# 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

.

I added the circles on the left. The next two diagrams and notes are otherwise taken exactly from the miai values list page.

### 4.42 (4 5/12)

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.

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.

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.

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

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.)

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.

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.

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 is at least 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.

I phrase that 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 .

(In case any of this ends up being cited for whatever reason, I'm mentioning that my legal name is "Eric Demer".)