xela wrote:

I've heard the phrase "mast value" before, but not paid a great deal of attention. The page at

https://senseis.xmp.net/?Mast needs a bit more context to make sense. Is there something written down to link this to kos?

*Mast value* is a term coined by Berlekamp in "The economist's view of games" in

*Games of No Chance*, edited by Richard Nowakowski (Cambridge University Press, 1996), precisely to deal with hyperactive ko positions, such as approach kos, and 10,000 year kos. Up to this point thermographs had masts which represented their mean value when they were not big enough to play. But it is impossible (or nearly so) to derive mean values for hyperactive ko positions. I go into this in "Evaluating kos: A review of the research" in the proceedings of ICOB 2006 (Myongji University).

So what is mean value? Suppose that we have two simple gote such that Black to play can move to a local score of 6 pts. and White to play can move to a local score of 0. The combination has a value of 6 pts. (for Black). They are miai. If Black plays first in one of them, White can play first in the other, to guarantee a result of only 6 pts. If White plays first in one of them, Black can reply in the other, to guarantee a result of at least 6 pts. Those two guarantees establish a game theoretical value for the pair of 6 pts. The mean value for each is thus 6/2 = 3 pts. The value of the mast of the thermograph of each is the same as the mean value.

Now consider a simple ko which Black may win for a local score of 6 and White may win for a local score of 0. There are two such kos, depending upon who can win the ko in one move. Let's call the one that Black can win in one play, K, and the one that White can win in one play, L. Consider the combination, K + L. It is possible to show that Black can guarantee a local score of at least 6 and White can guarantee a local score of at most 6, assuming that play ends. So we may take the value of the combination as 6. OC, there is no mean value here, because they are two different kos. It is easy to see that ko threats do not matter in this case, although they may prolong the play.

Now suppose that we have 3 of K on the board. Black to play can win one, for a local score of 6, and then White can capture in one of the other two, for a local score of 6 in the pair. The result is a local score of 12. By the same token, White to play can capture in one of them and then Black can win one of the others, for a local score of 12. The game theoretical value of 3 Ks is 12, and the mean value of each is 12/3 = 4. Again, ko threats do not matter.

You can't do that for hyperactive ko positions, because for them ko threats do matter. In the early 1990s Berlekamp discovered a way to take ko threats into account, at least abstractly, with the idea of komaster. We can draw thermographs of such positions, but the value at the mast does not equal the mean value, because there is no mean value. So we call it simply the mast value.