tapir: General idea is, that you start a ko only when the moves in the ko are worth more than the average move elsewhere (ambient temperature).
The whole timing question is crucial but there isn't much help from pages here on SL. Especially for non-direct kos.
Example approach ko
24 pt. one-sided 1-move-approach ko - local tally is 4, average value is 6. Is it really the right time to start the ko when the ambient temperature hits 6? Average value is just average value, what is the value of the effectively first move you play (the approach move)? Compare with the situation after playing the approach move.
24 pt. direct ko, local tally is 3, average value is 8. Oh, wait the average value is 8, everything is still at stake, what was then the value of the winning the ko once? Turning it into a direct ko, pretty small. (Local count is 6 before, 8 after the approach move, the gain by the approach move is only 2.)
2+8+8 still makes 18 and an average gain of 6, but the unfortunate fact is the first (approach) move in an approach ko is an underperformer it gains less, while the later moves after the approach move is played are "hotter". So when do I start playing it? When average move value is 6, or 2, or between?
(What does losing the direct ko after playing an approach move means? Exchanging a 2 pt. move for an 8 pt. move? How can I omit this? How big have my threats in an approach ko to be?)
60 pt. ko - direct ko, local tally 3, average value 20, practically (one average value)+20+20
60 pt. 1 move approach ko, local tally 4, average value 15, practically 5+20+20
60 pt. 2 move approach ko, local tally 5, average value 12, practically 3+5+20+20
60 pt. 3 move approach ko, local tally 6, average value 10, practically 2+3+5+20+20
In other words, the approach moves are incredibly small in comparison to the ko.
Assuming ko mastery, we can disregard this altogether and just calculate average values instead of differentiated values. However, say in the 1 move approach ko, there are in a way two kos one fought for a 5 point gain, another one for the whole thing.
Now, isn't it likely that the threat situation is different for small threats and large threats? Now, when is it possible to fight an approach ko? Ideally, being ko master for both, but does this EVER happen in a real game? What if not, what if A is ko master for large, B for small kos? Does A then have to give away large threats for small ones, either way having to concede a compensation beyond mere ambient temperature for the final direct ko, because he had to use the large threats to win the fight for the approach move?
About timing: When you play the ko when the average value is right then the approaching side is actually losing points while playing approach moves countered by ordinary plays elsewhere according to CGT logic. (Hoping to gain them back when winning the ko later - hopefully you counted your threats well enough.) - Well, in a way it is kind of obvious that you better should win a ko after you played an approach move and should only play an approach move if you are assured of winning the direct ko thereafter (or resign if not).
Dieter: is this particular to approach kos? Take a simple endgame move, which is sente for one player but gote for the other. The tally is 1. The value of the sente itself is x. The value of the follow-up, if the sente were not answered is y>x. The sente player plays the move. However, the other player decides to go for mutual damage because there are moves around of value about y/2. Suddenly, the sente player is confronted with a move worth y, which is bigger than the original x, due to the opponent ignoring the sente. This sounds quite similar to what happens in an approach ko. The approach player plays the approach, the opponent plays a kothreat, the approacher ignores it and plays the direct ko, whose value is larger than the approach ko was, but this is compensated for by the opponent playing a kothreat, executing it and then have another free move.
tapir: Well, as is we have no clear answer to the rather simple case of ko threats to be played in an approach ko although we have tons of complicated terms. Say, in the simple approach ko case (1 approach move, 60pt. value of the whole ko) do you play threats in the initial fight about the approach move according to the average value (15) or according to the mathematical gain at stake in the approach move (5) or according to the difference between losing the whole ko now to the direct ko situation (10 = 20/2)?
My intuition would be that you can play far smaller threats during the "fight about the approach move" phase and this could boil down to a clear heuristic "do not waste ko threats sufficient for the direct ko phase in the approach ko phase, but play smaller ones". (In practice at my level I would probably gain a few points out of all those approach kos, by losing them, simply because the opponent ignores my surprisingly small threats for a 60pt. group at stake.) Now, if we just calculate the approacher winning the ko and the other player playing three moves elsewhere this simplifies things, but this is of little help as guidance for actual play (what is a sufficient threat now?) once you sit down and do fight a ko.
Instead of any practical advice we have a term such as "hyperactivity" ("A hyperactive position is one whose count depends on who is komaster.") when in practice there often is no komaster, but some form of compensation beyond gote moves elsewhere. Now, what is the count when there is no komaster? Shouldn't the surplus compensation (above gote moves elsewhere) be accounted for in the count? It makes perfect sense for Berlekamp to avoid needless complications for his model, but his terms are treated like advice to actual play (up to the term komaster in beginner exercises) resulting in endless confusion. Komaster in particular is often used as a term for the side who wins a ko, although the term is supposed to be something altogether different (see the enlightening discussion in the komaster forum). It should be made very clear all over the place, that a komaster, who can answer all primary and secondary ko threats, but has no tertiary ko threats himself, is a rare beast in actual gameplay (especially rare in early and midgame kos), even if useful for CGT purposes to get a grip on some ko fights.
Dieter: I fully agree with you that the taxonomy of ko on SL has exploded and is more of interest to theorists analyzing the game than it is of practical use to the average experienced player. In practice, I always play my threats in order of size, for an approach ko I will either postpone fighting it until I'm clearly thicker overall, or I'll fight it right away if the present aji and thickness are in my favour, ... Actually I'm quite happy already if I have an idea of the ko threat balance before I rush into one and if I can decide if the exchange favors me, when ignoring or capitalizing on a threat. In short, getting the basics of ko fighting right, will be sufficient for us. I believe ko master and ambient temperature are good terminology to turn intuitive concepts into food for debate, but they don't really give better heuristics.