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Bill: I copied everything below to the new forum pages. As JuhoP pointed out, that was not the thing to do. I had not considered the editing problems involved. With the exception of an anonymous question and my immediate response, I have deleted everything else I copied. My apologies.
I have left titles of some forum topics intact, in case people wish to move their own material there.
Meanwhile, I am making titled sections of this page, which would have been a better idea to start with.
(Later): I have been reassured by the powers that be that such problems can be fixed, if need be. So, I will not touch Juho's material. He can start a thread if he wants to. Otherwise, if no one else objects by Oct. 15, I will copy material from here to the forums.
I'm confused. Is the property of someone being Komaster internal to the ko, or is it a matter of external ko threats? It seems to me that in the first diagram Black does have to ignore White's ko threat with 7. Does this mean that Black is komaster because White has no ko threats (or none big enough)?
Bill: From an analytical point of view, one beauty of the komaster concept is that you do not have to work out exactly what it takes for a player to become komaster. The komaster has just enough ko threats to win the ko without having to ignore an opposing threat. In practice, of course, there is often a trade-off between winning the ko and allowing the opponent to complete a threat. This is called the ko exchange.
In the first diagram, if Black is komaster White has no ko threat to play at move 7. If White plays a small ko threat that Black has to ignore, Black is not komaster.
Jan: Hey Bill, when are you going to write Breakthrough to Komaster? I'm sure Ritchie Press will want to publish it?
It would be nice if someone could write down the answer to the question "How do I know who is komaster?". If I knew how to determine it I might understand it :). The above discusion really doesn't seem to clarify anything for me. It sounds an awful lot like the komaster is simply the person who has enough excess ko threats to win. There is also a lot of asigning points in the above, which seems difficult if not impossible without knowing the position on the rest of the board.
Bill: In the precise technical meaning of the term, for a player to have just exactly enough threats that are large enough so that she is komaster is against the odds. Usually one player or the other has more than enough large enough threats to win the ko without having to ignore an opposing threat, or neither player is komaster.
Simple examples can be given, however. The simplest is that of a simple one-step ko where neither player has a ko threat. The player to take the ko is komaster.
Charles It's a modelling term, anyway. The better question is 'how can I tell whether the results from the model apply to my game?'
Bill: The beauty of the komaster idea, IMO, is that for the first time it gave us a way to calculate the value of single kos in a manner similar to the way we calculate the value of other plays. For that, it's vagueness in terms of what conditions need to be met on the the go board for a player to be the komaster for any specific ko fight is a virtue, just as the vagueness for calculating the value of non-ko plays about other plays on the board is a virtue.
For non-ko plays we have a good idea of the conditions on other plays that have to be met so that we can have confidence in the theoretical results. I. e., that the largest play is the best play. Furthermore, we know that these conditions are typically met, so that playing the largest play is usually correct.
For placid kos, which have the same value regardless of who is komaster, we can be relatively confident. However, komaster calculations do not provide absolute limits, so we cannot be as confident about any ko as about non-kos.
For hyperactive positions, such as approach kos, mannen kos, and positions which threaten fights for large placid kos, the story is different. Being able to calculate their value gives some guidance, where before there was none, but little is known in general about conditions for komaster, so our level of confidence is much lower.
There are other ways, which I pioneered, of theorizing about such positions by analyzing kos and threats together in what I term a ko ensemble. However, that theory is so complex that it is not yet of any real practical use.
But is there some objective definition of a 'ko threat'? Isn't 'ko threat' just a relative term? Doesn't almost every reasonable move threaten some follow-up? So how is it possible to connect a ko without ignoring a threat? And doesn't this make 'komaster' quite a vague term?
Bill: I do not believe that there is a comprehensive formal definition of a ko threat. There are definitions for specific types of ko threats. Like sente, ko threat is a fuzzy term.
Yes, most reasonable moves have follow-ups. But that does not mean that the follow-up is a threat.
E. g., suppose that we have the following plays, as well as a simple ko:
A B / \ / \ C 0 6 0 / \ 10 6
If Black plays from A to C as a ko threat, White will surely ignore it. Either she will win the ko or respond in B.
Komaster is not a vague term. It is precise. However, as Charles points out, it belongs to a model. That model may apply to the current situation to varying degrees, or not at all. Maybe nobody is komaster.
