Weird and wonderful consequences of simple rules

Since this thread is getting long enough that I can’t remember myself what I’ve covered already, I’ve now added an index at the top for easier browsing.

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No-pass go

Consider the following ruleset:

  • Black and white take turns placing stones on the board.
  • Capture works as normal, suicide is illegal. *
  • Repeating a board state is illegal (positional superko).
  • Passing is not allowed - the first player without a legal move loses.

* It’s not game-breaking to allow suicide in no-pass go, but it would have a larger effect on the game than it does in regular go. In particular, it would affect the point values we’ll calculate at the end of this post.

This ruleset is significantly simpler than regular go in these respects:

  • Only one type of action is taken each turn: putting a stone on the board.
  • No condition such as “2 consecutive passes” for ending the game is needed.
  • There is no scoring phase! No need to define territory or alive/dead stones.

Despite the lack of mention of territory in the rules, territory is an emergent goal of the game, because the player with the smaller territory will usually run out of moves first! I say usually, because the value you can expect to get from your territories in no-pass go is a little bit different from in regular go. Let’s see how it plays out in some simple examples.

Example 1: Single point eyes

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On this board, black has 4 single point eyes, and white has 3 single point eyes. Since passing is not allowed and suicide is illegal, the only moves available to both players are filling in their own eyes. Which eyes they choose to fill does not matter much. The game might proceed in this way:
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The last move is quite bad since it is self-atari, but white has no other choice as she ran out of safe moves first. Black of course captures:
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White can keep playing on for a little while longer, but by now the situation is hopeless. Eventually we’re going to reach a position where white has no legal moves available:
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There’s no need to play it out to the bitter end like this: white can resign the moment she is forced to reduce her group to one eye ** (or at any earlier point where she can foresee that this is inevitable).

** The claim here is that if white is currently out of moves, the situation will not get any better after her group is captured, since black will be able to make at least as many moves as white underneath the captured stones. This is not true for arbitrary board graphs, but I suspect it is always true on grid boards.

What we saw here is that due to black having one more point than white, black was able to force white to run out of moves first. The outcome of the game in this case is the same as it would have been with territory scoring.

Note: If both players have the exact same number of moves available, then the player to move is the one who loses.

Example 2: Group tax

How many points do black and white have in the position below?
image
In territory scoring, we would count 4 points for black and 4 points for white. In no-pass go however, what actually matters is the number of moves each player can make before being forced to reduce a group to one eye.

Counting in this way, we see that black has two points, and white has zero points. In general, for every group that needs two eyes to live, we need to subtract two points from its territory.

This so-called “group tax” is a common property of simple rulesets (including no-pass go, no-pass go with prisoner return, and stone scoring).

Example 3: A two point eye

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In this scenario, it may look like both players have the same number of moves available.
Indeed, if black plays first and simply fills in his own eyes, he is going to run out of moves first:
image

But black has a better strategy! He should take the opportunity to play inside white’s two point eye:
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White is still able to make the same number of safe moves, but black was able to sneak in an extra one, so now black wins:
image

The takeaway is that larger eyes are worth less than they look. You would rather have two single point eyes than a single two point eye!

The value of a two point eye

Is it possible to count the territories in a way where we can easily see that black is winning here, without having to play out all the moves?
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We know that the two point eye is worth less than two moves. We also know that it is worth more than one move (we saw above that if black starts by filling one of his own eyes, white is winning, but this would not have been the case if the two point eye was a one point eye instead).

Could it be that a two point eye is worth precisely one and a half moves? There are a couple of ways of motivating that this is in fact the correct value. First consider this:

No matter what happens, white will always be able to make two moves inside her own two point eye.
(don’t worry about the group tax for now - we can ignore that while counting individual eyes and subtract it from the total at the end)

Depending on whether black plays first inside the eye or not, black will make either one or zero moves.

Since both white and black playing first in the eye is gote (there is no rush for the opponent to respond), we can treat these two possibilities as equally likely. Thus the expected number of moves for black is one half.

White does not care about the total number of moves she can make, but only about how many more moves than black she can make. From her perspective, the value of the 2-point eye is:
[2 white moves] - [0.5 black moves] = [1.5 white moves]

For a more concrete approach, we can verify that 1.5 + 1.5 = 3 by analyzing this position:
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Let’s set aside the square points as eyes and only focus on the left half of the board. Black clearly has 3 moves available in the top left, and white will make 4 moves in total by filling in the bottom left.

Furthermore, no matter who plays first black will be able to make exactly one move in the bottom left.
(when one of the players plays inside one of the two point eyes, the other player will respond in the other)

Both players have 4 moves in total, so the position is balanced, and whoever moves first will lose. The outcome is the same as if white had simply had three single point eyes.


