# Analysis of Ultimate Go

__Keywords__: Rules

*(There is a discussion section at the bottom: please comment there. Thanks in advance!)*

## I. **Abstract**

Rulesets of Go are roughly divided into two flavors: “area scoring” (e.g. Chinese rules) and “territory scoring” (e.g. Japanese rules). Area-scoring rulesets tend to be quite simple, however they usually share the “dame parity” disadvantage: in very close games the result might depend on the parity of the number of neutral (or dame) points. Territory-scoring rulesets manage to dispose of the dame-parity issue (among other things), but at the cost of becoming quite involved, albeit following a clear underlying logic.

The Ultimate Go rules presented in this note aim to be simple, and at the same time solve the dame-parity issue (also known as the "free teire" or "score sharpness" issue). We take the very simple “No-pass Go with prisoner return” ruleset and add the device of ultimate prisoners, in order to solve the dame-parity issue in a way free of hypothetical play and\or a moderator. This device can be thought of as a simple and unambiguous realization of the “Taiwan Rule”.

We also give a procedure for expedited counting, that should (in most cases) allow players to automate the "counting phase" of the game.

## II. **Background and motivation**

Go is a game with very simple rules, or so goes the common wisdom. However, the various rulesets for the game of Go are not always so simple. In particular the game has two phases (moving and counting), the latter phase involving terms like ‘score’, ‘territory’ or ‘area’, and sometimes terms like ‘alive’ and ‘seki’. These terms, albeit beautiful and logical, all stem from one common underlying logic that derives from the basic notion of ‘capture’, so it feels like they should all be implicit in the ruleset, rather than explicit.

In addition simpler rulesets, which are usually of the “area scoring” flavor (e.g. Chinese rules), share the following disadvantage: in a very close game, right before the “dame filling” phase of the endgame, the result may depend on the parity of the number of dame points. This is undesirable, as it introduces a seemingly random element to a hard-fought game. A related issue is the one of free teire moves: when the number of dames is even a player can play a reinforcement move "for free", i.e. without risking even a single point (see Ikeda's book in the references).

The ruleset suggested in this note mentions none of the above terms. The only non-trivial concept is the one of capture (referred to
below as removal of “surrounded” stones from the board), and all other terms are implicit from an analysis of the ruleset, including:
score, territory, dame, seki, etc. The underlying logic, motivating the step back from the usual rulesets to these more fundamental
rules, is best explained by considering the following question: *why* should players thrive for more territory? Well, because it will
allow them to have a larger number of safe moves at the “counting phase” of the game, delaying the (almost) inevitable loss of one of
them.

The counting phase, where players only delay the inevitable, will be implicit from the game’s analysis, not explicit in the rules. In particular players may fail to agree whether this “counting phase” has begun, and yet no harm will come to the flow of play, nor to the result. In particular no “hypothetical play” phase will be necessary, even in case of such disagreement.

The main advantage of the suggested ruleset over simple rulesets of the “area scoring” flavor is that it manages to avoid the “dame parity” issue, and it does that with only a slight complication of the rules (via the device of ultimate prisoners).

## III. **Rules of Ultimate Go**

0. Ultimate Go is a game played between two players, **Black** and **White**. Each player
has an endless supply of identical **stones** of her color, and a **prisoner bowl**.

1. The **board** consists of 19 horizontal and 19 vertical **lines**, and the stones are placed on
their 19*19 **intersections**.

2. A stone on the board is said to be **surrounded** if there is no connected path along the
lines from that stone to an empty intersection which avoids all opponent stones.

3. Before the game starts the board is empty, as are the prisoner bowls. Each player takes
one stone from her opponent’s supply and places it next to her prisoner bowl. That stone is
the **ultimate prisoner** of that player. Players move alternately, Black moving first.

4. A **placement** consists of a player placing a stone from her supply on an empty intersection,
then checking which opponent stones are surrounded, and if such stones exist - removing all of them
from the board to her prisoner bowl.

