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Value Of A Monkey Jump
  Difficulty: Advanced   Keywords: EndGame

[Diagram]
Diag.: Monkey jump.

In Monkey Jump, unkx80 wonders what the value of this sequence is. I will try to answer this question.


[Diagram]
Diag.: Zero-diagram

To know the value of the moves, we will have to look at the difference in outcome between various results. To make that easier, we will define some absolute values for the outcome. To do so, I have taken the result to the left arbitrarily as my zero-point - any result better than this for black is given a positive value, and result worse is given a negative value.


[Diagram]
Diag.: White plays first.

We will have to compare the result with the result we would get if white played first. For this result, it is important to know whether the hane of white 3 is sente. If it is, we may assume that white plays it at some time in the future, and the outcome is -6 (white has the 4 marked points more than in the zero diagram, while black has played 2 moves in what is there considered his territory). If white 3 is not sente, white 3 (now giving a value of -5) and black 5 (giving a value of -3) must be regarded equally likely, so the outcome is the average of the 2, being -4.


[Diagram]
Diag.: black plays after the monkey jump

Likewise, after the monkey jump, we have to look at the results when black plays next and when white plays next. If black plays next, he will connect at 1 here. White's territory is 1 smaller than in the 'zero diagram', while black's 0.5 larger (depending on who gets to play at A), so this is worth +1.5.


[Diagram]
Diag.: white plays after the monkey jump

More complicated is the situation when white captures at 1. If black answers at A, black gets one point for the marked point, one for the marked white stone, -1 for the captured black stone, -0.5 for white's possibility to get a point of territory there, and +1/6 for black's possibility to capture a stone in ko. Adding all up, the score is +2/3.

If white next plays at A, we can regard this as sente if the hane at 3 in the 'white plays first' diagram was sente (black will lose quite a bit if he allows white to capture once more), leading to a result of -2. If it is not sente, the value is again the average between the result if black plays next (-2) and the result if white plays next (-2 2/3), being -2 1/3.

So what is now the result after white's capture? If the hane were not sente, it is simply the average between black moving next and white moving next, which is -5/6. If it is sente, we will assume that the value is -2, that is, we assume that white's original capture is gote.

Comment: W 1 is sente in either case. So assume W 1 - B a.
(Suppose that the value after W 1 is -5/6. Then B a is worth 1 1/2, moving from -5/6 to + 2/3. But if W 1 is gote, the value before W 1 is the average of 1 1/2 and -5/6, or 1/3, and W 1 is worth 1 1/6, less than B a. Impossible if W 1 is gote, so W 1 is sente.) -- BillSpight

Adding up everything, the value after the monkey jump is +1/3 if the hane is not sente, and -1/4 if the hane is sente. As the values after a white move are -4 and -6 respectively, the outcome is that the monkey jump is worth a bit over 4 points if the hane is not sente, and slightly under 6 if it is. In both cases in sente.

Andre Engels waiting for people to correct his calculation errors...

Bernhard Herwig? As Bill pointed out, whites capture is sente. So it becomes whites "right" to play that move. Andre already calculated the score of the position after white 1 and black a: +2/3. This is now the expected position after whites monkey jump. So the score after the monkey jump is 2/3. Compare this to the value -4 or -6 if white plays the reverse sente. You get: The value of the monkey jump is 4 2/3 or 6 2/3 (depending if whites Hane Tsugi is sente).

Bill: All of that assumes that the monkey jump is sente in the first place. Given the diagram, that is an unlikely assumption. Further discussion below.

unkx80: Thank You!



I think that I followed Andre's answer and agree. The one thing that I found unusual was the zero-diagram. I have not seen this approach before. I am more used to something like the one below. Is there something about Andre's choice that aids in the analysis or does it come down to person preference?

[Diagram]
Diag.: Alternative Zero-diagram

Starting point B territory versus W territory


[Diagram]
Diag.: But now the Big Question :-)

The real question that I always have is whether B should play 1 here rather than the monkey jump in cases like this? Is this better or worse and why?

--DaveSigaty



Please allow me to answer DaveSigaty using a similar way Andre Engels did, except that I shall be using DaveSigaty's notation. I'll agree that it's my own personal preference, but I guess everybody will agree that we could simplify life and do away with negative numbers! :)

[Diagram]
Diag.: White plays first.

White plays first at 1.

Case I: If the hane at 3 is sente, then black is left with 6 points and white has 10 points of territory.

