EdLee wrote:Sorry for being dense; maybe I wasn't being clear.
Assume I'm a total beginner.
I need to start from first principles:
I don't know what the term "value of a position" means.
Example:
Does it make any sense to ask what's the value of this position ?
Yes.
EdLee wrote:And if there's a numerical answer, does it depend on B or W,
or is it a value of this local shape as a whole ?
Thanks.
$$
$$ |---------------
$$ | . X . X O . .
$$ | X X X X O . .
$$ | O O O O O . .
$$ | . . . . . . .
[go]$$
$$ |---------------
$$ | . X . X O . .
$$ | X X X X O . .
$$ | O O O O O . .
$$ | . . . . . . .[/go]
It depends upon the rules. Under modern Japanese rules Black has 2 pts. of territory. Under ancient Japanese rules, which had a group tax, the local score was 0. In no pass go the score is also 0.
As we know, teaching total beginners about dead stones (none of which are in this example) and about stopping play with territory scoring is not always easy.
Re: Think and Grow Old
Posted: Fri Jul 27, 2018 1:18 am
by jlt
@Ed, Bill, Tryss:
I am 30k in endgame theory, but here is my understanding, I hope I don't get it wrong.
The game tree is from the point of view of both players. Each node has at most two children : "black plays first" or "white plays first".
When there are two branches, the probability that black plays first is 0.5.
The value of a position is the expected value of ((number of points for black) minus (number of points for white)).
We assume perfect endgame play for both players.
When playing perfect endgame, the general principle is to start from the biggest play. If there are still 4 points gote elsewhere on the board, players don't consider position "c". When there are two (or an even number of) positions with 3 points gote (including c), all these positions are miai so, as explained by Bill in his previous post, the expected number of points for black is 1.5.
Therefore, saying that each branch from position "c" has probability 0.5 means that when there is exactly one 3 points gote position left, the probability that it's black turn is 0.5. This is not a "huge" assumption, but it's still something we have to admit.
Re: Think and Grow Old
Posted: Fri Jul 27, 2018 1:40 am
by Bill Spight
jlt wrote:@Ed, Bill, Tryss:
I am 30k in endgame theory, but here is my understanding, I hope I don't get it wrong.
The game tree is from the point of view of both players. Each node has at most two children : "black plays first" or "white plays first".
There can be more than one node per player. There can even be no node for a player.
When there are two branches, the probability that black plays first is 0.5.
You can assign a probability to a play, but it is not necessary to do so, and if you do the probability is not necessarily 0.5.
The value of a position is the expected value of ((number of points for black) minus (number of points for white)).
Sometimes.
We assume perfect endgame play for both players.
That is not necessary. Besides, perfect play depends upon the rest of the board, which is not given. Lines of play that identify the value of a position are called orthodox.
When playing perfect endgame, the general principle is to start from the biggest play.
Usually the biggest play is best.
If there are still 4 points gote elsewhere on the board, players don't consider position "c". When there are two (or an even number of) positions with 3 points gote (including c), all these positions are miai so, as explained by Bill in his previous post, the expected number of points for black is 1.5.
Two gote plays can have the same value without being miai in the strict sense. And, OC, their positions need not have the same territorial value. The values of plays and positions are different things.
Therefore, saying that each branch from position "c" has probability 0.5 means that when there is exactly one 3 points gote position left, the probability that it's black turn is 0.5. This is not a "huge" assumption, but it's still something we have to admit.
No, we don't. All we have to say is that the value of a position does not depend upon whose turn it is.
Posted: Fri Jul 27, 2018 1:58 am
by EdLee
Hi jlt,
The value of a position is the expected value of (( pts ) - ( pts )).
Thanks. This magic (very human) definition is what I didn't know and was looking for.
Who came up with this definition ?
( Who picked (B-W) and not (W-B), etc. ? )
Re:
Posted: Fri Jul 27, 2018 2:31 am
by Tryss
EdLee wrote:Hi jlt,
The value of a position is the expected value of (( pts ) - ( pts )).
Thanks. This magic (very human) definition is what I didn't know and was looking for.
Who came up with this definition ?