5. A placement is **legal** if the placed stone is not surrounded in the resulting board position,
and furthermore that board position has not occurred in the game.

6. A **delay** consists of a player returning either one of her prisoners (i.e. a stone from her
prisoner bowl) or her ultimate prisoner to her opponent’s supply. In the latter case this delay
is called the **ultimate delay**.

7. A **move** consists of either a legal placement or a delay.

8. If a player plays her ultimate delay second (i.e. after her opponent played his ultimate delay)
then she should return her ultimate prisoner **next** to her opponent’s supply. From now on her
opponent may **declare a draw** by taking that stone from next to his supply and moving it into
his supply.

9. At a player’s turn she must do one of the following: make a move; declare a draw, in which case the game ends with a draw; resign the game, in which case her opponent wins.

Remarks:

• The above rules are the original version, however by the analysis below they are equivalent to the current (and simpler) version found in Ultimate Go. Briefly, in the current version the first ultimate delay is forced to be the first delay of the game, and the second ultimate delay is allowed only when the player has no (regular) prisoners.

• In order to give a **Komi** of K a player should surrender K stones to her opponent’s prisoner bowl
before game starts.

• In order to play with **Handicap** H Black should give White a Komi of H-1, and then White should delay
in her first H-1 moves. If the simpler version of the rules (see Ultimate Go) is used then by the "first" delay there we mean the one coming after these H-1 delays, of course.

• As usual with (Go) rulesets, players may agree on anything that they deem will leave the result unchanged, e.g. removing dead stones, expediting the "counting phase" by automatic moves (see VIII below), automatic prisoner exchange, etc. The ruleset allows a player to prolong the game (either because of misjudging the status of the game, or intentionally), but this prolongation is usually as bad as any other ruleset allows. In a close game a loser with such prolongation intentions may harm her own position instead of resigning, and the winner might be forced to capture one of her groups (and prolong the game longer than what other rulesets allow). However, this disadvantage is more than compensated by the simplicity of the rules.

• The sole purpose of adding ultimate prisoners is for the outcome of a close game not to depend on
the parity of the number of neutral intersections. This independence is true even if it’s agreed that
a draw result actually leads e.g. to a White win, of course. If we omit ultimate prisoners we call the resulting
ruleset **Fundamental Go**. This latter ruleset is actually mentioned very briefly in Sensei’s Library as
No pass Go with prisoner return (see References below), as the author discovered while writing this note.

• The above rules generally favor ‘territory scoring’ over ‘area scoring’, e.g. seki eyes don’t count as territory – because you can’t play in them safely (though in some Seki-related subtleties these rules do "agree" with area scoring, e.g. a false eye in a Seki). However filling dames is part of the game (as is “counting” for that matter). The rules go one further than Japanese ones, and dictate a 2-point “group tax” on each non-Seki living group.

• There is *no* “pass”, so when it’s Black’s turn the number of black stones “in the game” (i.e. on the
board + prisoners) equals precisely the number of white stones in the game, and when it’s White’s turn
the difference is precisely 1. This is true assuming no Komi, of course.

• Every game has a finite number of moves: the number of placements is bounded by 3^(19*19) since a board position is just a function from the set of intersections to {empty, black, white}, and the number of delays is clearly bounded by the number of placements + 2. In Fundamental Go a draw is impossible, hence on every node of the game tree either Black or White has a winning strategy. This “strategy” is just a huge book containing answers to all possible future moves by the opponent.

• Rule 5 favors “positional superko” mainly for simplicity of statement, but also since “situational superko”
would practically allow a “reverse delay” of surrendering one prisoner to the opponent, which seems undesirable.
Since delays are always allowed (even after such a reverse delay) the same situation (board position + turn)
*can* be repeated in the situational superko variant, making this variant even less desirable. For an example showing the two superko variants are *strategically* different - see Strategic difference between Superko variants.