In conclusion, the no-pass endgame can be analyzed in exactly the same way as the regular endgame, using Combinatorial Game Theory. To play no-pass go well, it would be useful to have memorized the values of common small eyes, as well as the value of moving next in these areas, to be able to play the moves in the correct order.

What would a high-level game of no-pass go on a 19x19 board look like? The difference wouldn’t be too drastic in the opening stages, but as the game progressed, you might notice a tendency of the players to prefer “narrower” territories, where they can more effectively limit the number of moves the opponent can make inside in the future. Towards the end of the game, possibly even before the borders have been completely finished, you would start to see some “invasions” that are dead in gote, but give enough extra moves to be more valuable than a regular endgame play. And once the borders have been finalized, there would be a drawn-out eye-filling stage, where players make moves both in their own and in the opponents territory.

This final stage is theoretically interesting, but from a human perspective it’s a bit dull compared to the more exciting opening and middle game. Even for enthusiasts of simple rulesets such as myself, no-pass go is in some sense “too simple” - but I still find it very useful to have some understanding of it, for a couple of reasons:

  1. As we’ve seen previously, there are rare beasts such as Molasses ko and Unremovable ko that (under superko rules) can transform a regular game of go into no-pass go.
  2. There are go variants such as Gonnect and Redstone which employ no-pass rules.
  3. Even though no-pass go by itself isn’t a very good go ruleset, there is a simple “patch” to get rid of the eye-filling stage, while still keeping the rules short. This is “no-pass go with prisoner return”, which we’ll take a look at in a future post.
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This is partly what I find happens in Redstone also, when it hits the no-pass go style endgame. Sometimes a doomed invasion can be sente-enough to even remove potential eyes and chunks of eyespaces while dieing :slight_smile: That could happen before a gote endgame of hane-and-connect type move.

I wonder if it’s the same value when you can capture your own stones in Redstone. In comparison Black can get two moves in a two point eyespace belonging to White, but White will be guaranteed at least one move no matter what, or two if they play first :slight_smile:

Very interesting post overall :smiley:

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Very interesting, is there a way to implement komi? White allowed N passes?

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Yes, N passes for white is one fine way to give komi! If for some reason one really wants to stay away completely from passes, one can simulate something almost equivalent by setting up a position where white has N guaranteed moves available.

For instance, using a third color of stones, we could setup a 9x9 game with 7 komi on a 13x13 board:


I’ve made sure to keep some black stones around the white stones, so that white is definitely lost if she is forced to reduce that group to one eye. (if white is only surrounded by blue stones, it’s not obvious that black will be able to make more moves than white in the empty area after white is captured)

Of course the third color isn’t really necessary, it’s just a trick to be able to set it up on a single square board. You can also let the board be the union of a regular 19x19 board and a board like this on the side:

I say that this almost simulates N free passes, because there is one important difference between passes and these guaranteed moves: passes do not change the board state. So for instance if a Molasses ko happens in the game, white would rather have moves than passes.

Conversely, it might be possible to construct some position (perhaps involving Frozen life) where white would rather have passes than moves.

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I think the problem with no-pass go - and to an extent, strict superko which also lacks proper passes - is that it’s too different from real go (beyond the obvious territory differences).

Nearly a century ago, before formal passes were invented, games ended on verbal agreement. There were various disputes whether it’s ok to leave certain kos unresolved (for extra point) with excess threats. The invention of passes eliminated most such questions with clean logic (passes can eventually deplete finite ko threats, force ko stabilization).

Passes are also important for analysis with environmental coupons. Adding such coupon stack could change things too much unless the game already had infinite 0-point moves (passes as fully functional moves) and there was no “zugzwang” to begin with. This is essential in modern go.

Passes are also related to the (mis-?) interpretation of the ko concept (like “recreation of a position”). Since the ko rule only forbids immediate ko recapture, the question whether passes lift (simple) ko bans does not arise. After a pass, a recapture is not immediate anymore - so if the board changed or not is irrelevant. Imo this is what 1-Eye-Flaw illustrates (and how one recreation of a position may not even lead to perpetual repetition).

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If you think of it as a go variant this isn’t necessarily a bad thing.

If you want it as a competitor for the “one true modern rules of go” then yes you want it to be the same as the other most popularly used rules.

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Sure, and some people are quite fond of no-pass-go as a theoretical / experimental tool. What I meant to emphasize is that in modern go, the role of passes is deeper and goes beyond what may seem at first glance (without even mentioning their central role in Japanese L/D).

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Here is a silly position I wasted my evening constructing:

image
Click here for interactive board

Black to play. How will the game continue if both players try to optimize their score?

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My thoughts
  • White has 24 ko threats in the huge bamboo jungle seki
  • Black has 23 liberties in his group on the northeast
  • Black has 4 ko threats at S15, P19, P11, O11.
  • However, each of these Black ko threats wastes one or two liberties, so Black might not want to play them. Instead, Black can play moves in the bamboo jungle to remove ko threats for White.

