once and for all.

Indeed, those calculations show that (under the previous hypothesis), White cannot lose "because of area counting". Of course, the reader will have already guessed that a few weaknesses subsist in this analysis . First, the goban must be "odd-sized" (9×9,13×13, etc.) ; this is (and has always been) the case in serious play. Then, the analysis is no longer correct for handicap games (as the parity argument above depends of the handicap), but this is not a large inconvenience in practice (though... : in Grenoble 2003, David Wu lost by half a point a three stones handicap game (against Titi), which he would therefore have won in japanese scoring (and that he believed having won in area counting too, but this is another story, related to a misconception about the value of komi in this case )) as in any case handicap official games are much less frequent (and codified) that even games. The komi must also be odd (5.5 or 7.5). This is still the case now, but this is liable to change…(see on that question the next section). Last but not least, there must be no dame left (or an even number). But the only (reasonable) case where this could happen is seki (between one-eyed groups). Since this case is not so frequent, it will probably only exceptionally combine with half-pointers. Without doubt, this combination will appear as rarely as the other two major differences between area and territory scoring, which will conclude this part of the discussion.

First, area scoring allows getting points in seki. This is not so easy : here is the simplest case :

a and b would be neutral points according to Japanese rules ; with area counting White gets 2 points here. One can note again that Japanese rules are more "logical" (White has barely escaped here, so he got b only by the narrowest margin ), but much harder to formalize  : what exactly is a seki ?

The other difference lies in the possibility of unilateral dames, i.e. playable by only one player. The opponent must then pass (and pay the usual penalty if one counts the Japanese way) while they are filled. A few examples of this are known (and they can get quite tricky to play optimally when one meets them for the first time) : it may, for instance, happen in connection with a "half point" ko in the last moves of yose, when one player has many more threats than the other : she can leave the ko open, then fill it, while her opponent can only pass (or play inside her own territory, which is no better).

Unilateral dames appear also in some "shape sekis". For instance, in the opposite figure, White can play at any time moves 2 and 4 without destroying the seki, but Black cannot play at all, as if she plays 2, for instance, White answers at 4 and the black group dies ("bulky five"). Area scoring adds therefore 2 points to White territory, and counting the Japanese way (aka "fast counting method") gives the same result (if White plays correctly, i.e. after all external dames are filled) by making Black pay one or two pass stones (depending on who filled the last external dame). It is for those situations that the following clarifications (not completely explained in the official French document) were made : The game must end by a last exchange [Black passes, White passes]; a player who passes in any other exchange is penalised by one point, this situation happening usually in the last exchange: [Black plays, White passes] (which should therefore be followed by [Black passes, White passes] (without penalties) to be completely rigorous).

Choosing the right komi

Actually, another "philosophical" question has not yet been discussed : what exactly is one supposed to prove by winning a Go game? It is always possible to argue that only the pleasure of having played good moves (or at least better moves than the opponent) is what matters. But if the point is to prove an expertise greater than the opponent's, it is clear that initial conditions must give equal chances to both (one will nevertheless read with profit the fascinating analysis of Roger Caillois in "On games and men" , especially the section contrasting games of agon and games of alea). Yet the game is obviously biased towards Black (in the worst case, if Black's first stone would cause her to lose the game (very implausible), she needs only to pass to get at least a draw).

There are two solutions to this problem : play a succession of games, alternating colors, or compensate one way or another for the initial advantage. As the first solution can be quite inconvenient in practice, Japanese players, with some reluctance, invented (in the 19th century) and then adopted (circa 1920) the idea of komi (or "compensation"), i.e. the custom of giving White some points to compensate for the disadvantage of playing second. Alas, the choice of the correct komi is difficult. Most national associations (following the then all-powerful Nihon Ki-in) opted for a fixed komi, the same for all official games, fixed at 5.5 points (around the sixties) ; the choice of a non integer komi being of course made to prevent draws. We have seen above that this value is compatible with the eventual correction needed for area counting.