## IV. **Analysis of Fundamental Go**

Our aim is to describe the optimal use of ultimate prisoners. When played correctly (which should be very easy for most Go players, at least in most games) the two ultimate delays will signal transitions between very different phases of the game - the game actually “ends” (in the usual sense of the word) after the first ultimate delay; what comes after that are “counting” moves; and these end with the second ultimate delay. In order to achieve that aim we first analyze the game without ultimate prisoners, i.e. Fundamental Go. Remember that in Fundamental Go in each node of the game tree one of the players has a winning strategy (in the sense of Game Theory).

At any stage of the game the current score is defined as follows: assume Black has a winning strategy. Clearly if White got enough “mid-game Komi”, i.e. Black surrendered enough stones to White’s prisoner bowl, White would have a winning strategy. As an extreme upper bound, if Black gave a mid-game Komi of K=360 White can win by simply delaying until board has 360 black stones and Black can’t move (assuming no initial Komi). Also the set of values K of mid-game Komi sufficing for a White win is closed upwards – if White can win with K extra prisoners, she can certainly win with more than K.

Now we define the **score** as follows:

score := min { K | White has a win with a mid-game Komi of K } – 0.5

If White has a winning strategy then we define similarly:

score := - min { K | Black has a win with a mid-game Komi of K } + 0.5

Clearly the score is positive when Black has a win and negative when White has a win. Its absolute value |score| is the average between the minimal mid-game Komi giving the losing player a victory, and the maximal mid-game Komi keeping the winner unchanged (this maximum might be 0).

Note that this definition is usually theoretical, but when the game nears its “end” (in the usual sense of the word) it
becomes quite practical to evaluate the score. Also note that the score is a function of the **complete situation**, defined
to include the following data: (a) whose turn is it; (b) the board position; and (c) the set of previous board positions
in the game. Note that the prisoner-difference defined as #(black prisoners) - #(white prisoners) can be deduced from (a)
and (b) (and the Komi, if any). The absolute number of prisoners is irrelevant – adding one prisoner to each player clearly
does not affect the score, since if a player had a win before the addition she can simply wait with her extra prisoner, and
use it to delay precisely after her opponent used *his* extra prisoner to delay.

We claim that no move by Black can increase the score (and no move by White can decrease it). Assume for contradiction Black
has a move that increases the score by at least 1. After giving enough mid-game Komi to the correct player that means White
had a win before Black’s move (score = -0.5), but is losing after it (score >= +0.5) – a contradiction, since a win by White
means she has a winning answer to *every* move by Black. A move is called **optimal** if it keeps the score fixed. The best way
to think of this is as follows: trying to play optimally means playing as if one doesn’t know the value (or direction) of the
Komi. The **cost** of a move is the (non-negative) amount by which it changes the score. So after a Black move the score decreases
by the move’s cost, and after a White move the score increases by the move’s cost.

Note that, somewhat surprisingly, at some point in the game a player might have *no* optimal moves (this is unlike a minimax score).
In detail, this happens when a player has no prisoners, and every legal placement puts one of her own living groups in atari.
In such a situation the player should resign the game, and the score (in favor of the winner) can then be usually computed as
0.5 plus the number of safe moves the winner can still make (this number might be 0).

## V. **Phases of a game of Fundamental Go**

Assume that the player to make a move has a one-time option to play a “reverse delay”, i.e. to surrender a prisoner to the opponent.
Clearly the cost of such a reverse delay equals the cost of a delay, and it’s introduced only because sometimes (when prisoner bowl
is empty) no delay is possible. We call this cost of a delay \ reverse delay the current **sente level** - again, this is a function
of the *complete* situation (defined above), and is purely theoretical until the endgame. This term might sound non-fitting in certain
occasions, but it should again (like the term 'optimal') be thought of as the urgency of play assuming the player doesn't know the Komi.