Black of course starts by playing atari at S17. White connects at R18. Black takes the ko at T18.

Repeat:

  • White makes a ko threat in the bamboo jungle
  • Black answers the threat
  • White takes the ko
  • Black removes one threat in the jungle
  • White removes a liberty in the northeast
  • Black takes the ko.

It seems that Black will win the ko easily, with about 11 liberties remaining?

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I don’t think black can voluntarily connect any of the bamboos/groups to each other in that bamboo jungle, because of a shortage of liberties.

I think it has this weird property that when you connect one of them, white can threaten to push through all of them one after the other and eventually there’s a shortage of liberties with some atari.

So I think they’re all unremovable ko threats for white is the idea.

(I could make this a hide details response, but I feel like having something like that somewhat clearly presented only helps people to attempt it anyway)

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I feel a bit ashamed because I’m currently playing a 13x13 correspondence game in which I have exactly such a shape with two bamboo joints in a row and winning a semeai by one liberty, and I’ve spent the whole game reminding myself at every move “Do not connect. Do not connect. Do not connect.”

This game is still ongoing and so far I have successfully not connected my bamboo joints and I’m still winning the semeai, and now @antonTobi shows us this problem and the first thing I say is “Black can connect the bamboos!”

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The only reason I remember it is because it came up before in other places like earlier

and explained in this one Weird and wonderful consequences of simple rules - #15 by antonTobi

otherwise I’d probably not realise it, only that it was discussed :slight_smile:

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Reasoning
  • I miscounted, White has only 23 ko threats in the bamboo jungle, not 24. The 24th ko threat would be suicide.
  • Black has 23 outside liberties.

The sequence is now:

Black atari S17, White connects R18, Black takes the ko.

Repeat:

  • White makes a ko threat in the jungle.
  • Black answers the threat.
  • White takes the ko.
  • Black wastes a move by playing either at P11 or by throwing a stone in White’s territory.
  • White takes a liberty.
  • Black takes the ko.

This goes on for 22 cycles until Black has only 1 outside liberty left, plus one liberty in the ko, and White has 1 ko threat left. White makes the ko threat, Black answers, White takes the ko and puts Black in atari.

Black can now choose to either waste a move, in which case Black’s group in the northeast dies and the seki remains a seki; or play an auto-atari threat in the bamboo jungle.

If Black plays the auto-atari threat, White must answer the threat, so White captures the jungle but Black lives in the Northeast.

So the two possible results are:

  • A. The jungle remains a seki and Black dies in the Northeast.
  • B. Black dies in the jungle but lives in the Northeast.

Scores:

  • A. Black 0 points; White 36 points in the Southeast plus 0 in the jungle plus 44 points in the Northeast plus 22 stones that Black wasted in White’s territory: White wins by 102 points on the board.
  • B. Black 12 points in the Northeast; White 36 points in the Southeast plus 56x2+48 points in the jungle plus 21 stones that Black wasted in White’s territory: White wins by 205 points on the board.
Final answer

Black should accept their death in the Northeast, so that White wins by only 102 points on the board.

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I was also thinking...

that as you mentioned

black would lose the ko, but I’m wondering was there supposed to be some other point to the puzzle? Like in particular the formation in the bottom right is peculiar.

So I was thinking something like

you can kind of make a nakade but it doesn’t look like it works.

I don’t know if there was supposed to be some idea like that, where white just runs out of ko threats to give Black two moves in the bottom at the right time.

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@shinuito is on the right track!

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Interesting! In that case, it looks like only the 24th black move is a ko threat, putting White in atari, which I think is not soon enough to be helpful?

Gaining one move

But Black can get a ko threat at the 23rd move if Black focuses on trying to make two eyes:
image

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Indeed, this is the solution I had in mind!

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Very nice! The best sequence works out to be identical in all the major rulesets, right?

So, the final result (in area scoring rules) is that black gets a 52 points better score (26 white intersections are now black), compared to simply passing and dying in the upper-right, yet still white wins. If the position is slightly adjusted or some reverse komi is announced so that black can win but only using this crazy sequence, it is a beautiful puzzle (that way it can be presented in the simple: “black to win” statement).

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I believe so! And with a big enough board, this could obviously be extended to give black any number of moves in a row in a local area.

Yeah, it is perhaps unsatisfying that the correct sequence gives such a lopsided result. I went through some other variations on smaller boards first, but didn’t really come up with anything I was happy with, and in the end just settled for this one, ignoring the score thing.

I invite anyone who is interested to play around with it and share what other variations you come up with. The basic formula of a ko with N approach moves + N unremovable ko threats is easy enough to adjust to different N, so then you just need some position where black can achieve something with N moves in a row (and here there is more room for creativity - it could be that the order of moves matters due to captures, or it could be that multiple sets of moves “work” but one of them is best for points).

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