( Who picked (B-W) and not (W-B), etc. ? )
B-W is just the score (territory).
$$B Black win by 9
$$ ----------------------
$$ | . O X X X X X X X X .|
$$ | O O 1 O . O . O . X X|
$$ | . O X X X X X X X X .|
$$ ----------------------
[go]$$B Black win by 9
$$ ----------------------
$$ | . O X X X X X X X X .|
$$ | O O 1 O . O . O . X X|
$$ | . O X X X X X X X X .|
$$ ----------------------[/go]
$$B Black win by 6
$$ ----------------------
$$ | . O X X X X X X X X .|
$$ | O O 2 O 3 O . O . X X|
$$ | . O X X X X X X X X .|
$$ ----------------------
[go]$$B Black win by 6
$$ ----------------------
$$ | . O X X X X X X X X .|
$$ | O O 2 O 3 O . O . X X|
$$ | . O X X X X X X X X .|
$$ ----------------------[/go]
$$B Black win by 3
$$ ----------------------
$$ | . O X X X X X X X X .|
$$ | O O 2 O 4 O 5 O . X X|
$$ | . O X X X X X X X X .|
$$ ----------------------
[go]$$B Black win by 3
$$ ----------------------
$$ | . O X X X X X X X X .|
$$ | O O 2 O 4 O 5 O . X X|
$$ | . O X X X X X X X X .|
$$ ----------------------[/go]
$$B draw
$$ ----------------------
$$ | . O X X X X X X X X .|
$$ | O O 2 O 4 O 6 O . X X|
$$ | . O X X X X X X X X .|
$$ ----------------------
[go]$$B draw
$$ ----------------------
$$ | . O X X X X X X X X .|
$$ | O O 2 O 4 O 6 O . X X|
$$ | . O X X X X X X X X .|
$$ ----------------------[/go]
You can also decide to chose W-B, that doesn't really change anything (but you need to stay consistant).
Posted: Fri Jul 27, 2018 4:30 am
by EdLee
Hi Tryss,
You can also decide to chose W-B, that doesn't really change anything (but you need to stay consistant).
Thanks. Yes, that part is clear, no problem.
My problem was I knew almost nothing about formal end game theory, and when the question "what's the value of this position?" came up, everyone else who was familiar with the basics of end game theory knew the established magic convention (B-W), whereas I had no clue what the term meant. jlt was very helpful to provide the definition and other fundamentals.
Re:
Posted: Fri Jul 27, 2018 8:27 am
by Bill Spight
EdLee wrote:Hi Tryss,
You can also decide to chose W-B, that doesn't really change anything (but you need to stay consistant).
Thanks. Yes, that part is clear, no problem.
My problem was I knew almost nothing about formal end game theory, and when the question "what's the value of this position?" came up, everyone else who was familiar with the basics of end game theory knew the established magic convention (B-W), whereas I had no clue what the term meant. jlt was very helpful to provide the definition and other fundamentals.
You may have noticed that I did not assume that White's territory was negative in these notes. I did assume that the stones surrounding the corridor are alive.
Re: Think and Grow Old
Posted: Fri Jul 27, 2018 8:29 am
by Bill Spight
@ Tami
This discussion has grown quite a bit. I didn't mean to hijack your thread. If you'd like, we can move this discussion elsewhere.
Re: Think and Grow Old
Posted: Fri Jul 27, 2018 1:56 pm
by Tami
Bill Spight wrote:@ Tami
This discussion has grown quite a bit. I didn't mean to hijack your thread. If you'd like, we can move this discussion elsewhere.
I don't mind. It was my own fault for asking the question I'm just trying to make sense of your reply, mathematics not being my thing. Thank you very much for taking the time. It might take me some time to understand fully.
Re: Think and Grow Old
Posted: Fri Jul 27, 2018 2:31 pm
by Bill Spight
Tami wrote:
Bill Spight wrote:@ Tami
This discussion has grown quite a bit. I didn't mean to hijack your thread. If you'd like, we can move this discussion elsewhere.
I don't mind. It was my own fault for asking the question I'm just trying to make sense of your reply, mathematics not being my thing. Thank you very much for taking the time. It might take me some time to understand fully.
Thanks.
The math is easy: count, add, subtract, divide by 2. The first example is about gote. Next comes sente, and that's it.
Re: Think and Grow Old
Posted: Fri Jul 27, 2018 10:07 pm
by jlt
@Bill,
I am trying to clarify my mind. I understand from the above messages that
Bill has a definition of a value of a position that makes no reference to probabilities nor perfect play. For instance, in the example of position "c", the value is 1.5 because if there are two copies of "c" on the board (and no points elsewhere), then black can get 3 points no matter what white does.
Tryss has calculated the value of a position using the language of probabilities.
The two approaches are equivalent (i.e. the numerical value of a position is the same using both approaches).