But changes are always possible, and, as a matter of fact, the new Japanese komi (of 6.5 points, this was decided in Spring 2002) will no longer give the same winner in both rules in every case. Anyway, those fashion inspired komis are clearly not a secure guarantee of fairness. In the 70's, many players (including myself) proposed procedures for determining the komi based on the classical problem of fair cutting of a cake (one cuts, and the other chooses his piece), known under the generic appellation of "komi bidding". My best proposal, obtained after much trials and errors, and discussions on rec.games.go (and which has not been seriously contested yet) is the following :

1)      Every player makes publicly known (for instance at registration time in a tournament) a number (integer or half-integer) called his "favorite komi" (fk)

2)      For every official even game (in a tournament where this system is applied, of course), the "real komi" (rk) is the mean of the "favorite komis" of both opponents.

3)      The player with the larger "favorite komi" begins (and plays with black). In case of equality of fk, colors are drawn randomly…

Commentaries :

1)      Using this method, let's say that A declared a fk of 8 and B a fk of 4. This means they feel the game is fair with those respective komis. The procedure gives a real komi of 6, and A begins. A should be satisfied, as she must pay only 6 points to be Black while she thought this advantage should cost her 8 points ; B should be equally satisfied, as he gets a compensation of 6 points, and was prepared to receive only 4.

2)      Some similar systems have been criticized for introducing a psychological dimension (a poker game) in the bidding period  : let's say A knows that C hates playing with White, and is willing to declare 12 as fk to be sure to get Black. A has then only to propose 11.5 to get White and a komi of 11.75 points, while she could not hope for more than 8 if C knew A's fk in advance. This is the main reason for point 1) above : no such kind of manipulation is really possible, as every player must declare his fk before knowing who will be his opponents.

3)      Of course, if rk is an integer, the problem of drawn games ("jigos") is back. But is this such a large inconvenience in practice ?

Solutions

If a player doesn't wait for all the dame to be played before defending, he loses one point (or more ) by playing inside his own territory, as his score doesn't change, while his opponent can score one more point, if only by playing a dame himself. On the other hand, if some problem should materialize after the dame have been filled, adding the defensive move will lose one point (the dame left to the opponent); this proves the score will not change, except in the case of the last dame (which will be detailed later).

White captures 4 stones (playing a1, say), Black cuts back at b2,  White plays a2, sacrificing the 3 stones a2-a1-b1, Black captures them at a3, White plays atari at b3 and Black loses the a3 stone.

White puts the NW group in atari playing a3, Black plays (and captures) at a1, White plays the ko threat at j6, Black answers j5 (let's call this the position *), White captures at a2, Black must pass, White plays a3 and Black is not allowed to capture (at a1), as this would return to the position * !

Let's admit there are no more dames (they have just been filled in alternation, and there is no seki), and both players have just passed (for the first time). Dead stones have been taken off (and added to prisoners, if we are counting the Japanese way). The black territory (in the japanese sense, i.e the number of empty intersections) is b, the white territory is w, there are pb black prisoners (captured by White) and pw white prisoners. At this instant, there are therefore t=361-b-w stones on the board, x played by Black and y by White. If White played last, both players played the same number of moves, and so x+pb=y+pw, and x+y=t, so x=(t+pw-pb)/2 et y=(t+pb-pw)/2 ; those numbers being (of course) integers, this means that t+pw-pb is even. Or else, White passed first, and x+pn=y+pb+1, so x=(t+pw-pb+1)/2 et y=(t+pb-pw-1)/2. Japanese score (for Black) is j=b+pw-w-pb (minus komi). Chinese score is c= b+x-w-y (minus komi). One can check easily that this is exactly what the "rule of the first pass" takes into account, i.e. j=c if t+pw-pb is even, and j=c-1 if it is odd. Now, if White wins by one half point (in Japanese scoring, with a 5.5 points komi), it is because b-w+pw-pb= -5, so t+pw-pb=356-2b is even and there is no correction : White wins also by area counting.