Now the set of optimal moves can’t be empty – to see that introduce mid-game Komi so that the player to make a move has the closest win possible (i.e. |score| = 0.5), and let that player make one of her winning moves (and if a delay of a mid-game Komi stone is her only option – she can reverse delay). There are three distinct scenarios:

1. **Normal play phase**: the sente level is positive (i.e. the set of optimal moves consists only of placements). It’s of course very
hard to play optimally, although after a strong sente move (i.e. when the cost of a delay is very high), it might be easier. Of course
better players will tend to make moves with lower costs.

2. **Counting phase**: the sente level equals 0, and the set of optimal moves includes moves other than the reverse delay (a delay
and\or some placements). This can happen slightly before the usual counting starts, e.g. there might be an even number of neutral points,
or perhaps two miai moves on board making territory boundaries not completely defined yet (like *a* and *b* in diagram 1).

The number of moves each player will make in
this phase (after filling the even number of dames etc., and assuming players won't place dead stones) is roughly the “territory score” of
the losing player, i.e. the number of territory points (computed with a “group tax” of two points per non-seki living shape) plus the total
number of prisoners and dead stones. However, note that if players insist on playing optimally throughout the counting phase they might be *forced* to place dead stones - in diagram 2, after Black captures at *c*, White is forced to play a dead stone (unless he reaches an agreement with Black).

3. **You–should-resign-now phase**: the sente level is 0, and the set of optimal moves includes *only* the reverse delay. That means the
player has no prisoners, and all legal placements harm her position (i.e. all living groups are completely filled except for two eyes, all
Seki situations are played out for that player, an eternal Ko was just played, etc.). That player should resign the game, and the score can
then (roughly) be computed by counting how many moves the winner can safely make.

## VI. **Optimal use of ultimate prisoners**

The above analysis applies equally well to Ultimate Go if we ignore the “declare a draw” option (clearly adding a prisoner to each
player has no effect per se). Now consider the following question: when should the *second* ultimate delay be played? Clearly a
player should do that only if it’s her unique optimal move, i.e. we’ve almost reached phase 3 above. The result will thus be either
a win for her opponent (if he has at least one optimal move), or a draw (if he has none). Therefore, the second ultimate delay (if
played optimally, which is trivial as noted) signals the end of the counting phase (phase 2 above). Note that the result of the game
is affected by the “declare a draw” option only in the following scenario: the player who played an ultimate delay first is about to
lose (if we ignore this option) by the closest possible margin (i.e. |score| = 0.5).

Given the above it’s not hard to see that the *first* ultimate delay should be played only if a delay is one of the player’s optimal
moves (i.e. when the sente level is 0), since otherwise the cost of the ultimate delay is at least 1, and the (small) advantage of being
the first to play an ultimate delay is nullified. However, on the first occasion that a delay is optimal the player usually won’t have
to play the ultimate delay immediately - there might be other optimal moves after which the sente level becomes positive, e.g. filling
one dame out of an even number of dames (which changes the sente level from 0 to 1). After all dames are filled (and all miai moves are
played, etc.) that player *will* have to play an ultimate delay, lest her opponent will do just that (and gain the advantage of being
able to turn a tight loss into a draw). At any rate, the first ultimate delay signals the transition from the normal play phase (phase
1 above) to the counting phase. As mentioned this transition can be slightly fuzzy, but if the player waits till *not* playing the ultimate
delay is a mistake then it will signal the end of dame filling and the beginning of proper counting.

## VII. **Examples**

We give two examples where the *Fundamental Go score* (defined above) is +0.5, but in the first example the game ends in a draw (because the Ultimate-Go-adjusted territories are equal), while in the second example Black wins (because she has more "territory"). Note that the concept of "territory" implicit in Ultimate Go is different than what most territory-scoring rulesets dictate, e.g. some seki false eyes *do* count as points - see Seki false eye example.

= ultimate delay

= ultimate delay

= declare a draw

White could also play her ultimate delay earlier (instead of ).

If Black attempts to deny White the opportunity to declare a draw then she lets White fill the last dame, and of course loses:

= ultimate delay

= resign

In the second example the Fundamental Go score is also +0.5, but this time we want Black to win, since she has a territorial lead of 1 point. Indeed, the ultimate delay device does *not* allow White to force a draw this time.

