Two ways to count a finished board

Two rungs below, the ko rule turns out to be the hypothesis rather than the housekeeping: it is what puts Go inside the class of games every theorem on this site is about. One rung below, the same clause turns out to make a ko fight depend on a quantity counted somewhere else entirely.

Both of those are about which positions exist and which side runs out of moves. Neither is about the thing a game of Go actually ends with, which is a number.

Go has two ways of producing that number. Both are in daily use, they were written down by different people in different centuries, and a player who has learned one usually believes the other is the same rule stated differently.

It is not, and the difference has a parity in it.

The two conventions #

Territory scoring — the Japanese and Korean rule — counts the empty points a side surrounds, plus the stones it has captured. A stone placed on a dame, a neutral point belonging to nobody, scores nothing for anyone. A stone placed inside one’s own territory is worth minus one, because it fills a point that was being counted.

Area scoring — the Chinese rule — counts the empty points a side surrounds and its own stones on the board. A stone on a dame is worth one. A stone inside one’s own territory is worth nothing either way: the point stops being counted as empty and starts being counted as a stone.

So the two scores differ by exactly the number of stones each side has placed. On a board where the two sides place the same number, they agree. On a board where one side places one more, they differ by one.

The model, stated exactly #

Nothing here reads a board. The endgame is modelled as the accounting a Go player already reduces it to:

A position is a bag of independent plays. A gote worth v gives whoever takes it v points of territory and is then gone. A dame is a neutral point somebody has to fill: nobody gets territory for it, and under area scoring its filler gets one point for the stone. Players alternate, either may pass, and two passes end the game.

Every number below is that game solved exactly, twice, with the value of filling a dame as the only difference between the runs. Whether a real board reduces to a bag of independent plays is the same claim the temperature field makes when it prices an endgame, and nothing here establishes it.

The pass is in the model because Go’s ending rule is two passes, and because a scoring game with no pass in it is harder than the real thing rather than easier.

The parity, exactly #

Seventy-nine endgames, each solved under both conventions. Thirty-nine come out with identical scores and forty do not, and the split is not approximate:

Every endgame with an even number of dame scores the same under both conventions. Every endgame with an odd number scores differently, by exactly one point.

Thirty-nine of thirty-nine and nought of forty. That is the parity account confirmed as a theorem about the model rather than quoted as folklore, and the difference is never larger than a point because the difference is a count of stones and the dame are filled alternately.

Sixteen of the seventy-nine name a different winner — not a different margin, a different name at the top of the scoresheet. That is what a rules argument between two Go players is about, and it is a point either way, and it is the half of the finding everybody already knows.

The half nobody expects #

Twelve of the seventy-nine are played differently.

That is not something a scoring convention is supposed to do. A convention decides how the result is written down; it is not supposed to change which move is best. And here it does, and the reason is one line of the accounting.

Under area scoring a dame is worth a point to whoever fills it, so it is a one-point play and has to be taken in its turn against everything else on the board. Under territory scoring a dame is worth nothing, so it is left until last and filled when there is nothing else to do.

So the two rule sets disagree about the order of the endgame. On a board with a one-point gote and a dame on it, a player under Japanese rules takes the gote; a player under Chinese rules is indifferent, because the two are worth the same.

This is the finding worth carrying, and the reason it belongs on this site rather than in a rules dispute. A convention that changes the score is a convention about bookkeeping. A convention that changes the best move is a different game, and the two rule sets have been treated as the same game for a very long time. That is the same discovery the octal codes force about a rule table, one field over: two rules differing in a single bit are not two versions of one game, and nothing about how similar they look predicts how similar their answers are.

Where the extra stone comes from #

The account above says the two scores differ by the number of stones each side places, and that sentence carries the whole result, so it is worth deriving rather than quoting.

Take a finished board and count it both ways. Territory scoring gives a side its surrounded empty points; area scoring gives it those points and its stones. So area minus territory, for one side, is that side’s stone count — and the difference between the two scores, which is a difference of differences, is the difference between the two stone counts.