Let me add that the explanation I gave of komaster is not its formal definition. For that, see http://www.msri.org/publications/books/Book29/files/ber.pdf. I tried to explain the term for the average go player. :-)
RobertJasiek: I am not sure whether I understand "komaster". Is this, for a particular ko,
Bill: None of the above, although they may also be the case. The idea of komaster is an abstraction that allows us to evaluate certain types of ko positions. Any particular ko threat situation may well not fit the bill exactly.
Bill:Probably the simplest example of komaster is this. There is a simple, direct (one stage) ko. Neither player has a ko threat. In that case the player who can take the ko is its komaster.
Bill:Another example: There is the same ko, but if one player takes the ko the other has a single humungous ko threat that must be answered by correct play. That player is the komaster, not the one who takes the ko.
RobertJasiek: So komaster requires perfect play, then looks who wins the ko, and then calls this person the komaster? Virtual-force works vice versa: First enable one player to recapture in the ko without making a threat (unless that player's previous move was a ko-capture elsewhere), then watch the move-sequences, and then (if this was the "something" of the aim) we will have found out whether the player could win the ko.
Santo: Robert, I think there is a lot of confusion here (I also realize this was a fairly old discussion, but as it stands it might keep confusing new visitors). I'll try to clear up a few of these, although anybody with a better understanding of CGT can correct me if I'm wrong. I think some of this should be cleared up and moved to the main article.
I see you wrote "I had a look into Berlekamp's paper, do not care much for the details of thermography [...]". The formal definition of komaster is inherently game theoretical, and it is quite impossible to separate it from the thermography. Berkelamp's komaster paper is quite readable and reasonable if you're familiar with both CGT ("Winning Ways" or "On Numbers and Games" stuff) and go (so that you're familiar with kos, which are mostly outside the scope of classical CGT). But I think that the best way to understand the concept, for a go player without a background in combinatorial game theory, is not as an absolute term, but as a *strategy*.
That is, a player "is not" a komaster in real go. In the paper, a player is a komaster by specially *modifying the rules of the game*. That's never the case in real go, where you don't have "by rules" the komaster privilege. So that's not the best way to think about it. A player may *choose* however to adopt *the komaster strategy*. The komaster strategy is: playing the ko by responding to every single one of the opponent's threats (see next paragraphs for clarification about this), then winning the ko immediately after the opponent plays its first non-threat (gote) move. Of course, it might be impossible to perform the komaster strategy, because it requires a certain minimum amount of large enough ko-threats. The whole analysis assumes that it is possible and the player has chosen to do so, which are strategical considerations that might not be perfect play in a real game, but are assumed once the players *chooses* to play the komaster strategy (for that ko).
So, what's the deal with this particular strategy? The beauty is that it is possible to put a precise numeric value on a ko, in the CGT sense, when we assume that a certain player will play it following the komaster strategy (that is what Bill means when saying "this player is the komaster": it means we are assuming that he will be carrying out the komaster strategy. It is a given in the analysis, be it perfect play or not), which was impossible otherwise, due to the ko-exchange resulting from ignoring a ko-threat.
To show why ignoring a ko-threat (which is what the komaster strategy specifically avoids) breaks the whole CGT analysis, let's state the simple main idea of CGT analysis. The idea is that we can decompose the game into independent simpler components, and assign a definite number to each one. So for example a component might have a value of +3/4, another one -1/4, and the last one +1 (let's suppose a positive game is a black advantage, and a negative is a white advantage). The count of the game is then simply +3/4 - 1/4 + 1 = 1.5 in black's favour, according to CGT. That means that for example, if it is blacks turn, then he is automatically guaranteed a win by following "the CGT strategy" (typically thermostrat. It basically boils down to the Orthodox Strategy in go of playing the next largest miai counting move, and responding to the opponent's sente moves). This nice guarantee that if it is your turn and the count favours you, playing orthodoxly you will certainly win, is a very nice and simple CGT property.
Now, in calculating the CGT values of a subgame, local sente moves are basically assumed to be immediately responded (it follows from CGT thermography). But if those sente moves are played as ko-threats and ignored during a ko fight, then this assumption is broken, and the CGT count and miai value assigned to this (non-ko) component will completely break down.
As a simple example, imagine a string of 5 stones on the side of the board, with only two liberties: one is a dame that the opponent can fill at any time, and the other is a "false eye", that will safely connect this string to the main live group, and which the opponent can't fill without capturing the string. This is a typical example of endgame teire appearing in some form in almost every game. The count, according to CGT, of this position, is exactly zero points, and the miai value is zero. This is fairly intuitive to go players: this situation is "boring" and nobody is interested in playing there, there is nothing to win. Just at the end of the game, the false-eye player will eventually defend and play the teire, even if only once the opponent fills the dame. So the local score is no points for anybody, no matter what happens. So when taking the sum to get the count for the whole board, we take this to be zero, and count 0 + c1 + c2 + c3 ... + cn = R, with R the total count.
