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
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:
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:
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:
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?
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
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:
But black has a better strategy! He should take the opportunity to play inside white’s two point eye:
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:
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?
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:
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:
- 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.
- There are go variants such as Gonnect and Redstone which employ no-pass rules.
- 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.