Case II: Otherwise, we assume that black and white played at 3 and 5 respectively and black has 8 points and white has 10 points.


[Diagram]
Diag.: Black plays first.

Black plays first, and the sequence ends at 4. I shall assume that white gets the atari at 'a' in sente, so black will have to connect at 'b'. So black has 8 points and white has 7 points.

So what is the value of black 1?

Case I: Black's territory gained by 8 - 6 = 2 points and white's territory is reduced by 10 - 7 = 3 points. Thus the value is 2 + 3 = 5 points.

Case II: Black's territory is unchanged at 8 points, but white's territory is reduced by 10 - 7 = 3 points. The value is 3 points.

Conclusion: The value of black 1 is worth 5 points if the hane is sente, 3 points otherwise. Comparing it with the monkey jump I proposed (<6 points or >4 points depending on whether the hane is sente), the monkey jump is definitely better.

Any objections? ...

--unkx80



BillSpight:

[Diagram]
Diag.: Value of the monkey jump

The values of the monkey jump, and of the alternatives, a, b, and c, depend on surrounding conditions. Let us suppose that if W c, White's HaneTsugi is sente, and that the White wall on the third line extends indefinitely. (In that case we know that eventually the 1-space jump on the first line is worth 4 points, and can be treated as sente.)

The calculations are complicated, and I will just summarize my findings here. (No guarantee that I haven't made a mistake! In fact, since my original posting, Prof. Teigo Nakamura, 6-dan, has showed me some of my mistakes. I have revised accordingly. -- And since then, both he and I have found other mistakes. This is the second revision.) Real positions will be different, anyway. Who is komaster makes a difference, of course. I will assume neither player has ko threats, to give a neutral result.


[Diagram]
Diag.: Monkey jump (1 tenuki)

W 2 tenuki.

To evaluate the position after B 3, we play out the subsequent sente sequences, W 4 - B 5 and B 7 - W 10. (B 7 is the 4-point play that we can treat as sente.) The net score is +3 (for Black.)


[Diagram]
Diag.: Monkey jump (White replies)

B 4 tenuki.

W 1 threatens W 2. Afterwards, W 5 - B 6 is sente. The local count is -5 1/3. A play at 'a' has a miai value of 2/3.


[Diagram]
Diag.: Monkey jump (What is the count?)

If Black plays the net count is +3; if White plays it is -5 1/3. The count now is the average, -1 1/6, and the miai value of a play is 4 1/6.


[Diagram]
Diag.: Monkey jump (White plays first)

After W 1, the HaneTsugi is a 3 point sente for White. The count is -12.

The original count is the average of -12 and -1 1/6, or -6 7/12, and this monkey jump has a miai value of 5 5/12.



[Diagram]
Diag.: Value of the monkey jump(2)

In this case if White plays first, there is no sente follow-up. That means that W a originally garners 2 points less, and that reduces the miai value of the monkey jump by 1 point, to 4 5/12.


[Diagram]
Diag.: Monkey jump(2) White's response

White's response is different, however. If W 5 instead of W 1, White does not threaten W 4 right away, because there is no sizable follow-up threat.
B 4 - W 5 is sente later, however, and it comes to the same thing in the end.



The small monkey jump:

[Diagram]
Diag.: Small monkey jump

Without going into any detail, the small monkey jump is worse than the large monkey jump in this situation. I reckon the count at slightly more than -3, while the count after the large monkey jump is a little less than -1. The difference is almost 2 points.

Two points can be accounted for as the difference in White territory from intruding 1 point less. The actual difference is less than 2 points because the small monkey jump is better connected than the large monkey jump.

I was surprised to find that the difference was so large. But it seems that Black's followups to the small monkey jump are worth less than those to the large monkey jump. The reason seem to be that when Black intrudes further, the weakness at 'a' becomes more critical. OC, the large monkey jump has a corresponding weakness, but it is already very weak, and Black's extensions do not make it much weaker.

The kosumi:


[Diagram]
Diag.: Kosumi

The kosumi is worth about the same as the small monkey jump. It is very solid, and B 1 - W 2 is later sente.



The crawl:

[Diagram]
Diag.: Crawl

The crawl of B 3 gains 3 points. It takes away 2 points from White and adds one to Black.
B 1 gains 4 points, however, since a White play there takes away the 2 marked Black points in addition when White plays HaneTsugi with sente.





This is a copy of the living page "Value Of A Monkey Jump" at Sensei's Library.
(C) the Authors, published under the OpenContent License V1.0.