Now ask what fixes that. Every play in the endgame places one stone, the players alternate, and the game ends when nothing is left. So the two stone counts differ by at most one, and they differ by exactly one when the total number of plays is odd. The gote are shared out by the play; the dame are what remains at the end; and since a dame is worth nothing under territory scoring, nobody has any reason to take one early, so the dame are filled last and their parity is the parity of the leftover.

That is why the finding is a parity rather than a size. It is not that the conventions are approximately the same and disagree by a small amount. It is that they disagree by a stone, a stone is one point under one of them and nothing under the other, and whether there is an extra stone at all is decided by an even-or-odd question about the neutral points.

Why the parity is not the whole story #

A reader who has followed the parity account will expect the twelve to be a subset of the sixteen, and they are not.

The score difference is decided by the parity of the dame, which is fixed before anybody moves. The play difference is decided by whether a dame ties or beats something else on the board, which is a fact about what else is there — and a position can have an even number of dame, score identically under both conventions, and still have two different best moves.

That is the sharp version. The two conventions can agree about the answer and disagree about how to get it, and a player who reasons that the rules differ by at most a point and can therefore be ignored has drawn a conclusion about the score from a premise about the score, and the premise says nothing about the play.

It also has a consequence for the parity account itself. The parity of the dame is not a constant of the position — a player can change it, by taking a sente sequence that gains a tempo, and under area scoring that is worth doing and under territory scoring it is not. So the two conventions disagree about whether the parity is worth fighting over, and the parity is the thing they disagree about.

The count, and what a count of this size means #

Seventy-nine endgames is a small number and it is worth saying what it is a number of.

The sweep takes every bag of up to three scoring plays drawn from values of one, two and three, against nought to three neutral points, and solves each of the seventy-nine exactly under both conventions. Each solve is a search over the remaining plays with passing allowed, so the answers are optimal play rather than a rule of thumb applied twice.

What that range is good for is establishing the mechanism: that the score gap is a parity, that it is never more than a point, and that a difference in best move exists at all. What it is not good for is any claim about frequency on a real board, where an endgame has dozens of plays in it, values that are not small integers, sente sequences that change the parity, and half-point plays that this model has no way to express.

So the honest form of every number here is: over a stated family of accountings, this many. The parity claim is stronger than that, because it is a statement the model proves rather than samples — thirty-nine of thirty-nine and nought of forty is a clean split, and a clean split over an exhaustive range is the shape of a mechanism rather than of a coincidence. The counts of different winners and different plays are the weaker kind, and are quoted as counts.

What this costs the theory #

The rung two below is about the ko rule making Go a finite game, and it earns its place because a finite game has values. This one is about what happens after the values run out.

Both conventions score a game with a number and neither is a normal-play game, so everything this site’s apparatus is built on is unavailable to both. What is worth noticing is that the two conventions are not equally far away.

Area scoring is closer to a normal-play game than territory scoring is. Under area rules every stone placed is worth a point, so making a move is never worse than passing, and the incentive condition Milnor’s mean-value theory needs is nearly automatic. Under territory rules a stone in one’s own territory costs a point, so a player can be worse off for having to move — which is a zugzwang, and it is exactly what the older theory assumed away. A pass is what makes that condition unbreakable, and both conventions have one; what area scoring adds is that a player who does move is never punished for it.

That is not a small preference. It says the two conventions sit on opposite sides of the condition the entire classical theory of scoring games needs, and a theorem proved for one of them is not a theorem about the other.

Why players disagree about this and are both right #

There is a genuine argument between the two conventions and it is not about which is more elegant.

Territory scoring rewards efficiency: a player who achieves the same territory with fewer stones is not penalised, since the stones are not counted. Area scoring rewards occupation: a player is credited for every point of the board they hold, however they hold it. Those are different games in the sense that different play is correct in them, and the twelve positions above are the smallest instances of the difference.

