Think and Grow Old

Bill Spight · **#61**

EdLee wrote:

Hi Bill,

Bill Spight · **#62**

OK. Tryss worked out the answer.

I am going to add some discussion, and explain why these positions and plays are gote.

Click Here To Show Diagram Code: [go]$$B $$ ---------------------- $$ | a O . O . O . O . X . $$ | . O X X X X X X X X . $$ | . O 1 O . O . O . X . $$ | . O X X X X X X X X . $$ | b O 2 O . O . O . X . $$ | . O X X X X X X X X . $$ | . O 2 O 3 O . O . X . $$ | . O X X X X X X X X . $$ | c O 2 O 4 O . O . X . $$ | . O X X X X X X X X . $$ | . O 2 O 4 O 5 O . X . $$ | . O X X X X X X X X . $$ | . O 2 O 4 O 6 O . X . $$ | . O X X X X X X X X . $$ | . . . . . . . . . . .[/go]

and

are each gote, and provide the final local score, 3 pts. for Black after :b5:

and 0 pts. after :w6:

So for the average value of "c" we take the average of those two two scores, or 1½ pts. for Black, on average.

Similarly, for the average value of "b" we can take the average of the value after :b3:

(6 pts. for Black) and the value after :w4:

(1½ pts. for Black). That gives us an average value of 3¾ pts. for Black.

Likewise, for the average value of "a" we can take the average of the value after :b1:

(9 pts. for Black) and the value after :w2:

(3¾ pts. for Black). That gives us an average value of 6⅜ pts. for Black.

Having figured the values of the different positions, let's go back and figure how much each play gains, on average. As we have seen, all the values are in terms of Black territory.

:b5:

starts from a position worth 1½ pts. on average and moves to a position worth 3 pts., for an average gain of 1½ pts. :w6:

starts from a position worth 1½ pts. on average and move to a position worth 0, also for an average gain of 1½ pts. Each player's move gains the same (on average).

:b3:

starts from a position worth 3¾ pts. on average and moves to a position worth 6 pts., for an average gain of 2¼ pts. :w4:

starts from a position worth 3¾ pts. on average and moves to a position worth 1½ pts. on average, also for an average gain of 2¼ pts., as expected.

starts from a position worth 6⅜ pts. on average and moves to a position worth 9 pts., for an average gain of 2⅝ pts. :w2:

starts from a position worth 6⅜ pts. on average and moves to a position worth 3¾ pts. on average, also for an average gain of 2⅝ pts.

Note: Go players usually don't talk about these value as averages, but they are, and so to be careful I am doing so.

We say that all of the non-final positions, "a", "b", and "c" are gote, and all of the moves :b1:

to

are gote. How come?

Well, :b5:

and

are obviously gote, because they are single plays that move to a scorable position. So are :b1:

and

. What about :w2:

and

?

gains on average 2¼ pts., while :b5:

gains on average 1½ pts. This method of evaluating plays is helpful because as a rule the best gote play is one that gains the most, on average. Normally there will be a gote play elsewhere on the board that gains less than 2¼ pts. (on average) and more than 1½ pts. (on average) for Black. In that case, then, Black will usually do better to make that play than :b5:

, and so White can play :w4:

with gote. Sometimes :b5:

will be Black's best play after :w4:

, and so

will be played with sente. However, we still classify :w4:

as gote.

You may verify that :w2:

gains more than :b3:

, on average, and so we classify :w2:

as gote, too.

Note that the difference in average gains between :w2:

and

is only ⅜ of a point, so :b2:

may well be the best reply to :w2:

and then White may play :w2:

with sente. That does not alter the classification, however.

Bill Spight · **#63**

Here is another problem.

Click Here To Show Diagram Code: [go]$$B $$ ---------------------- $$ | a O . O . O O O . X . $$ | . O X X X X X X X X . $$ | . O 1 O . O O O . X . $$ | . O X X X X X X X X . $$ | b O 2 O . O O O . X . $$ | . O X X X X X X X X . $$ | . O 2 O 3 O O O . X . $$ | . O X X X X X X X X . $$ | . O 2 O 4 O O O . X . $$ | . O X X X X X X X X . $$ | . . . . . . . . . . .[/go]

Evaluate the positions, "a" and "b" and all of the plays.

jlt · **#64**

Let's try.

