# Ricky Demer's analysis regarding the monkey jump's value

I find myself skeptical of the claimed values for the monkey jump being correct when

- if White prevents it, a subsequent hane-connect by White would be gote

and

- Black barely has ko threats and Black will not get any bigger ko threat

.

# Text to be discussed

I added the circles on the left. The next two diagrams and notes are otherwise taken exactly from the section showing the monkey jump in the article *Miai Values List / 2.00 and more*.

### 4.42 (4 5/12)

# Analysis

## White prevents the monkey jump

After White prevents the monkey jump, the remaining play is just the 1-point hane-connect. Thus, the resulting temperature is strictly less than 2, so the players are assumed equally likely to get it.

In other words, preventing the monkey jump reaches this count in gote.

## Simplifying the position

Putting black stones at the s here only stops their intersections from being black territory and gives Black up to three extra removable ko threats. Aside from the obvious shift by 2, that does not affect the count.

Aside from the same shift by 2, making the s **here** Black can’t make things any worse for Black. Thus, making them Black in both cases can only **increase** the difference between this diagram and the previous diagram.

## White’s response to the monkey jump

### Black does not block the descent

If is tenuki, then I consider White immediately playing and White protecting by connecting at . I assume that despite protecting that way, the area to the right of will be solid white territory. Under that assumption, at the cost of one net move, White has done locally two points better than having prevented the monkey jump.

If Black continues locally anywhere other than , then I consider White responding immediately with . I assume that the area to the right of is solid white territory (even if is in there). Under that assumption, at the cost of **zero** net moves, White has done locally at least one point better than having prevented the monkey jump.

### Black blocks the descent (White throws in)

The remaining case is Black continuing with blocking the descent. In that case, I consider White responding immediately with .

### Black blocks but does not capture

If is tenuki then I consider White immediately playing and White protecting and by connecting at . I assume that, despite protecting and that way, the area to the right of and is solid white territory. Under that assumption, at the cost of one net move, White has done locally two points better than having prevented the monkey jump.

If Black continues locally anywhere other than at , then I consider White immediately connecting with . I assume that the area to the right of is solid white territory (even if is in there). Under that assumption, at the cost of **zero** net moves, White has done locally at most 1 point worse than having prevented the monkey jump. (Recall that the value of preventing the monkey jump was supposed to be over 4.)

### Black blocks and captures the throw-in

The remaining case is Black continuing by capturing . In that case, I consider White responding immediately with and then White being willing to let the temperature drop. Note that

- ``6-"(supposed_value_of_preventing_monkey_jump)" = 6-4 5/12 = 1 7/12`` .

Suppose that with temperature strictly above ``1 7/12``, Black plays the next local move, and that that move is somewhere **other than** the five marked points. In that case, I consider White responding immediately at *w*. I assume that that will make the s and the area to the right of them solid white territory (even if is in there). Under that supposition and assumption, a white play at *b* would get **at least** ``4`` points more than the territory I just mentioned, so Black can’t do better than as in the diagram and playing *b* at temperature strictly above ``1 7/12``. That would leave White **locally** at most ``6`` points worse than preventing the monkey jump. However, ``6`` points is strictly less than the temperature at which Black got *b* plus the supposed value of preventing the monkey jump. (I.e., it would not be worth White having spent ``4 5/12`` to prevent this.)

Suppose that with temperature strictly above ``1 7/12``, Black plays the next local move, and that that move is filling the ko. In that case, I consider White responding immediately with . I assume that that will make the s and the area to the right of them solid white territory. Under that supposition and assumption, *b* is **at least** a 3-point play, so the “*so Black ... prevent this*” from the previous diagram’s analysis applies here too.

### Black blocks and captures, White recaptures

Note that ``2*2 17/24 = 1+4 5/12 = 1+"(supposed_value_of_preventing_monkey_jump)"`` .

I assume that Black is **not** komonstrous. If Black lets the temperature drop below ``2 17/24``, then I consider White taking the ko until Black ignores that or does something locally other than just retaking.

Suppose that with temperature strictly below ``2 17/24``, Black ignores . In that case, I consider White immediately playing and White protecting by connecting at . I assume that that will make the area to the right of and solid white territory. Under that supposition and assumption, at the cost of ``2`` net moves at temperature strictly less than ``2 17/24``, White has done locally ``1`` point better than having prevented the monkey jump.

Suppose that with temperature strictly below ``2 17/24``, Black responds locally to . In that case, I consider White immediately filling at . I assume that that will make the area to the right of and solid white territory (even if is in there). Under that supposition and assumption, at the cost of one net move at temperature less than ``3``, White has done locally at most ``1`` point worse than having prevented the monkey jump. (Recall that the value of preventing the monkey jump was supposed to be over ``4``.)