There is a third convention this site has already met, and it is the one the whole disagreement is measured against. The normal-play convention says the player who cannot move loses, and under it both of these endgames have the same answer and the answer is the parity of the number of plays — which is a correct statement about a question nobody in a game of Go is asking. That is the whole cost of scoring restated: the theory that has an arithmetic has no numbers in it, and the two conventions that have numbers have no arithmetic.

The practical consequence Go players know is that the two rule sets almost always agree, and “almost always” here has a number attached to it: on the range swept, forty-nine of seventy-nine agree about the winner and sixty-seven of seventy-nine agree about the best move. A rule set that agrees with another five times in six is a rule set nobody will bother distinguishing, until the sixth time.

And the sixth time is where the famous disputes come from. A game decided by half a point under one convention and drawn under the other is not a scoring error; it is two rule sets giving the answers they were built to give.

What the picture cannot show #

No board is drawn, because none is modelled. The plays here are numbers in a bag, and the step from a real Go position to that bag — deciding which regions are settled, what each is worth, which points are neutral — is the whole of an endgame player’s skill and none of it is here.

Nor is there any ko. The two rungs below are about the one mechanism that makes a Go position not a bag of independent parts, and this model has thrown it away in order to have parts at all. A ko in the endgame would couple the bag to the rest of the board, and the sweep would be a sweep over the wrong object.

The best move is reported as a best move and not as the only one. Where the two rule sets agree, they may still agree for different reasons, and where a convention is indifferent between two plays the figure prints both. A player is not being told to take the dame; they are being told the dame is worth as much, which is the whole of the difference.

And the two conventions differ in more clauses than the one modelled here. Seki, dead stones in seki, the treatment of prisoners at the end, and the rules for a game that never ends all differ between the rule sets, and each of those has produced its own famous dispute. What is modelled is the single clause about stones, which is the clause the parity comes from.

The convention, named #

The essay is about two conventions and the model uses a third.

Territory and area are the ones being compared, and the comparison is exact for the accounting stated.

The third is the one every other field of this site runs on: normal play, where the player who cannot move loses. Nothing here uses it, and that is the point — the model has a pass in it, so nobody is ever stuck, so the last-move convention has nothing to attach to. Every number on this page is a score, and none of them is a value in the sense the rest of the site means.

What that costs is stated once and applies to the whole page. There is no group here, no comparison by subtraction, and no way to add one of these endgames to a position from another game. What there is, instead, is a number at the end — which is what everybody who plays Go for stakes wanted in the first place.

The surprise: the convention decides the move, not just the score #

The expected finding was the parity, and the parity is real: an odd number of neutral points is one stone’s difference, and one stone is one point under area scoring and nothing under territory scoring.

The unexpected one is that a dame is a one-point play under one convention and worth nothing under the other, so the two rule sets sort the endgame into different orders. That is a difference in the game and not in the scoresheet, and it means a strong player trained under one rule set is playing a slightly different game from a strong player trained under the other — not by a point at the end, but move by move all the way through.

The general shape is worth carrying past Go. A scoring convention is not a layer on top of a game; it is part of the move rule, because what a move is worth is what decides which move to make. Any two conventions that price the same move differently are two games, however similar their totals, and the totals agreeing five times in six is exactly what makes that hard to notice.

It is also a specific instance of the thing this whole field exists to say. The theory this site is built on has one convention in it and the convention is invisible from inside; a reader who has never seen an alternative reads the player who cannot move loses as a fact about games rather than as a choice. Go supplies two alternatives to each other, both of them ancient, both of them in use, and the difference between them is measurable on a board with one neutral point on it.

Where the ladder goes next #

go has three rungs and each is a clause of the rules doing work nobody expected of it: the ko rule buying finiteness, the same rule creating a resource counted off the board, and the scoring convention deciding what a move is worth.

The rung above is the one this model throws away in its second line. A Go endgame is treated here as a bag of independent plays, and the whole difficulty of a real endgame is that it is not one until somebody has argued that it is — regions that look settled and are not, a ko that ties two corners together, a group whose life depends on a point counted elsewhere. That is a rung about when a Go position becomes a sum, and the answer is a fact about the board rather than about the rules.