Bill Spight · **#65**

@ jlt

So far, so good.

Now, what is the value of "a"?

jlt · **#66**

I wrote in my previous message: the value of position a is 6.75

Bill Spight · **#67**

jlt wrote:

I wrote in my previous message: the value of position a is 6.75

But in that case :w2:

loses 0.25 pt. on average when Black replies.

jlt · **#68**

I see my mistake. Let's try again. From position "a", if it's Black's turn then he plays :b1:

and gets 10 points. If it's White's turn, then White plays the sente move :w2:

, so Black has to respond with :b3:

and gets 7 points. So the value of position "a" is 8.5 points. From this, we can deduce that the value of :b1:

is 1.5 points and that the value of :w2:

is 5 points.

Bill Spight · **#69**

jlt wrote:

I see my mistake. Let's try again. From position "a", if it's Black's turn then he plays :b1:

and gets 10 points. If it's White's turn, then White plays the sente move :w2:

, so Black has to respond with :b3:

and gets 7 points. So the value of position "a" is 8.5 points. From this, we can deduce that the value of :b1:

is 1.5 points and that the value of :w2:

is 5 points.

If the value of "a" is 8.5 pts. (on average understood) then, as you say, the value of :w2:

is 5 pts. But 5 pts. is greater than 3.5 pts., so that would mean that :w2:

is gote, not sente.

Try again.

jlt · **#70**

I try again. Suppose there is a very large number of copies of "a" on the board (the board being larger than 19x19). If White plays first, then there is a sequence of plays :w2:

... and Black gets 7 points for each copy of "a". If Black plays first, then he plays :b1:

and we are in the preceding situation, so Black gets 10 points for the first copy of "a" and 7 points for each subsequent copy. Since we assumed there is a very large number of copies of "a", it means that Black gets 7 points on average, so the value of position "a" is 7 points.

The value of :b1:

is 10-7=3 points, and the value of :w2:

is 7-3.5=3.5 points.

Bill Spight · **#71**

jlt wrote:

I try again. Suppose there is a very large number of copies of "a" on the board (the board being larger than 19x19). If White plays first, then there is a sequence of plays :w2:

... and Black gets 7 points for each copy of "a". If Black plays first, then he plays :b1:

and we are in the preceding situation, so Black gets 10 points for the first copy of "a" and 7 points for each subsequent copy. Since we assumed there is a very large number of copies of "a", it means that Black gets 7 points on average, so the value of position "a" is 7 points.

The value of :b1:

is 10-7=3 points, and the value of :w2:

is 7-3.5=3.5 points.

Bravo!

BTW, John Conway developed a method to get these results in the 1970s. He was unaware that go players had figured out how to evaluate positions at least 200 years ago.

jlt · **#72**

Let's analyze further in terms of optimal play. Denote for instance by aab a board position with two copies of "a" and one copy of "b". For instance, starting from aab, playing :b1:

means that Black plays :b1:

in one of the two copies of "a".

From ab, if White plays first then the two possible sequences are :w2:

(7 points for Black) and :w4:

(10 points for Black), so only :w2:

is optimal (7 points for Black) .

From abb, if Black plays first then either he starts with :b1:

and then

so Black gets 17 points, or he starts with :b3:

and we get 7 points for Black+position ab, so Black gets 14 points. So, only :b1:

is optimal (even though the value of :b1:

is less than the value of :b3:

).

Similarly, from ab, if Black plays first, the optimal sequence is :b3:

(14 points).

From aa, if Black plays first, the optimal sequence is :b1:

(17 points).

From aa, if White plays first, the optimal sequence is :w2:

(14 points).

From aab, if White plays first, there are two optimal sequences: :w2:

and

(17 points). Note that even though :w2:

is supposedly "sente", in the first optimal sequence, Black must not answer :w2:

with

. In other words, and if I didn't make calculation mistakes, if both players play perfectly, then answering a sente move is not necessarily optimal.