However, our assumption that the score of that position will be 0 might be completely destroyed if a ko arises: then, a player may fill the dame as a ko-threat, and if it is not answered, then the local score will not be zero anymore, so the whole sum is altered. The theory does not work anymore.
You see in this example that CGT basically "ignores ko threats", which is a wonderful simplification that allows us to put a definite number of points into a position, and gives mathematical guarantees. If a ko fight appears, then the ko threats on the whole board must be considered, and can't be ignored anymore, to get a score around the ko. Also, the situation is not anymore "simply calculate the values and add them". That will not give an accurate count at all. This is much more complicated. Ko is hard, that's the way it is.
But in the specific case that we assume (no matter how we know that. The simplest possibility is simply that we are playing while making the analysis, and we *choose* to play this way) that a player follows the komaster strategy, then this problem does not arise. Sente moves will always be answered, never ignored to win the ko. And then it is possible to put a definite count number into the ko, and keep the original count for each of the other components, and add them together, and if it is our turn and the count is in our favour, then we will be still assured a win mathematically by following what is basically the Orthodox Strategy.
If it is not clear that any player has enough ko threats to successfully play by the komaster strategy, a ko-exchange will naturally arise. But we can at least put some bounds on the situation, by analyzing both extremes: first supposing that one player successfully carries out the komaster strategy, and then supposing that the other player does so, and finally comparing both counts. The real count in this case will naturally be somewhere in between (the fact that the komaster pays twice the temperature as a tax should be taken into account in this analysis). I believe this is what Bill means when he says for example, that "The beauty of the komaster idea, IMO, is that for the first time it gave us a way to calculate the value of single kos in a manner similar to the way we calculate the value of other plays."
RobertJasiek: Thanks! I agree that much of this should be promoted to the parent page.
What, in this respect, are "neutral threat environments"?
Bill: Komaster does not require perfect play. You can make either player komaster and do the analysis. But if you want to find out who, if anybody, is komaster, then you have to look at the ko threat situation and assume perfect play.
RobertJasiek: How, by definition, does komaster apply to superkos? Examples?
Santo: The Komaster analysis has no pretention to the analysis of long cycles like triple Ko. Those are simply out of scope, as far as CGT values are concerned. So, by definition, the concept does not apply.
Bill: I looked at the virtual-force definition on that page, and I do not understand it.
RobertJasiek: What is it that you do not understand?
Bill: The English. Give it its own page and explain more.
RobertJasiek: Let me make a start:
Bill: I suppose that I gave that impression, because I was talking to go players. By Berlekamp's original definition, however, the komaster is allowed to take the ko back immediately. The komaster must then make a second play in the ko.
Bill: If you are talking about an iterated ko, the komaster is not required to make a second ko capture (unless that is the only play in the ko), and if she does she is not required to make a third, or to make another play in the ko.
Bill: My statement that the komaster has just enough threats is an attempt to explain the concept to go players. It is not part of the mathematical definition. If virtual-force allows a player to gain by delaying the win of the ko, then that player is komonster.
Bill: Komaster says nothing about virtual threats. Don't confuse what I said to explain the concept to go players with the mathematical definition.
RobertJasiek: Are komaster and virtual-force just different models or can we say that the latter is the easier model? I think that also virtual-force is a tool to enable us assessing the value of a ko: We get the outcomes of a ko if either player moving first may virtual-force and we can count how often a player took advantage of virtual-force by recapturing immediately. This suffices for determining the ko's value (although so far only for basic kos).
Bill: I think that we can say that komaster is the simpler model. :)
RobertJasiek: What about writing the formal definition here? Maybe then I get a chance to understand it. Afterwards we can compare simplicity again :)
Bill: I did not give the formal definition here, but gave a link to Berlekamp's exposition, because this is a go site, not a math site. :)
RobertJasiek: Go is also about maths and go. So SL discussion should also be about that instead of self-censoring it (and other go-related things). - I had a look into Berlekamp's paper, do not care much for the details of thermography, and tried to understand ko master neverlessless. But his description of ko master is buried among too much informal text; I do not know what shall belong to a definition and what not. I am missing a paragraph starting with "Definition: A komaster is [...]".