= ultimate delay

= ultimate delay

= resign (this is a "1-point win", since White only needs 1 extra prisoner to force a draw)

If White attempts to gain by ultimate-delaying first this doesn't help of course - she actually loses by 2 points instead of 1 (i.e. she needs more extra prisoners to force a draw).

= ultimate delay

= resign (and this time White needs 2 extra prisoners to force a draw, so it's a "2-point win" for Black).

## VIII. **Expedited counting**

We present a procedure for expedited "counting" (suitable e.g. for computer servers) which should "work" in most cases, i.e. not change the score, hence the outcome (win\loss\draw). The procedure is phrased to conform with the current (and simpler) version of the ruleset found in Ultimate Go. This procedure is "work in progress", i.e. it might need some tuning to work in unusual positions. However remember that it's **not** part of the rules, just a way to expedite the counting phase when things are clear enough (which will usually be the case).

After the first delay (which we assume was an optimal move, so there was nothing further to achieve, e.g. the number of dames is even) players should agree on dead stones. The designation of dead stones must of course satisfy the usual condition: a dead stone can only "see" opponent non-dead stones and\or friendly dead stones (two stones *see* each other if there is a path along the lines between them avoiding all other stones, in particular adjacent stones see each other).

We call a placement **safe** if the group of the placed stone has at least two liberties. We define similarly a **safe-enough** placement such that "liberties" is replaced by "extended liberties", where a dead opponent stone is counted as an extended liberty.

We call a safe placement **dame-filling** when the placed stone sees non-dead stones of both colors. We call a safe-enough placement **territory-filling** when the placed stone sees only non-dead friendly stones and\or dead opponent stones.

Now play can continue automatically along the following guidelines (we define all placed stones from now on to be non-dead, of course):

0. If the player has dame-filling placements (which are safe by definition), and in addition all previous moves of that player during the procedure were dame-filling, then the player should make one of these placements.

1. Otherwise, i.e. if one of the conditions in (0) fail: if the player has territory-filling placements (safe-enough by definition) then the player should make one of these placements.

2. Otherwise: if the player has prisoners she should delay.

3. Otherwise: if the player can declare a draw she should do so.

4. Otherwise: the player should resign, and (the absolute value of) the score should be the number of extra moves her opponent can now make according to the procedure (here "declare a draw" is counted as a move).

Remarks:

• Note that if the above procedure ends with a player resigning then the score will always be non-zero. First, if the resigning player (call her Alice) was the first to delay, then the fact she couldn't declare a draw means that her opponent (call him Bob) has at least one prisoner, therefore can at least delay once, hence |score| is at least 1. On the other hand, if Bob was the first to delay then, since Alice has no prisoners (otherwise she would have delayed instead of resigning), Bob has at least one "move" which is declaring a draw, therefore again |score| is at least 1.

• For the reason why dame-filling moves must be safe (and not just safe-enough) - see diagram 2 above. As to why a player should *not* go back to dame-filling after she makes a territory-filling move (even if possible) - take diagram 2, and change it so that both Seki eyes have two dead stones in them.

• Implementation of the above procedure (e.g. on a computer server) might look as follows: if the last move by Alice was a delay then Bob will have an option titled "propose to score". If Bob chooses this option then his timer is halted, and he gets to designate dead stones. Alice will now have three options: (i) "agree to score", in which case the procedure above is run automatically using the designated dead stones; (ii) "disagree", in which case Bob's turn resumes (including his timer, of course); (iii) "amend dead stones", in which case Alice proposes a different set of dead stones, and then Bob gets to either "agree" or "disagree".

• There exist positions where the above automatic procedure will not necessarily work, i.e. might change the score. For example, assume Alice was first to delay, and she has a one-sided dame point (alternately Bob has at least two such one-sided dames). In such a position players will need to fill all dames before applying the procedure. However, if players trust each other then Bob can make a territory-filling move (or delay), then Alice can fill her one-sided dame, and *then* players can apply the procedure. One can amend the procedure by also designating one-sided dames, in order to accomodate such positions automatically.

