There are several reasonable ways of implementing a Pie rule in go. Probably the simplest and most popular options are:
A. The first player sets komi. The second player decides which color to play.
B. The first player makes the first move (i. e. places one black stone). The second player decides which color to play.
Some people think komi is inelegant, which makes option B attractive since it allows to make the game somewhat fair without having to use komi.
But of course there’s no guarantee that there is one first black move which gives a fair game with 0 komi. In fact, on the 19x19 board it seems that (according to KataGo) everything above the first line is at least slightly good for black, while everything on the first line is at least slightly good for white.
(This entire post assumes Japanese-like rules, since it’s unlikely that any 0-komi game on an odd-sized board will actually end in a draw under Chinese rules, and I like to use “draw under perfect play” as a proxy for “fair”, although they are obviously not always the same thing)
In particular, if black opens on 2-2 (probably the worst second line move?), KataGo estimates that black is ahead by between 1 and 2 points. This wasn’t that surprising to me, 2-2 is bad but not super bad, so it makes sense that it doesn’t completely erase black’s first-move advantage.
But I was surprised to see that if black opens on 1-10 (probably the best first line move?), then the white advantage is similarly small!
Comparing this with the evaluation of the empty board means that the value of K19 is roughly 5 points!
I’m using a recent KataGo network but my computer is very slow, so maybe someone else would like to check this setup with more playouts. It would also be interesting to see a KataGo game from this opening, to maybe better understand how K19 can be used.
So there might not be a balanced 1-stone pie on 19x19, but maybe we can find one on some smaller board? I started analyzing 7x7, and sort of expected KataGo to be able to play “almost perfectly” with something like 100k playouts. But since some sequences can end in close capturing races involving ko’s, it’s really hard (even for KataGo it seems) to predict the result without lots and lots of reading.
So take this with a huge grain of salt, but:
As far as I can tell, both 2-3 and 2-4 are reasonable candidates for fair pies on 7x7. Here is a demo board showing a possible drawing line for each one of them. The games are mostly calm and simple (the aforementioned complications are hiding in other variations), but this tesuji is pretty cool:
D3 seems to be the only move which maintains the draw.
It would be interesting to see someone with a faster computer do a deeper analysis of these starting moves (2-3 and 2-4 on 7x7, with Japanese rules 0 komi), and find something that my KataGo is missing