Bill Spight · **#73**

jlt wrote:

Let's analyze further in terms of optimal play. Denote for instance by aab a board position with two copies of "a" and one copy of "b". For instance, starting from aab, playing :b1:

means that Black plays :b1:

in one of the two copies of "a".

From ab, if White plays first then the two possible sequences are :w2:

(7 points for Black) and :w4:

(10 points for Black), so only :w2:

is optimal (7 points for Black) .

From abb, if Black plays first then either he starts with :b1:

and then

so Black gets 17 points, or he starts with :b3:

and we get 7 points for Black+position ab, so Black gets 14 points. So, only :b1:

is optimal (even though the value of :b1:

is less than the value of :b3:

).

Similarly, from ab, if Black plays first, the optimal sequence is :b3:

(14 points).

From aa, if Black plays first, the optimal sequence is :b1:

(17 points).

From aa, if White plays first, the optimal sequence is :w2:

(14 points).

From aab, if White plays first, there are two optimal sequences: :w2:

and

(17 points). Note that even though :w2:

is supposedly "sente", in the first optimal sequence, Black must not answer :w2:

with

. In other words, and if I didn't make calculation mistakes, if both players play perfectly, then answering a sente move is not necessarily optimal.

Interesting questions.

However, Tami asked about the definition of sente, and those questions take us far afield from that. For now, I'll post a hidden reply, but if you want to continue that discussion, please start a new thread. Thanks.

Bill Spight · **#74**

jlt solved the second problem.

Let's go over it and see what it can tell us about sente.

Click Here To Show Diagram Code: [go]$$B $$ ---------------------- $$ | a O . O . O O O . X . $$ | . O X X X X X X X X . $$ | . O 1 O . O O O . X . $$ | . O X X X X X X X X . $$ | b O 2 O . O O O . X . $$ | . O X X X X X X X X . $$ | . O 2 O 3 O O O . X . $$ | . O X X X X X X X X . $$ | . O 2 O 4 O O O . X . $$ | . O X X X X X X X X . $$ | . . . . . . . . . . .[/go]

The value of "b" is, OC, the average of the values of the position after :b3:

and the position after :w4:

, namely, 7/2 = 3½ pts. for Black.

To find the value of "a" we try the average of "b" and the position after :b1:

, which gives us 13½/2 = 6¾ pts. for Black.

Now we calculate the values of the plays. :b3:

and

each gain 3½ pts. :b1:

and

each gain 3¼ pts. Wait a second! In that case :b3:

gains more than :w2:

. When White plays :w2:

Black will reply with :b3:

. Then Black will get 7 pts., which is more than 6¾ pts. So :w2:

is sente, and the value of "a" is 7 pts. for Black.

That means that :b1:

gains only 3 pts., while :w2:

gains 3½ pts. That means that normally, when there are plays elsewhere on the board than gain more than 3 pts. but less than 3½ pts., White will be able to play :w2:

with sente. Go players say that White has the privilege of playing sente (in that case). Even if that is not the case, we still classify "a" as sente for White.

As a rule, how much the sente gains is not mentioned, and :w2:

is called a 3 pt. sente. That way of speaking has confused many players, myself included ( :oops:

), leading us to think that the sente gains points. It does not — at least, sente in this sense does not; a play made with sente might. It is the reverse sente that gains points.

So what is our operational definition of sente?

We attempt to evaluate a position as gote, finding an average value, C, for the position. Then we calculate the average value of the plays as gote, G. Then when we compare G with the average value of the reply, H, to it, we discover that C is incorrect, because H > G. The result after the reply is better for the opponent than C. Then that play is sente, as is the position.

Note that it takes more than the fact H is greater than G; there are cases where there is an alternating sequence plays, all of which have values greater than or equal to G. We continue the sequence until we get to the last play which is greater than or equal to G. If the opponent makes that last play and the result is better for the opponent than C, then that sequence of play is a sente sequence, and the position is sente.

I hope that is clear. If not, please ask questions.

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