• The above procedure can be amended slightly to allow players to score **without knowing the dame parity**. The procedure will begin *before* the first delay (by mutual agreement, of course), and when a player reaches the territory-filling phase (i.e. she has no dame-filling moves), if she can be the *first* to delay in the game she should do so (i.e. a first delay precedes all territory-filling moves).

## IX. **Recap**

Rulesets of Go usually come in two flavors: “area scoring” and “territory scoring”. Area-scoring rulesets tend to be quite simple, although they also contain concepts which ought to be implicit rather than explicit. However, such rulesets share the “dame parity” disadvantage, which apart from posing an aesthetic problem can become a very practical problem in close games. Territory-scoring rulesets manage to dispose of the dame-parity issue (and other issues which are mainly a matter of taste, like “should a Seki eye be counted as territory?”), but at the cost of becoming quite involved, albeit following a clear underlying logic.

The Ultimate Go rules presented above aim to be simple, and at the same time solve the dame-parity issue. We take the “no-pass Go with prisoner return” ruleset, and add the device of ultimate prisoners in order to solve the dame-parity issue in a way free of a moderator and\or hypothetical play. This device can be thought of as a simple and unambiguous realization of the “Taiwan Rule”, of which the author learned while finishing this note (see Ikeda’s book in the References below).

## X. **References**

• Sensei’s Library “no pass Go” page: https://senseis.xmp.net/?NoPassGo .

• Ikeda’s book: https://gobase.org/studying/rules/ikeda/ (dame parity in section 3.5, the Taiwan Rule in section 3.6).

## Discussion:

### What is the status of these rules?

Could you perhaps add an introduction explaining whose rules these are and what their status is?
As it stands it is unclear whether these rules are your own attempt to formulate the ideal rule set or if they already have a certain standing.
The name *Ultimate Go* is perhaps also misleading, as it sounds a bit like a name for a Variant: perhaps it would be better to include *Rules* in the name of the page. Personally I also find that *Ultimate* sounds a little presumptuous, though I realise that it is hard to find a name for a rule set.

YoavYaffe: Thanks for the comments. These rules are my own attempt to formulate a ruleset which is as simple as possible, with no redundancy. My main "claim" here is that "score" is actually quite easily definable from the concept of "capture", therefore need not be explicit in the rules. I'll try to add an introduction making this clear.

While working on this ruleset I rediscovered the dame parity issue, hence the move from Fundamental Go (which is actually no-pass Go with prisoner return, as I found out) to Ultimate Go.

AFAIK the Ultimate Go ruleset I present here is novel, i.e. it is not an equivalent rewording of an existing "orthodox" ruleset. However I believe that the "strategic distance" between Ultimate Go and the common rulesets (e.g. Chinese rules or Japanese rules) is not much greater than the (small) strategic distance that does exist between these rulesets (the main difference is certainly the implied "group tax"). Therefore I agree these rules should not be considered a variant, but are suggested as an alternative "orthodox" ruleset of Go, capturing its spirit as much as present rulesets do.

### A wiki may be edited

Since Sensei’s Library is a wiki, I do not think you can reasonably expect a page like this to be reserved for you alone to edit, though it is fair to separate discussion from the main content. In particular you may find people wanting to format this more in accordance with the style elsewhere on the site and to add links where appropriate; I have indeed added a table of contents. An exception is sub-pages of your home page: you can expect these to be left to you. If you would like this page to be moved to __Yoav Yaffe/Ultimate Rules of Go__?, a Librarian can do this for you.

YoavYaffe: I did not expect this page not to be edited by other people - sorry if that was implied. Thanks for adding a table of contents, and any other changes\edits are of course welcome. I suggested that the discussion be kept apart from the main content - I find that pages where this happens are easier to comprehend.