Gote is worth half: or, an introduction to miai counting

When people start learning about endgame values in Go, they are usually first presented with positions that are gote for both sides and told how to calculate the value of playing there first. Then they see positions that are sente for one side and are told that they are worth double. Unfortunately this order of presentation obscures the justification for the doubling and doesn’t generalize well to other situations like ko. Here’s my attempt to present the math in a logical way.

An aside: when I write posts like these that present the reasoning that makes things the most clear for me, some people’s reaction is “Oh, I get it now!”, and some people’s reaction is “I’m even more confused than before”. If you’re in the second group, don’t sweat it, just read something else that makes more sense to you.

Sente

We’ll start with a sente position that we’ve used before.

Here are the very simple variations for both sides, with a twist that I’ll apply consistently throughout this post: extra stones are added elsewhere so that both sides have made the same total number of moves on the whole board. (Of course Black wouldn’t play in that particular location; it’s just a marker that they played somewhere else.)

I marked the extra stone here but I didn’t give it a move number, because I don’t want to give the impression that Black played it at any particular time. What’s important is that when we look at the whole board some time in the future after one of White’s moves (so that each player has played an equal number of stones during the game), Black has one more move elsewhere on the board in Diag. 1c than in Diag. 1b. That move was presumably useful! Comparing Diag. 1c to Diag. 1b, White is three points better (locally) but one move worse. If that move was worth more than three points, Black got the better end of the deal in retrospect.

As an aside, Diags. 1b and 1c aren’t two continuations from the same point in the game (a different player is going first each time!); they’re just indicating how the balance of power on the whole board will be different in a world in which Black plays first locally (at some point) compared to if White plays first locally (at some point).

Gote

In this respect gote is actually more complicated than sente, not less.

Again, I’ll place extra stones so that the whole board evens out.

Now there is a two-point difference between the two diagrams, and also a two-move difference. Comparing the second variation to the first, White has two extra points locally, but Black has one extra move elsewhere and White has one fewer. Whoever makes this local move is paying an opportunity cost of two moves elsewhere (losing their own and adding their opponent’s) in the global balance of power, not just one.

If we’re going to try to find a gote move that’s as valuable as a given sente move, that opportunity cost is twice as big, so the local benefit had better be twice as big as well to make up for it. If we have a $3$-point sente move available, a competing gote move has to be worth $6$ local points to be comparable.

The formula

Ultimately, you win a game of Go by earning more points per move played. You and your opponent each get the same number of moves per game, so the question is how efficiently you can use them.

The formula for the value of a boundary-play sequence is just $\frac{\text{point difference locally}}{\text{move difference elsewhere}}$. That’s it. Just points per move.

Each time you fail to make a move elsewhere, or give the opponent an extra one, you are paying an opportunity cost of one move’s worth of value. If you’re paying double the cost, your local play is only half as good.

(Of course this formula could say “move difference locally” rather than “move difference elsewhere”, and in fact that’s generally how it’s presented, but I like to phrase it this way because that’s how you’re actually paying: your opponent is playing useful moves on the rest of the board while you take your local profit.)

Miai and deiri

If you’ve encountered the mysterious term “miai counting”, that’s all this is. “$6$ points in gote” (in Japanese this is called deiri counting), is worth $3$ points per move. $6$ points in sente is worth $6$ points per move. So their miai values are $3$ and $6$ respectively.

Ko

Maybe you don’t care about any of this yet. Double the value of sente sequences, or halve the value of gote sequences, what’s the difference? They both mean that sente is twice as valuable as gote.

The first reason to use the miai counting scale is that it maps directly to what is actually going on (points per move). The second reason is a consequence of the first, which is that it generalizes better to other situations.

Here’s the simplest ko.

And here are the continuations, with extra evening-out stones.

This time there’s a three-move difference on the rest of the board! So the value of a move in the ko is $\frac13$. Perhaps you have previously encountered this value and found it arbitrary; now you know exactly where it comes from.

Let’s make a table if you’re not convinced yet that miai counting is the way to go:

Bigger kos

I threw in a two-stage ko at the end of that table. By now I’m sure you can work it out yourself, but here it is for completeness:

Yep, a $4$-move difference, so we divide by $4$. The local point difference is $4$ (Black could get $3$ points or White could get $1$), so each move in this ko is worth $1$ point.

Summing up

You probably learned that “sente is worth double”, at least at first, but the more proper phrasing is “gote is worth half”. My hope is that thinking in these terms will clarify the reasons for the relationship and allow you to generalize to more complicated situations more easily. Now go win some games by half a point.