The perfect game(s)

[emeraldemon]

In another thread, Laman said

for Kami no Itte: there is only one move, no style-choice left

This got me thinking. Let's say by "Kami no Itte" we mean perfect play in the game theory sense: every move gives the best possible outcome for that player. Sometimes there is only 1 move that gives the best outcome, and every other move is worse. But sometimes there are multiple moves that give the same result: in the endgame for example.

So what if both players play perfectly from the beginning? Is there a unique "perfect game"? My guess is probably not. At the very least, I again imagine that towards the endgame there will be miai options, creating several optimal solutions (although what if the perfect game ended in one of those Berlekamp-Wolfe monsters where each of the last 30 moves had to be played in the exact right place, or risk losing a point?)

Even before the endgame, it seems very possible to me that there could be points where a player has two or more options, each of which leads to the same score with perfect play. There could in fact be thousands of perfect games, and it would still be impossible to stumble on one by chance.

We will probably never know exactly how exactly how many perfect games of go there are. There's something intuitively appealing about the idea of a single perfect game: a single razor-thin path where not a single step can be out of place. But there's also something pleasing to me about the possibility of many perfect games: maybe there's some calm and peaceful games, some intense battle games with huge sacrifices. Maybe some start 34 and some 44. Each game would have its own unique gems and brilliant tesuji. Perhaps each game might have its own "style".

As I said, we'll probably never know even one perfect game, so it's all idle speculation for now. But it's still interesting to me to think about.

(Bonus question for the rules lovers: would the perfect game depend on the ruleset?)

[Shaddy]

@The rules question:

Sekis work differently in Chinese and Japanese rules. The best way to play in certain corner LnD situations changes depending on the ruleset.

[Laman]

haha, wouldn't expect my tiny remark to seed another thread

this makes me think, what is exactly perfect play in go? the move whose game-subtree has most leaves marked as 'i win'? or the move after which i get the best possible winning / losing margin even against opponent's perfect play?
(sorry, it's 01:06 here, i am not as sharp as i will be 9 hours from now)

but you are probably right anyway. if i imagine a solved game like gomoku or draughts, its perfect play is not a single path but rather a tree with possible defence to every opponent's move - at least i haven't seen "this is the perfect game of draughts", so i suppose go will be similar

[jts]

I wonder if go is like tic-tac-toe... a triple ko lurking on every branch!

[daniel_the_smith]

...
this makes me think, what is exactly perfect play in go? the move whose game-subtree has most leaves marked as 'i win'? or the move after which i get the best possible winning / losing margin even against opponent's perfect play?...

The move whose sub-tree gives your opponent the fewest winning (or lowest scoring if it's won/lost at this point) end positions.

[jokkebk]

this makes me think, what is exactly perfect play in go?

I would assume the one where "worst case" sub-branch has the highest score for you. Assuming there are no triple ko -type killers lurking, any other interpretation would require opponent to play imperfectly, so that move could be considered a "trick move" (i.e. it will not give the best result if opponent plays perfectly).

And once one perfect game is known for a given board size and ruleset, the komi can be adjusted accordingly, and perfect play is just a move which results in a tie against the opponent.

[hyperpape]

You can define perfect play in many different ways. Any strategy that forces a win is the simplest way, and what I learned in my baby course on game theory. But you can also consider maximizing the score against an opponent who is as good as possible at minimizing your score.

You can even define perfect play as the most elegant play that forces a win, so long as moves can be ranked by elegance.

[kivi]

Let's say whenever two gods play with 6 points komi, it leads to jigo.
So when it is 6.5 points komi, some of the mentioned definitions for perfect play implies that there is no perfect play for black, as he cannot win against perfectly playing white. Other definitions imply that black should play as if komi is 6, and lose by half point.

With this I have a problem, not because I try to reach perfect play, but because it has implications for even at our level. Let's say there is some endgame moves left, and it is few enough so that you are able to read everything. You conclude perfect play from both parties lead to half a point loss for you. You follow the perfect play path as if komi is 6 points, which would have lead to jigo (or 5.5 komi, you win by 0.5: that is the same path). But the perfect path happens to be so straight forward. E.g., when your moves are sente white's responses are obvious. When white gets sente, there is always only one big move on the board. He never gets to make a non-trivial choice.

I am not saying you should make ridiculous overplays, trying to kill the unkillable or invade the non-invadable. But at least try to give white a position where he needs to choose between two moves one worth only so little more than the other. Mix the order of moves to get one extra point. Do something. Your plan may fail against god, and you lose by more points, but at least you tried to win. It is so unimaginative to play like a god and certainly lose by a small margin.

[daniel_the_smith]

Let's say whenever two gods play with 6 points komi, it leads to jigo.
So when it is 6.5 points komi, some of the mentioned definitions for perfect play implies that there is no perfect play for black, as he cannot win against perfectly playing white. Other definitions imply that black should play as if komi is 6, and lose by half point.

With this I have a problem, not because I try to reach perfect play, but because it has implications for even at our level. Let's say there is some endgame moves left, and it is few enough so that you are able to read everything. You conclude perfect play from both parties lead to half a point loss for you. You follow the perfect play path as if komi is 6 points, which would have lead to jigo (or 5.5 komi, you win by 0.5: that is the same path). But the perfect path happens to be so straight forward. E.g., when your moves are sente white's responses are obvious. When white gets sente, there is always only one big move on the board. He never gets to make a non-trivial choice.

I am not saying you should make ridiculous overplays, trying to kill the unkillable or invade the non-invadable. But at least try to give white a position where he needs to choose between two moves one worth only so little more than the other. Mix the order of moves to get one extra point. Do something. Your plan may fail against god, and you lose by more points, but at least you tried to win. It is so unimaginative to play like a god and certainly lose by a small margin.

In real games, everyone has to be magicwand sometimes. Making an inferior, but harder to read move is often a good strategy against non-deity level players.

[Toge]

this makes me think, what is exactly perfect play in go? the move whose game-subtree has most leaves marked as 'i win'? or the move after which i get the best possible winning / losing margin even against opponent's perfect play?
(sorry, it's 01:06 here, i am not as sharp as i will be 9 hours from now)

- First one is heuristical method of evaluation, the second one assumes that the game is already solved. If I remember correctly, 6x7 board is the largest solved board. Sensei's Library reports 6x6 ends with black winning by 4 points with perfect play.