The remaining case is Black playing a while the temperature is at least ``2 17/24``. (I am aware that Black playing the lower-right would let White crush the ko. However, I would need this analysis for the other three s anyway.) In that case, I consider White responding immediately by taking ko with . For this case only, I assume Black’s external ko threats are all small.

(The following analyses apply regardless of which of the four s is at. I simply picked the one that does not give White local threats.)

Suppose that Black responds locally to . In that case, I consider White in turn responding immediately by filling the ko. I assume that the area to the right of and will then be solid white territory (even if is in there). Under that supposition and assumption, at the cost of **zero** net moves, White will have done locally at most ``1`` point worse than having prevented the monkey jump. (Recall that the value of preventing the monkey jump was supposed to be over ``4``.)

Suppose that Black instead plays a small external ko threat. In that case, I consider White ignoring it and White immediately playing and White protecting by connecting at . I assume that the area to the right of and will be solid white territory. If Black does not now carry out Black’s ko threat, then there was no point in Black using the ko threat, so I further assume that Black now does carry out the ko threat. Under that supposition and those assumptions, Black spent one net move at temperature at least ``2 17/24`` to carry out Black’s ko threat, in return for White being **locally** ``2`` points better than having prevented the monkey jump.

# Conclusion

Compared to having prevented the monkey jump, White’s overall score from preventing the monkey jump at temperature ``4 5/12`` is ``-4 5/12``. On the other hand, White’s overall score from the above diagram will be ``T-S+2``. Recall that if Black lets the temperature drop below ``2 17/24``, then White scores better than preventing the monkey jump by starting the ko **before** Black enlarges it. If ``T >= 2 17/24``, then the only way one can have ``T-S+2 <= -4 5/12`` is if ``9 1/8 <= S`` .

Since ``2* 4 5/12 < 9 1/8`` , it follows that Black having extremely small secondary ko threats is not sufficient for it to be worth White preventing the monkey jump at temperature ``4 5/12``, assuming Black is not komonstrous.

# Subtleties excluded

I phrase the above conclusion carefully, to avoid subtleties such as the following diagram, which is based on https://www.littlegolem.net/jsp/game/game.jsp?gid=1833985&nmove=113 .

What is the miai value of ? (Note that the diagram is **not** hyperactive.)

## Discussion of Subtleties

Bill: When Black is komaster White to play gains ⅔ pt. by playing in reverse sente. The miai value is thus ⅔. When White is komaster Black throws in at between ambient temperature 1⅔ and 1 to force White to take and win the ko while Black takes profit elsewhere. The count remains the same at -2⅓, but the local temperature (miai value) depends on who is komaster. I dubbed such ko positions as active.

RickyDemer: No. When White is komaster Black throws in at between ambient temperature 1⅔ and 2+(1/9), which is too high for White to find it worthwhile to take and win the ko while Black takes profit elsewhere.

Bill: But when Black throws in, White gets an extra point for winning the ko. That is what raises the local temperature by 1 pt. to 1⅔.

At temperature 2 suppose Black throws in. White takes. Black plays elsewhere, gaining 2 points, and so does White. Then Black takes the ko back. If we are using Berlekamp’s original komaster rule, White simply fills the ko, leaving a ⅓ pt. ko. The net gain of all of these plays is zero. The fact that Black can play with sente above temperature 1⅔ does not raise the local temperature (miai value). At the local temperature the players are indifferent who plays the ko.

RickyDemer: It raises the temperature **by** 1 point, not **to** 1 point. At temperature below 1⅔ (in particular, at temperature between 1⅔ and 1), a White komaster limits Black to one play elsewhere in return for making the local outcome -4, which is better for White than an overall outcome of -2⅓. (Thus, Black should throw in at higher temperature, rather than waiting until the temperature is between 1⅔ and 1.)

Bill: You are arguing against something that I never claimed. The White komaster thermograph is a crooked line, rising vertically from a count of -3 at temperature 0 until temperature 1, when it makes a 45° turn to the left, until it reaches a count of -2⅓ at temperature 1⅔, at which point it rises vertically to infinity. The base of the mast at a count of -2⅓ indicates the local temperature, which is 1⅔. That’s what you asked about to start with.

(Later: I see that originally I did not explicitly state that the miai value when White is komaster is 1⅔. Sorry if that was not clear.)

# Author’s declaration

*In case any of this ends up being cited for whatever reason, I’m mentioning that my legal name is “Eric Demer”.*