Even though Bill says we don't assume perfect play, any general endgame theory supposes that both players play perfectly (in the sense that both players want to maximize their number of points). Let me give an example. Suppose black is 10 points behind, and it's black's turn. There are still 3 places on the board where players can get points, all in gote. Place A is 10 points (a clever tesuji), place B is 2 points (take a stone) and place C is 1 point (close a territory). A general endgame theory will say that black has to play A, then white plays B, then black plays C. However, if black follows the theory, he loses the game, so instead he plays B, hopes that white doesn't see the tesuji A and plays C instead, and finally black plays A: thus, black can win the game assuming imperfect play from white, even though the general endgame theory says he should have lost the game.
If one of the statements above is wrong, any explanation would be very much appreciated!
Re: Think and Grow Old
Posted: Sat Jul 28, 2018 2:05 am
by Bill Spight
I'm off to bed soon, but let me just make a quick clarification or two.
First, the evaluation of go positions and plays does not produce a general theory of the endgame. Far from it. But it is a good place to start your analysis. Doing so gives you information that will usually tell you where to play. But not always.
Second, you can use probabilities to understand this kind of evaluation. The go books do not teach that, and it is not necessary, but it is one way of looking at it. You can think of the value of a position as a kind of expected value. But you can also think of finding the average value of a position as a form of defuzzification. Historically, however, go players figured out how to evaluate positions and plays without reference to either probability or fuzzy logic.
Third, you can view the evaluation of go positions in minimax terms. I redefined go evaluation in such terms 20 years ago in my paper on the evaluation of multiple kos, which I presented at a computers and games conference in Japan. But that definition depends upon concepts that you do not need for these evaluations. That definition is algebraic. As I mentioned to Tami, all we need here is arithmetic.
In any event, we do not need to know perfect play in any sense to evaluate positions. We can use every possible line of play in a position, alternating or not. For instance, instead of playing first at the mouth of a corridor, we could start at the other end. We would then eliminate such lines of play from consideration. It makes our job easier if we know that they are wrong in the first place, but it is not necessary. We can use evaluation to find lines of play that are in some sense, correct. We do not have to assume perfect play.
Re: Think and Grow Old
Posted: Sat Jul 28, 2018 3:03 am
by Elom
EdLee wrote:...Seems the 0.5 probability is a huge assumption ? (I'm OK with it being a magic number, just to simplify this example; I'm OK if the exact probability is not very relevant at this stage of learning... )...
Maybe probabilities are assumptions by definition. With no evidence to favour the likelihood of either side having turn, no assumption tops equal distribution of chance.
Extra explanation for go beginners:
The isolated board configuration of the endgame in analysis consists of multiple variables. 18 of them are board-state variables with three possible permutations: black, white or blank. However, there is one more variable, which may be thought of as 'time', having only the two permutations, black's turn or white's turn.
Strictly speaking, endgame study might be about deriving the score of a position when the variable for time is not known. Making copies of position is common, with probabilities a shorthand, I guess.
If we know black plays first and colours alternate, corridor a is nine points for black, zero points for white. If players don't alternate, and we've no way to know who plays after black, it's still nine points (so this position is 'solved'?). If we know white plays first and colours alternate, corridor a is six points for black, zero points for white (white connects, black blocks). If players don't alternate and we've no way to know who plays after white, more calculation is necessary until the score of the position doesn't change depending upon who's turn it is (making some assumptions here).
We could know the time variable's state, and not know a board variable's state. Here:
$$
$$ |---------------
$$ | A X . X O . .
$$ | X X X X O . .
$$ | O O O O O . .
$$ | . . . . . . .
[go]$$
$$ |---------------
$$ | A X . X O . .
$$ | X X X X O . .
$$ | O O O O O . .
$$ | . . . . . . .[/go]
Let us say we know it is white to play and the colours alternate, so we'll know what state the time variable's in every configuration from here on. A Schrödinger's cat situation in which we don't know if the corner consists of a black stone or a blank space occurs. There's a 50% probability of a score of two points for black, zero points for white, and an equal chance of zero points for black, and 15 points for white. So the score of the position is one point for black, seven-and-a-half points for white.
Re: Think and Grow Old
Posted: Sat Jul 28, 2018 4:11 am
by jlt
@Bill:
I am still under the impression that using minimax is equivalent to assuming perfect play, but let's not worry too much about that minor point. I am more interested in what you have in mind about sente; so far we only had gote examples.
Posted: Sat Jul 28, 2018 10:26 pm
by EdLee
Hi Bill,
I did assume that the stones surrounding the corridor are alive.
Thanks. I didn't know what's the "value of a position";
but it seemed like everybody else did.
I was just hoping someone would give the definition, which jlt did.
I was just throwing stones down to see what was happening.
Kind of like shooting particles at a thin gold film.
All we have to say is that the value of a position does not depend upon whose turn it is.
pts ) - ( 
pts )).